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2,100 |
Some basic information on information-based complexity theory
|
math.NA
|
Numerical analysts might be expected to pay close attention to a branch of
complexity theory called information-based complexity theory (IBCT), which
produces an abundance of impressive results about the quest for approximate
solutions to mathematical problems. Why then do most numerical analysts turn a
cold shoulder to IBCT? Close analysis of two representative papers reveals a
mixture of nice new observations, error bounds repackaged in new language,
misdirected examples, and misleading theorems.
Some elements in the framework of IBCT, erected to support a rigorous yet
flexible theory, make it difficult to judge whether a model is off-target or
reasonably realistic. For instance, a sharp distinction is made between
information and algorithms restricted to this information. Yet the information
itself usually comes from an algorithm, so the distinction clouds the issues
and can lead to true but misleading inferences. Another troublesome aspect of
IBCT is a free parameter $F$, the class of admissible problem instances. By
overlooking $F$'s membership fee, the theory sometimes distorts the economics
of problem solving in a way reminiscent of agricultural subsidies.
The current theory's surprising results pertain only to unnatural situations,
and its genuinely new insights might serve us better if expressed in the
conventional modes of error analysis and approximation theory.
|
math
|
2,101 |
Perspectives on information-based complexity
|
math.NA
|
The authors discuss information-based complexity theory, which is a model of
finite-precision computations with real numbers, and its applications to
numerical analysis.
|
math
|
2,102 |
Mathematical pressure volume models of the cerebrospinal fluid
|
math.NA
|
Numerous mathematical models have emerged in the medical literature over the
past two decades attempting to characterize the pressure and volume dynamics
the central nervous system compartment. These models have been used to study he
behavior of this compartment under such pathological clinical conditions s
hydrocephalus, head injury and brain edema. The number of different pproaches
has led to considerable confusion regarding the validity, accuracy or
appropriateness of the various models. In this paper we review the mathematical
basis for these models in a mplified fashion, leaving the mathematical details
to appendices. We show at most previous models are in fact particular cases of
a single basic differential equation describing the evolution in time of the
cerebrospinal fluid pressure (CFS). Central to this approach is the hypothsis
that the rate change of CSF volume with respect to pressure is a measure of the
compliance of the brain tissue which as a consequence leads to particular
models epending on the form of the compliance funtion. All such models in fact
give essentially no information on the behavior of the brain itself. More
recent models (solved numerically using the Finite Element Method) have begun
to address this issue but have difficulties due to the lack of information
about the mechanical properties of the brain. Suggestions are made on how
development of models which account for these chanical properties might be
developed.
|
math
|
2,103 |
Good rotations
|
math.NA
|
Numerical integrations in celestial mechanics often involve the repeated
computation of a rotation with a constant angle. A direct evaluation of these
rotations yields a linear drift of the distance to the origin. This is due to
roundoff in the representation of the sine s and cosine c of the angle theta.
In a computer, one generally gets c^2 + s^2 <> 1, resulting in a mapping that
is slightly contracting or expanding. In the present paper we present a method
to find pairs of representable real numbers s and c such that c^2 + s^2 is as
close to 1 as possible. We show that this results in a drastic decrease of the
systematic error, making it negligible compared to the random error of other
operations. We also verify that this approach gives good results in a realistic
celestial mechanics integration.
|
math
|
2,104 |
Numerical Analysis of Two-Phase Flow in Gas-Dynamic Filter
|
math.NA
|
This paper presents numerical and analytical investigation of gas flow in
gas-dynamic filter - a device for cleaning gas from solid particles with
counter flow of large water particles in order to prevent their release to the
atmosphere.
Ideal and viscous gas flows are considered. It is assumed, that gas flow is
stationary, incompressible and plane, thus in the case of ideal gas stream
function is considered, and in its terms boundary conditions are formulated. To
determine stream function Dirichlet problem for Laplace equation is solved.
Numerical solution is obtained using five-point scheme, and analytical - by
conformal mapping. It is demonstrated that numerical solution fits very
accurately with the analytical one. Then in the already known gas flow field
trajectories of particles of different size are calculated in Lagrange
formulation, taking into account dust particles as well as filtering water
drops.
Trajectories of particles of several different sizes under different modes of
filter operation were analysed. The adequacy of real and computed flow is
demonstrated.
Computation of the flow is done using full Navier-Stokes equations. The
possibility of formation of rotation and separation zones in the flow is
demonstrated.
|
math
|
2,105 |
Approximate Models of Dynamic Thermoviscoelasticity Describing Shape-Memory-Alloy Phase Transitions
|
math.NA
|
We consider problems of dynamic viscoelasticity taking into account the
coupling of elastic and thermal fields. Efficient approximate models are
developed and computational results on thermomechanical behaviour of
shape-memory-alloy structures are presented.
|
math
|
2,106 |
Numerical Calculations Using Maple: Why & How?
|
math.NA
|
The possibility of interaction between Maple and numeric compiled languages
in performing extensive numeric calculations is exemplified by the Ndynamics
package, a tool for studying the (chaotic) behavior of dynamical systems.
Programming hints concerning the construction of Ndynamics are presented. The
system command, together with the application of the black-box concept, is used
to implement a powerful cooperation between Maple code and some other numeric
language code.
|
math
|
2,107 |
A multi-level algorithm for the solution of moment problems
|
math.NA
|
We study numerical methods for the solution of general linear moment
problems, where the solution belongs to a family of nested subspaces of a
Hilbert space. Multi-level algorithms, based on the conjugate gradient method
and the Landweber--Richardson method are proposed that determine the "optimal"
reconstruction level a posteriori from quantities that arise during the
numerical calculations. As an important example we discuss the reconstruction
of band-limited signals from irregularly spaced noisy samples, when the actual
bandwidth of the signal is not available. Numerical examples show the
usefulness of the proposed algorithms.
|
math
|
2,108 |
Rates of convergence for the approximation of dual shift-invariant systems in $l_2(Z)$
|
math.NA
|
A shift-invariant system is a collection of functions $\{g_{m,n}\}$ of the
form $g_{m,n}(k) = g_m(k-an)$. Such systems play an important role in
time-frequency analysis and digital signal processing. A principal problem is
to find a dual system $\gamma_{m,n}(k) = \gamma_m(k-an)$ such that each
function $f$ can be written as $f = \sum < f, \gamma_{m,n} > g_{m,n}$. The
mathematical theory usually addresses this problem in infinite dimensions
(typically in $L_2(R)$ or $l_2(Z)$), whereas numerical methods have to operate
with a finite-dimensional model. Exploiting the link between the frame operator
and Laurent operators with matrix-valued symbol, we apply the finite section
method to show that the dual functions obtained by solving a finite-dimensional
problem converge to the dual functions of the original infinite-dimensional
problem in $l_2(Z)$. For compactly supported $g_{m,n}$ (FIR filter banks) we
prove an exponential rate of convergence and derive explicit expressions for
the involved constants. Further we investigate under which conditions one can
replace the discrete model of the finite section method by the periodic
discrete model, which is used in many numerical procedures. Again we provide
explicit estimates for the speed of convergence. Some remarks on tight frames
complete the paper.
|
math
|
2,109 |
A Levinson-Galerkin algorithm for regularized trigonometric approximation
|
math.NA
|
Trigonometric polynomials are widely used for the approximation of a smooth
function $f$ from a set of nonuniformly spaced samples
$\{f(x_j)\}_{j=0}^{N-1}$. If the samples are perturbed by noise, controlling
the smoothness of the trigonometric approximation becomes an essential issue to
avoid overfitting and underfitting of the data. Using the polynomial degree as
regularization parameter we derive a multi-level algorithm that iteratively
adapts to the least squares solution of optimal smoothness. The proposed
algorithm computes the solution in at most $\cal{O}(NM + M^2)$ operations ($M$
being the polynomial degree of the approximation) by solving a family of nested
Toeplitz systems. It is shown how the presented method can be extended to
multivariate trigonometric approximation. We demonstrate the performance of the
algorithm by applying it in echocardiography to the recovery of the boundary of
the Left Ventricle.
|
math
|
2,110 |
On the fourth-order accurate compact ADI scheme for solving the unsteady Nonlinear Coupled Burgers' Equations
|
math.NA
|
The two-dimensional unsteady coupled Burgers' equations with moderate to
severe gradients, are solved numerically using higher-order accurate finite
difference schemes; namely the fourth-order accurate compact ADI scheme, and
the fourth-order accurate Du Fort Frankel scheme. The question of numerical
stability and convergence are presented. Comparisons are made between the
present schemes in terms of accuracy and computational efficiency for solving
problems with severe internal and boundary gradients. The present study shows
that the fourth-order compact ADI scheme is stable and efficient.
|
math
|
2,111 |
Stochastic trace formulas
|
math.NA
|
The spectrum of the evolution Operator associated with a nonlinear stochastic
flow with additive noise is evaluated by diagonalization in a polynomial basis.
The method works for arbitrary noise strength. In the weak noise limit we
formulate a new perturbative expansion for the spectrum of the stochastic
evolution Operator in terms of expansions around the classical periodic orbits.
The diagonalization of such operators is easier to implement than the standard
Feynman diagram perturbation theory. The result is a stochastic analog of the
Gutzwiller semiclassical spectral determinant with the ``$\hbar$'' corrections
computed to at least two orders more than what has so far been attainable in
stochastic and quantum-mechanical applications, supplemented by the estimate
for the late terms in the asymptotic saddlepoint expansions.
|
math
|
2,112 |
Methods for the approximation of the matrix exponential in a Lie-algebraic setting
|
math.NA
|
Discretization methods for ordinary differential equations based on the use
of matrix exponentials have been known for decades. This set of ideas has come
off age and acquired greater urgency recently, within the context of geometric
integration and discretization methods on manifolds based on the use of
Lie-group actions.
In the present paper we study the approximation of the matrix exponential in
a particular context: given a Lie group $G$ and its Lie algebra $g$, we seek
approximants $F(tB)$ of $\exp(tB)$ such that $F(tB)\in G$ if $B\in g$. Having
fixed a basis of the Lie algebra, we write $F(tB)$ as a composition of
exponentials of the basis elements pre-multiplied by suitable scalar functions.
|
math
|
2,113 |
Condition number bounds for problems with integer coefficients
|
math.NA
|
An apriori bound for the condition number associated to each of the following
problems is given: general linear equation solving, minimum squares,
non-symmetric eigenvalue problems, solving univariate polynomials, solving
systems of multivariate polynomials. It is assumed that the input has integer
coefficients and is not on the degenerate locus of the respective problem (i.e.
the condition number is finite). Then condition numbers are bounded in terms of
the dimension and of the bit-size of the input.
In the same setting, bounds are given for the speed of convergence of the
following iterative algorithms: QR without shift for the symmetric eigenvalue
problem, and Graeffe iteration for univariate polynomials.
|
math
|
2,114 |
Lower bounds for some decision problems over C
|
math.NA
|
Lower bounds for some explicit decision problems over the complex numbers are
given.
|
math
|
2,115 |
Ultimate Polynomial Time
|
math.NA
|
The class $\mathcal{UP}$ of `ultimate polynomial time' problems over $\mathbb
C$ is introduced; it contains the class $\mathcal P$ of polynomial time
problems over $\mathbb C$.
The $\tau$-Conjecture for polynomials implies that $\mathcal{UP}$ does not
contain the class of non-deterministic polynomial time problems definable
without constants over $\mathbb C$.
This latest statement implies that $\mathcal P \ne \mathcal{NP}$ over
$\mathbb C$.
A notion of `ultimate complexity' of a problem is suggested. It provides
lower bounds for the complexity of structured problems.
|
math
|
2,116 |
A novel methodology of weighted residual for nonlinear computations
|
math.NA
|
One of strengths in the finite element (FE) and Galerkin methods is their
capability to apply weak formulations via integration by parts, which leads to
elements matching at lower degree of continuity and relaxes requirements of
choosing basis functions. However, when applied to nonlinear problems, the
methods of this type require a great amount of computing effort of repeated
numerical integration. It is well known that the method of weighted residual is
the very basis of various popular numerical techniques including the FE and
Galerkin methods. This paper presents a novel methodology of weighted residual
for nonlinear computation with objectives to avoid the above-mentioned
shortcomings. It is shown that the presented nonlinear formulations of the FE
and Galerkin methods can be expressed in the Hadamard product form as in the
collocation and finite difference methods. Therefore, the recently developed
SJT product approach can be applied in the evaluation of the Jacobian matrix of
these nonlinear formulations. This also provides possibility to introduce the
nonlinear uncoupling technique to the FE and Galerkin nonlinear computing.
Furthermore, the present scheme of weighted residuals also greatly eases the
use of the least square and boundary element methods to nonlinear problems
|
math
|
2,117 |
An efficient step size selection for ODE codes
|
math.NA
|
We give an algorithm for efficient step size control in numerical integration
of non-stiff initial value problems, based on a formula tailormade to methods
where the numerical solution is compared with a solution of lower order.
|
math
|
2,118 |
Generalized linearization of nonlinear algebraic equations: an innovative approach
|
math.NA
|
Based on the matrix expression of general nonlinear numerical analogues
presented by the present author, this paper proposes a novel philosophy of
nonlinear computation and analysis. The nonlinear problems are considered an
ill-posed linear system. In this way, all nonlinear algebraic terms are instead
expressed as Linearly independent variables. Therefore, a n-dimension nonlinear
system can be expanded as a linear system of n(n+1)/2 dimension space. This
introduces the possibility to applying generalized inverse of matrix to
computation of nonlinear systems. Also, singular value decomposition (SVD) can
be directly employed in nonlinear analysis by using such a methodology.
|
math
|
2,119 |
Reliable operations on oscillatory functions
|
math.NA
|
Approximate $p$-point Leibniz derivation formulas as well as interpolatory
Simpson quadrature sums adapted to oscillatory functions are discussed. Both
theoretical considerations and numerical evidence concerning the dependence of
the discretization errors on the frequency parameter of the oscillatory
functions show that the accuracy gain of the present formulas over those based
on the exponential fitting approach [L. Ixaru, "Computer Physics
Communications", 105 (1997) 1--19] is overwhelming.
|
math
|
2,120 |
Relationship formula between nonlinear polynomial equations and the corresponding Jacobian matrix
|
math.NA
|
This paper provides a general proof of a relationship theorem between
nonlinear analogue polynomial equations and the corresponding Jacobian matrix,
presented recently by the present author. This theorem is also verified
generally effective for all nonlinear polynomial algebraic system of equations.
As two particular applications of this theorem, we gave a Newton formula
without requiring the evaluation of nonlinear function vector as well as a
simple formula to estimate the relative error of the approximate Jacobian
matrix. Finally, some possible applications of this theorem in nonlinear system
analysis are discussed.
|
math
|
2,121 |
Pseudo-Newton method for nonlinear equations
|
math.NA
|
In order to avoid the evaluation of the Jacobian matrix and its inverse, the
present author recently introduced the pseudo-Jacobian matrix with a general
applicability of any nonlinear systems of equations. By using this concept,
this paper proposes the pseudo-Newton method
|
math
|
2,122 |
Implicit Integration of the Time-Dependent Ginzburg-Landau Equations of Superconductivity
|
math.NA
|
This article is concerned with the integration of the time-dependent
Ginzburg-Landau (TDGL) equations of superconductivity. Four algorithms, ranging
from fully explicit to fully implicit, are presented and evaluated for
stability, accuracy, and compute time. The benchmark problem for the evaluation
is the equilibration of a vortex configuration in a superconductor that is
embedded in a thin insulator and subject to an applied magnetic field.
|
math
|
2,123 |
Antisymmetry, pseudospectral methods, and conservative PDEs
|
math.NA
|
`Dual composition', a new method of constructing energy-preserving
discretizations of conservative PDEs, is introduced. It extends the
summation-by-parts approach to arbitrary differential operators and conserved
quantities. Links to pseudospectral, Galerkin, antialiasing, and Hamiltonian
methods are discussed.
|
math
|
2,124 |
A modified BFGS quasi-Newton iterative formula
|
math.NA
|
The quasi-Newton equation is the very basis of a variety of the quasi-Newton
methods. By using a relationship formula between nonlinear polynomial equations
and the corresponding Jacobian matrix. presented recently by the present
author, we established an exact alternative of the approximate quasi-Newton
equation and consequently derived an modified BFGS updating formulas.
|
math
|
2,125 |
A new definition of nonlinear statistics mean and variance
|
math.NA
|
This note presents a new definition of nonlinear statistics mean and variance
to simplify the nonlinear statistics computations. These concepts aim to
provide a theoretical explanation of a novel nonlinear weighted residual
methodology presented recently by the present author.
|
math
|
2,126 |
Fast and accurate multigrid solution of Poissons equation using diagonally oriented grids
|
math.NA
|
We solve Poisson's equation using new multigrid algorithms that converge
rapidly. The novel feature of the 2D and 3D algorithms are the use of extra
diagonal grids in the multigrid hierarchy for a much richer and effective
communication between the levels of the multigrid. Numerical experiments
solving Poisson's equation in the unit square and unit cube show simple
versions of the proposed algorithms are up to twice as fast as correspondingly
simple multigrid iterations on a conventional hierarchy of grids.
|
math
|
2,127 |
On the Geometry of Graeffe Iteration
|
math.NA
|
A new version of the Graeffe algorithm for finding all the roots of
univariate complex polynomials is proposed. It is obtained from the classical
algorithm by a process analogous to renormalization of dynamical systems. This
iteration is called Renormalized Graeffe Iteration.
It is globally convergent, with probability 1. All quantities involved in the
computation are bounded, once the initial polynomial is given (with probability
1). This implies remarkable stability properties for the new algorithm, thus
overcoming known limitations of the classical Graeffe algorithm. If we start
with a degree-$d$ polynomial, each renormalized Graeffe iteration costs
$O(d^2)$ arithmetic operations, with memory $O(d)$. A probabilistic global
complexity bound is given. The case of univariate real polynomials is briefly
discussed. A numerical implementation of the algorithm presented herein allowed
us to solve random polynomials of degree up to 1000.
|
math
|
2,128 |
Tangent Graeffe Iteration
|
math.NA
|
Graeffe iteration was the choice algorithm for solving univariate polynomials
in the XIX-th and early XX-th century. In this paper, a new variation of
Graeffe iteration is given, suitable to IEEE floating-point arithmetics of
modern digital computers. We prove that under a certain generic assumption the
proposed algorithm converges. We also estimate the error after N iterations and
the running cost. The main ideas from which this algorithm is built are:
classical Graeffe iteration and Newton Diagrams, changes of scale
(renormalization), and replacement of a difference technique by a
differentiation one. The algorithm was implemented successfully and a number of
numerical experiments are displayed.
|
math
|
2,129 |
High Accuracy Method for Integral Equations with Discontinuous Kernels
|
math.NA
|
A new highly accurate numerical approximation scheme based on a Gauss type
Clenshaw-Curtis Quadrature for Fredholm integral equations of the second kind,
whose kernel is either discontinuous or not smooth along the main diagonal, is
presented. This scheme is of spectral accuracy when the kernel is infinitely
differentiable away from the main diagonal, and is also applicable when the
kernel is singular along the boundary, and at isolated points on the main
diagonal. The corresponding composite rule is described. Application to
integro-differential Schroedinger equations with non-local potentials is given.
|
math
|
2,130 |
Prediction of large-scale dynamics using unresolved computations
|
math.NA
|
We present a theoretical framework and numerical methods for predicting the
large-scale properties of solutions of partial differential equations that are
too complex to be properly resolved. We assume that prior statistical
information about the distribution of the solutions is available, as is often
the case in practice. The quantities we can compute condition the prior
information and allow us to calculate mean properties of solutions in the
future. We derive approximate ways for computing the evolution of the
probabilities conditioned by what we can compute, and obtain ordinary
differential equations for the expected values of a set of large-scale
variables. Our methods are demonstrated on two simple but instructive examples,
where the prior information consists of invariant canonical distributions
|
math
|
2,131 |
The influence of the flow of the reacting gas on the conditions for a Thermal Explosion
|
math.NA
|
The classical problem of thermal explosion is modified so that the chemically
active gas is not at rest but is flowing in a long cylindrical pipe. Up to a
certain section the heat-conducting walls of the pipe are held at low
temperature so that the reaction rate is small and there is no heat release; at
that section the ambient temperature is increased and an exothermic reaction
begins. The question is whether a slow reaction regime will be established or a
thermal explosion will occur. The mathematical formulation of the problem is
presented. It is shown that when the pipe radius is larger than a critical
value, the solution of the new problem exists only up to a certain distance
along the axis. The critical radius is determined by conditions in a problem
with a uniform axial temperature. The loss of existence is interpreted as a
thermal explosion; the critical distance is the safe reactor's length. Both
laminar and developed turbulent flow regimes are considered. In a computational
experiment the loss of the existence appears as a divergence of a numerical
procedure; numerical calculations reveal asymptotic scaling laws with simple
powers for the critical distance.
|
math
|
2,132 |
The Kolmogorov-Obukhov Exponent in the Inertial Range of Turbulence: A Reexamination of Experimental Data
|
math.NA
|
In recent papers Benzi et al. presented experimental data and an analysis to
the effect that the well-known "2/3" Kolmogorov-Obukhov exponent in the
inertial range of local structure in turbulence should be corrected by a small
but definitely non-zero amount. We reexamine the very same data and show that
this conclusion is unjustified. The data are in fact consistent with incomplete
similarity in the inertial range, and with an exponent that depends on the
Reynolds number and tends to 2/3 in the limit of vanishing viscosity. If
further data confirm this conclusion, the understanding of local structure
would be profoundly affected.
|
math
|
2,133 |
A numerical scheme for impact problems
|
math.NA
|
We consider a mechanical system with impact and n degrees of freedom, written
in generalized coordinates. The system is not necessarily Lagrangian. The
representative point of the system must remain inside a set of constraints K;
the boundary of K is three times differentiable.
At impact, the tangential component of the impulsion is conserved, while its
normal coordinate is reflected and multiplied by a given coefficient of
restitution e between 0 and 1. The orthognality is taken with respect to the
natural metric in the space of impulsions.
We define a numerical scheme which enables us to approximate the solutions of
the Cauchy problem: this is an ad hoc scheme which does not require a
systematic search for the times of impact. We prove the convergence of this
numerical scheme to a solution, which yields also an existence result.
Without any a priori estimates, the convergence and the existence are local;
with some a priori estimates, the convergence and the existence are proved on
intervals depending exclusively on these estimates.
This scheme has been implemented with a trivial and a non trivial mass
matrix.
|
math
|
2,134 |
Optimal prediction and the Klein-Gordon equation
|
math.NA
|
The method of optimal prediction is applied to calculate the future means of
solutions to the Klein-Gordon equation. It is shown that in an appropriate
probability space, the difference between the average of all solutions that
satisfy certain constraints at time t=0, and the average computed by an
approximate method, is small with high probability.
|
math
|
2,135 |
Compact Central WENO Schemes for Multidimensional Conservation Laws
|
math.NA
|
We present a new third-order central scheme for approximating solutions of
systems of conservation laws in one and two space dimensions. In the spirit of
Godunov-type schemes,our method is based on reconstructing a
piecewise-polynomial interpolant from cell-averages which is then advanced
exactly in time. In the reconstruction step, we introduce a new third-order as
a convex combination of interpolants based on different stencils. The heart of
the matter is that one of these interpolants is taken as an arbitrary quadratic
polynomial and the weights of the convex combination are set as to obtain
third-order accuracy in smooth regions. The embedded mechanism in the WENO-like
schemes guarantees that in regions with discontinuities or large gradients,
there is an automatic switch to a one-sided second-order reconstruction, which
prevents the creation of spurious oscillations. In the one-dimensional case,
our new third order scheme is based on an extremely compact point stencil.
Analogous compactness is retained in more space dimensions. The accuracy,
robustness and high-resolution properties of our scheme are demonstrated in a
variety of one and two dimensional problems.
|
math
|
2,136 |
Optimal Prediction for Hamiltonian partial differential equations
|
math.NA
|
Optimal prediction methods compensate for a lack of resolution in the
numerical solution of time-dependent differential equations through the use of
prior statistical information. We present a new derivation of the basic
methodology, show that field-theoretical perturbation theory provides a useful
device for dealing with quasi-linear problems, and provide a nonlinear example
that illuminates the difference between a pseudo-spectral method and an optimal
prediction method with Fourier kernels. Along the way, we explain the
differences and similarities between optimal prediction, the representer method
in data assimilation, and duality methods for finding weak solutions. We also
discuss the conditions under which a simple implementation of the optimal
prediction method can be expected to perform well.
|
math
|
2,137 |
Mathematical Modeling of Boson-Fermion Stars in the Generalized Scalar-Tensor Theories of Gravity
|
math.NA
|
A model of static boson-fermion star with spherical symmetry based on the
scalar-tensor theory of gravity with massive dilaton field is investigated
numerically.
Since the radius of star is \textit{a priori} an unknown quantity, the
corresponding boundary value problem (BVP) is treated as a nonlinear spectral
problem with a free internal boundary. The Continuous Analogue of Newton Method
(CANM) for solving this problem is applied.
Information about basic geometric functions and the functions describing the
matter fields, which build the star is obtained. In a physical point of view
the main result is that the structure and properties of the star in presence of
massive dilaton field depend essentially both of its fermionic and bosonic
components.
|
math
|
2,138 |
Numerical Investigation of a Dipole Type Solution for Unsteady Groundwater Flow with Capillary Retention and Forced Drainage
|
math.NA
|
A model of unsteady filtration (seepage) in a porous medium with capillary
retention is considered. It leads to a free boundary problem for a generalized
porous medium equation where the location of the boundary of the water mound is
determined as part of the solution. The numerical solution of the free boundary
problem is shown to possess self-similar intermediate asymptotics. On the other
hand, the asymptotic solution can be obtained from a non-linear boundary value
problem. Numerical solution of the resulting eigenvalue problem agrees with the
solution of the partial differential equation for intermediate times. In the
second part of the work, we consider the problem of control of the water mound
extension by a forced drainage.
|
math
|
2,139 |
Holistic finite differences accurately model the dynamics of the Kuramoto-Sivashinsky equation
|
math.NA
|
We analyse the nonlinear Kuramoto-Sivashinsky equation to develop an accurate
finite difference approximation to its dynamics. The analysis is based upon
centre manifold theory so we are assured that the finite difference model
accurately models the dynamics and may be constructed systematically. The
theory is applied after dividing the physical domain into small elements by
introducing insulating internal boundaries which are later removed. The
Kuramoto-Sivashinsky equation is used as an example to show how holistic finite
differences may be applied to fourth order, nonlinear, spatio-temporal
dynamical systems. This novel centre manifold approach is holistic in the sense
that it treats the dynamical equations as a whole, not just as the sum of
separate terms.
|
math
|
2,140 |
Optimal Prediction of Stiff Oscillatory Mechanics
|
math.NA
|
We consider many-body problems in classical mechanics where a wide range of
time scales limits what can be computed. We apply the method of optimal
prediction to obtain equations which are easier to solve numerically. We
demonstrate by examples that optimal prediction can reduce the amount of
computation needed to obtain a solution by several orders of magnitude.
|
math
|
2,141 |
The Characteristic Length Scale of the Intermediate Structure in Zero-Pressure-Gradient Boundary Layer Flow
|
math.NA
|
In a turbulent boundary layer over a smooth flat plate with zero pressure
gradient, the intermediate structure between the viscous sublayer and the free
stream consists of two layers: one adjacent to the viscous sublayer and one
adjacent to the free stream. When the level of turbulence in the free stream is
low, the boundary between the two layers is sharp and both have a self-similar
structure described by Reynolds-number-dependent scaling (power) laws. This
structure introduces two length scales: one --- the wall region thickness ---
determined by the sharp boundary between the two intermediate layers, the
second determined by the condition that the velocity distribution in the first
intermediate layer be the one common to all wall-bounded flows, and in
particular coincide with the scaling law previously determined for pipe flows.
Using recent experimental data we determine both these length scales and show
that they are close. Our results disagree with the classical model of the "wake
region".
|
math
|
2,142 |
Geometrically Graded h-p Quadrature Applied to the Complex Boundary Integral Equation Method for the Dirichlet Problem with Corner Singularities
|
math.NA
|
Boundary integral methods for the solution of boundary value PDEs are an
alternative to `interior' methods, such as finite difference and finite element
methods. They are attractive on domains with corners, particularly when the
solution has singularities at these corners. In these cases, interior methods
can become excessively expensive, as they require a finely discretised 2D mesh
in the vicinity of corners, whilst boundary integral methods typically require
a mesh discretised in only one dimension, that of arc length.
Consider the Dirichlet problem. Traditional boundary integral methods applied
to problems with corner singularities involve a (real) boundary integral
equation with a kernel containing a logarithmic singularity. This is both
tedious to code and computationally inefficient. The CBIEM is different in that
it involves a complex boundary integral equation with a smooth kernel. The
boundary integral equation is approximated using a collocation technique, and
the interior solution is then approximated using a discretisation of Cauchy's
integral formula, combined with singularity subtraction.
A high order quadrature rule is required for the solution of the integral
equation. Typical corner singularities are of square root type, and a
`geometrically graded h-p' composite quadrature rule is used. This yields
efficient, high order solution of the integral equation, and thence the
Dirichlet problem.
Implementation and experimental results in \textsc{matlab} code are
presented.
|
math
|
2,143 |
Gauß Cubature for the Surface of the Unit Sphere
|
math.NA
|
Gau{\ss} cubature (multidimensional numerical integration) rules are the
natural generalisation of the 1D Gau{\ss} rules. They are optimal in the sense
that they exactly integrate polynomials of as high a degree as possible for a
particular number of points (function evaluations). For smooth integrands, they
are accurate, computationally efficient formulae.
The construction of the points and weights of a Gau{\ss} rule requires the
solution of a system of moment equations. In 1D, this system can be converted
to a linear system, and a unique solution is obtained, for which the points lie
within the region of integration, and the weights are all positive. These
properties help ensure numerical stability, and we describe the rules as
`good'. In the multidimensional case, the moment equations are nonlinear
algebraic equations, and a solution is not guaranteed to even exist, let alone
be good. The size and degree of the system grow with the degree of the desired
cubature rule. Analytic solution generally becomes impossible as the degree of
the polynomial equations to be solved goes beyond 4, and numerical
approximations are required. The uncertainty of the existence of solutions,
coupled with the size and degree of the system makes the problem daunting for
numerical methods.
The construction of Gau{\ss} rules for (fully symmetric) $n$-dimensional
regions is easily specialised to the case of $U_3$, the unit sphere in 3D.
Despite the problems described above, for degrees up to 17, good Gau{\ss} rules
for $U_3$ have been constructed/discovered.
|
math
|
2,144 |
Partitioning Sparse Graphs using the Second Eigenvector of their Graph Laplacian
|
math.NA
|
Partitioning a graph into three pieces, with two of them large and connected,
and the third a small ``separator'' set, is useful for improving the
performance of a number of combinatorial algorithms. This is done using the
second eigenvector of a matrix defined solely in terms of the incidence matrix,
called the graph Laplacian. For sparse graphs, the eigenvector can be
efficiently computed using the Lanczos algorithm. This graph partitioning
algorithm is extended to provide a complete hierarchical subdivision of the
graph. The method has been implemented and numerical results obtained both for
simple test problems and for several grid graphs.
|
math
|
2,145 |
A holistic finite difference approach models linear dynamics consistently
|
math.NA
|
I prove that a centre manifold approach to creating finite difference models
will consistently model linear dynamics as the grid spacing becomes small.
Using such tools of dynamical systems theory gives new assurances about the
quality of finite difference models under nonlinear and other perturbations on
grids with finite spacing. For example, the advection-diffusion equation is
found to be stably modelled for all advection speeds and all grid spacing. The
theorems establish an extremely good form for the artificial internal boundary
conditions that need to be introduced to apply centre manifold theory. When
numerically solving nonlinear partial differential equations, this approach can
be used to derive systematically finite difference models which automatically
have excellent characteristics. Their good performance for finite grid spacing
implies that fewer grid points may be used and consequently there will be less
difficulties with stiff rapidly decaying modes in continuum problems.
|
math
|
2,146 |
Irregular Input Data in Convergence Acceleration and Summation Processes: General Considerations and Some Special Gaussian Hypergeometric Series as Model Problems
|
math.NA
|
Sequence transformations accomplish an acceleration of convergence or a
summation in the case of divergence by detecting and utilizing regularities of
the elements of the sequence to be transformed. For sufficiently large indices,
certain asymptotic regularities normally do exist, but the leading elements of
a sequence may behave quite irregularly. The Gaussian hypergeometric series 2F1
(a, b; c; z) is well suited to illuminate problems of that kind. Sequence
transformations perform quite well for most parameters and arguments. If,
however, the third parameter $c$ of a nonterminating hypergeometric series 2F1
is a negative real number, the terms initially grow in magnitude like the terms
of a mildly divergent series. The use of the leading terms of such a series as
input data leads to unreliable and even completely nonsensical results. In
contrast, sequence transformations produce good results if the leading
irregular terms are excluded from the transformation process. Similar problems
occur also in perturbation expansions. For example, summation results for the
infinite coupling limit k_3 of the sextic anharmonic oscillator can be improved
considerably by excluding the leading terms from the transformation process.
Finally, numerous new recurrence formulas for the 2F1 (a, b; c; z) are derived.
|
math
|
2,147 |
New Numerical Algorithm for Modeling of Boson-Fermion Stars in Dilatonic Gravity
|
math.NA
|
We investigate numerically a models of the static spherically symmetric
boson-fermion stars in scalar-tensor theory of gravity with massive dilaton
field. The proper mathematical model of such stars is interpreted as a
nonlinear two-parametric eigenvalue problem with unknown internal boundary. We
employ the Continuous Analogue of Newton Method (CANM) which leads on each
iteration to two separate linear boundary value problems with different
dimensions inside and outside the star, respectively. Along with them a
nonlinear algebraic system for the spectral parameters - radius of the star
$R_{s}$ and quantity $\Omega $ is solved also.
In this way we obtain the behaviour of the basic geometric quantities and
functions describing dilaton field and matter fields which build the star.
|
math
|
2,148 |
Extrapolation Methods for Improving the Convergence of Oligomer Calculations to the Infinite Chain Limit of Quasi-Onedimensional Stereoregular Polymers
|
math.NA
|
Quasi-onedimensional stereoregular polymers as for example polyacetylene are
currently of considerable interest. There are basically two different
approaches for doing electronic structure calculations: One method is
essentially based on concepts of solid state theory. The other method is
essentially a quantum chemical method since it approximates the polymer by
oligomers consisting of a finite number of monomer units. In this way, the
highly developed technology of quantum chemical molecular programs can be used.
Unfortunately, oligomers of finite size are not necessarily able to model those
features of a polymer which crucially depend of its in principle infinite
extension. In such a case extrapolation techniques can be extremely helpful.
For example, one can perform electronic structure calculations for a sequence
of oligomers with an increasing number of monomer units. In the next step, one
then can try to determine the limit of this sequence for an oligomer of
infinite length with the help of suitable extrapolation methods. Several
different extrapolation methods are discussed which are able to accomplish an
extrapolation of energies and properties of oligomers to the infinite chain
limit. Calculations for the ground state energy of polyacetylene are presented
which demonstrate the practical usefulness of extrapolation methods.
|
math
|
2,149 |
Shock capturing by anisotropic diffusion oscillation reduction
|
math.NA
|
This paper introduces the method of anisotropic diffusion oscillation
reduction (ADOR) for shock wave computations. The connection is made between
digital image processing,in particular, image edge detection, and numerical
shock capturing. Indeed, numerical shock capturing can be formulated on the
lines of iterative digital edge detection. Various anisotropic diffusion and
super diffusion operators originated from image edge detection are proposed for
the treatment of hyperbolic conservation laws and near-hyperbolic hydrodynamic
equations of change. The similarity between anisotropic diffusion and
artificial viscosity is discussed. Physical origins and mathematical properties
of the artificial viscosity is analyzed from the kinetic theory point of view.
A form of pressure tensor is derived from the first principles of the quantum
mechanics. Quantum kinetic theory is utilized to arrive at macroscopic
transport equations from the microscopic theory. Macroscopic symmetry is used
to simplify pressure tensor expressions. The latter provides a basis for the
design of artificial viscosity. The ADOR approach is validated by using
(inviscid) Burgers' equation in one and two spatial dimensions, the
incompressible Navier-Stokes equation and the Euler equation. A discrete
singular convolution (DSC) algorithm is utilized for the spatial
discretization.
|
math
|
2,150 |
A Note on Regularized Shannon's Sampling Formulae
|
math.NA
|
Error estimation is given for a regularized Shannon's sampling formulae,
which was found to be accurate and robust for numerically solving partial
differential equations.
|
math
|
2,151 |
Enhanced inverse-cascade of energy in the averaged Euler equations
|
math.NA
|
For a particular choice of the smoothing kernel, it is shown that the system
of partial differential equations governing the vortex-blob method corresponds
to the averaged Euler equations. These latter equations have recently been
derived by averaging the Euler equations over Lagrangian fluctuations of length
scale $\a$, and the same system is also encountered in the description of
inviscid and incompressible flow of second-grade polymeric (non-Newtonian)
fluids. While previous studies of this system have noted the suppression of
nonlinear interaction between modes smaller than $\a$, we show that the
modification of the nonlinear advection term also acts to enhance the
inverse-cascade of energy in two-dimensional turbulence and thereby affects
scales of motion larger than $\a$ as well. This latter effect is reminiscent of
the drag-reduction that occurs in a turbulent flow when a dilute polymer is
added.
|
math
|
2,152 |
Approximation by quadrilateral finite elements
|
math.NA
|
We consider the approximation properties of finite element spaces on
quadrilateral meshes. The finite element spaces are constructed starting with a
given finite dimensional space of functions on a square reference element,
which is then transformed to a space of functions on each convex quadrilateral
element via a bilinear isomorphism of the square onto the element. It is known
that for affine isomorphisms, a necessary and sufficient condition for
approximation of order r+1 in L2 and order r in H1 is that the given space of
functions on the reference element contain all polynomial functions of total
degree at most r. In the case of bilinear isomorphisms, it is known that the
same estimates hold if the function space contains all polynomial functions of
separate degree r. We show, by means of a counterexample, that this latter
condition is also necessary. As applications we demonstrate degradation of the
convergence order on quadrilateral meshes as compared to rectangular meshes for
serendipity finite elements and for various mixed and nonconforming finite
elements.
|
math
|
2,153 |
Gauge techniques in time and frequency domain TLM
|
math.NA
|
Typical features of the Transmission Line Matrix (TLM) algorithm in
connection with stub loading techniques and prone to be hidden in common
frequency domain formulations are elucidated within the propagator approach to
TLM. In particular, the latter reflects properly the perturbative character of
the TLM scheme and its relation to gauge field models. Internal 'gauge' degrees
of freedom are made explicit in the frequency domain by introducing the complex
nodal S-matrix as a function of operators that act on external or internal
fields or virtually couple the two. As a main benefit, many techniques and
results gained in the time domain thus generalize straight away. The recently
developed deflection method for algorithm synthesis, which is extended in this
paper, or the non-orthogonal node approximating Maxwell's equations, for
instance, become so at once available in the frequency domain. In view of
applications in computational plasma physics, the TLM model of a relativistic
charged particle current coupled to the Maxwell field is treated as a
prototype.
|
math
|
2,154 |
Implicit integration of the TDGL equations of superconductivity
|
math.NA
|
This article is concerned with the integration of the time-dependent
Ginzburg--Landau (TDGL) equations of superconductivity. Four algorithms,
ranging from fully explicit to fully implicit, are presented and evaluated for
stability, accuracy, and compute time. The benchmark problem for the evaluation
is the equilibration of a vortex configuration in a superconductor that is
embedded in a thin insulator and subject to an applied magnetic field.
|
math
|
2,155 |
The frozen-field approximation and the Ginzburg-Landau equations of superconductivity
|
math.NA
|
The Ginzburg--Landau (GL) equations of superconductivity provide a
computational model for the study of magnetic flux vortices in type-II
superconductors. In this article we show through numerical examples and
rigorous mathematical analysis that the GL model reduces to the frozen-field
model when the charge of the Cooper pairs (the superconducting charge carriers)
goes to zero while the applied field stays near the upper critical field.
|
math
|
2,156 |
Discrete singular convolution and its application to computational electromagnetics
|
math.NA
|
A new computational algorithm, the discrete singular convolution (DSC), is
introduced for computational electromagnetics. The basic philosophy behind the
DSC algorithm for the approximation of functions and their derivatives is
studied. Approximations to the delta distribution are constructed as either
bandlimited reproducing kernels or approximate reproducing kernels. A
systematic procedure is proposed to handle a number of boundary conditions
which occur in practical applications. The unified features of the DSC
algorithm for solving differential equations are explored from the point of
view of the method of weighted residuals. It is demonstrated that different
methods of implementation for the present algorithm, such as global, local,
Galerkin, collocation, and finite difference, can be deduced from a single
starting point. Both the computational bandwidth and the accuracy of the DSC
algorithm are shown to be controllable. Three example problems are employed to
illustrate the usefulness, test the accuracy and explore the limitation of the
DSC algorithm. A Galerkin-induced collocation approach is used for a waveguide
analysis in both regular and irregular domains and for electrostatic field
estimation via potential functions. Electromagnetic wave propagation in three
spatial dimensions is integrated by using a generalized finite difference
approach, which becomes a global-finite difference scheme at certain limit of
DSC parameters. Numerical experiments indicate that the proposed algorithm is a
promising approach for solving problems in electromagnetics.
|
math
|
2,157 |
Stochastic Optimal Prediction with Application to Averaged Euler Equations
|
math.NA
|
Optimal prediction (OP) methods compensate for a lack of resolution in the
numerical solution of complex problems through the use of an invariant measure
as a prior measure in the Bayesian sense. In first-order OP, unresolved
information is approximated by its conditional expectation with respect to the
invariant measure. In higher-order OP, unresolved information is approximated
by a stochastic estimator, leading to a system of random or stochastic
differential equations.
We explain the ideas through a simple example, and then apply them to the
solution of Averaged Euler equations in two space dimensions.
|
math
|
2,158 |
Conjugated filter approach for solving Burgers' equation with high Reynolds number
|
math.NA
|
We propose a conjugated filter oscillation reduction scheme for solving
Burgers' equation with high Reynolds numbers. Computational accuracy is tested
at a moderately high Reynolds number for which analytical solution is
available. Numerical results at extremely high Reynolds numbers indicate that
the proposed scheme is efficient, robust and reliable for shock capturing.
|
math
|
2,159 |
The inverse problem of the Birkhoff-Gustavson normalization and ANFER, Algorithm of Normal Form Expansion and Restoration
|
math.NA
|
In the series of papers [1-4], the inverse problem of the Birkhoff-Gustavson
normalization was posed and studied. To solve the inverse problem, the
symbolic-computing program named ANFER (Algorithm of Normal Form Expansion and
Restoration) is written up, with which a new aspect of the Bertrand and Darboux
integrability condition is found \cite{Uwano2000}. In this paper, the procedure
in ANFER is presented in mathematical terminology, which is organized on the
basis of the composition of canonical transformations.
|
math
|
2,160 |
A backward Monte-Carlo method for solving parabolic partial differential equations
|
math.NA
|
A new Monte-Carlo method for solving linear parabolic partial differential
equations is presented. Since, in this new scheme, the particles are followed
backward in time, it provides great flexibility in choosing critical points in
phase-space at which to concentrate the launching of particles and thereby
minimizing the statistical noise of the sought solution. The trajectory of a
particle, Xi(t), is given by the numerical solution to the stochastic
differential equation naturally associated with the parabolic equation. The
weight of a particle is given by the initial condition of the parabolic
equation at the point Xi(0). Another unique advantage of this new Monte-Carlo
method is that it produces a smooth solution, i.e. without delta-functions, by
summing up the weights according to the Feynman-Kac formula.
|
math
|
2,161 |
Numerical Analysis of the Non-uniform Sampling Problem
|
math.NA
|
We give an overview of recent developments in the problem of reconstructing a
band-limited signal from non-uniform sampling from a numerical analysis view
point. It is shown that the appropriate design of the finite-dimensional model
plays a key role in the numerical solution of the non-uniform sampling problem.
In the one approach (often proposed in the literature) the finite-dimensional
model leads to an ill-posed problem even in very simple situations. The other
approach that we consider leads to a well-posed problem that preserves
important structural properties of the original infinite-dimensional problem
and gives rise to efficient numerical algorithms. Furthermore a fast multilevel
algorithm is presented that can reconstruct signals of unknown bandwidth from
noisy non-uniformly spaced samples. We also discuss the design of efficient
regularization methods for ill-conditioned reconstruction problems. Numerical
examples from spectroscopy and exploration geophysics demonstrate the
performance of the proposed methods.
|
math
|
2,162 |
Non-Markovian Optimal Prediction
|
math.NA
|
Optimal prediction methods compensate for a lack of resolution in the
numerical solution of complex problems through the use of prior statistical
information. We know from previous work that in the presence of strong
underresolution a good approximation needs a non-Markovian "memory", determined
by an equation for the "orthogonal", i.e., unresolved, dynamics. We present a
simple approximation of the orthogonal dynamics, which involves an ansatz and a
Monte-Carlo evaluation of autocorrelations. The analysis provides a new
understanding of the fluctuation-dissipation formulas of statistical physics.
An example is given.
|
math
|
2,163 |
Asymptotic Summation of Slow Converging and Rapidly Oscillating Series
|
math.NA
|
Mean values of some observables describing quantum interaction between the
Bose field in a cavity and a movable mirror can be represented as expectations
of rapidly oscillating functions w.r.t. the Poisson measure with a large mean
value ($N\approx 10^{23}$) corresponding to the average number of photons in
laser beam. Straightforward summation of the series is impossible because over
$2\sqrt N$ summands make a significant contribution. We derive an analytical
expression approximating this sum with the error $O(N^{-1})$.
|
math
|
2,164 |
Approximate construction of rational approximations and the effect of error autocorrection. Applications
|
math.NA
|
Several construction methods for rational approximations to functions of one
real variable are described in the present paper; the computational results
that characterize the comparative accuracy of these methods are presented; an
effect of error autocorrection is considered. This effect occurs in efficient
methods of rational approximation (e.g., Pade approximations, linear and
nonlinear Pade-Chebyshev approximations) where very significant errors in the
coefficients do not affect the accuracy of the approximation. The matter of
import is that the errors in the numerator and the denominator of a fractional
rational approximant compensate each other. This effect is related to the fact
that the errors in the coefficients of a rational approximant are not
distributed in an arbitrary way but form the coefficients of a new approximant
to the approximated function. Understanding of the error autocorrection
mechanism allows to decrease this error by varying the approximation procedure
depending on the form of the approximant. Some applications are described in
the paper. In particular, a method of implementation of basic calculations on
decimal computers that uses the technique of rational approximations is
described in the Appendix. To a considerable extent the paper is a survey and
the exposition is as elementary as possible.
|
math
|
2,165 |
A unifying approach to software and hardware design for scientific calculations and idempotent mathematics
|
math.NA
|
A unifying approach to software and hardware design generated by ideas of
Idempotent Mathematics is discussed. The so-called idempotent correspondence
principle for algorithms, programs and hardware units is described. A software
project based on this approach is presented.
|
math
|
2,166 |
Approximate rational arithmetics and arbitrary precision computations
|
math.NA
|
We describe an approximate rational arithmetic with round-off errors (both
absolute and relative) controlled by the user. The rounding procedure is based
on the continued fraction expansion of real numbers. Results of computer
experiments are given in order to compare efficiency and accuracy of different
types of approximate arithmetics and rounding procedures.
|
math
|
2,167 |
Holistic projection of initial conditions onto a finite difference approximation
|
math.NA
|
Modern dynamical systems theory has previously had little to say about finite
difference and finite element approximations of partial differential equations
(Archilla, 1998). However, recently I have shown one way that centre manifold
theory may be used to create and support the spatial discretisation of \pde{}s
such as Burgers' equation (Roberts, 1998a) and the Kuramoto-Sivashinsky
equation (MacKenzie, 2000). In this paper the geometric view of a centre
manifold is used to provide correct initial conditions for numerical
discretisations (Roberts, 1997). The derived projection of initial conditions
follows from the physical processes expressed in the PDEs and so is
appropriately conservative. This rational approach increases the accuracy of
forecasts made with finite difference models.
|
math
|
2,168 |
Accuracy and convergence of the backward Monte-Carlo method
|
math.NA
|
The recently introduced backward Monte-Carlo method [Johan Carlsson,
arXiv:math.NA/0010118] is validated, benchmarked, and compared to the
conventional, forward Monte-Carlo method by analyzing the error in the
Monte-Carlo solutions to a simple model equation. In particular, it is shown
how the backward method reduces the statistical error in the common case where
the solution is of interest in only a small part of phase space. The forward
method requires binning of particles, and linear interpolation between the bins
introduces an additional error. Reducing this error by decreasing the bin size
increases the statistical error. The backward method is not afflicted by this
conflict. Finally, it is shown how the poor time convergence can be improved
for the backward method by a minor modification of the Monte-Carlo equation of
motion that governs the stochastic particle trajectories. This scheme does not
work for the conventional, forward method.
|
math
|
2,169 |
Universal numerical algorithms and their software implementation
|
math.NA
|
The concept of a universal algorithm is discussed. Examples of this kind of
algorithms are presented. Software implementations of such algorithms in C++
type languages are discussed together with means that provide for computations
with an arbitrary accuracy. Particular emphasis is placed on universal
algorithms of linear algebra over semirings.
|
math
|
2,170 |
On a Generalisation of Obreshkoff-Ehrlich Method for Simultaneous Extraction of All Roots of Polynomials Over an Arbitrary Chebyshev System
|
math.NA
|
New modifications of the methods for simultaneous extraction of all roots of
polynomials over an arbitrary Chebyshev system are elaborated. A cubic
convergence of iterations is proved. The method presented is a generalisation
of the classical methods of Obreshkoff and Ehrlich for simultaneous seeking of
all roots of algebraic equations. Numerical examples are provided.
|
math
|
2,171 |
A Generalization of Obreshkoff-Ehrlich Method for Multiple Roots of Algebraic, Trigonometric and Exponential Equations
|
math.NA
|
In this paper methods for simultaneous finding all roots of generalized
polynomials are developed. These methods are related to the case when the roots
are multiple. They possess cubic rate of convergence and they are as
labour-consuming as the known methods related to the case of polynomials with
simple roots only.
|
math
|
2,172 |
Generalization of Ehrlich-Kjurkchiev method for multiple roots of algebraic equations
|
math.NA
|
In this paper a new method which is a generalization of the
Ehrlich-Kjurkchiev method is developed. The method allows to find
simultaneously all roots of the algebraic equation in the case when the roots
are supposed to be multiple with known multiplicities. The offered
generalization does not demand calculation of derivatives of order higher than
first simultaneously keeping quaternary rate of convergence which makes this
method suitable for application from the practical point of view.
|
math
|
2,173 |
A Generalization of Obreshkoff-Ehrlich Method for Multiple Roots of Polynomial Equations
|
math.NA
|
In this paper we develop a new method which is a generalization of the
Obreshkoff -Ehrlich method for the cases of algebraic, trigonometric and
exponential polynomials. This method has a cubic rate of convergence. It is
efficient from the computational point of view and can be used for simultaneous
finding all roots if the roots have known multiplicities. This new method in
spite of the arbitrariness of multiplicities is of the same complexity as the
methods for simultaneous finding all roots of simple roots. We do not use
divided differences with multiple knots and this fact does not lead to
calculation of derivatives of the given polynomial of higher order, but only of
first ones.
|
math
|
2,174 |
Some Generalizations of the Chebyshev Method for Simultaneous Determination of All Roots of Polynomial Equations
|
math.NA
|
Iterative methods for the simultaneous determination of all roots of an
equation are dis-cussed. The multiplicities of the roots are assumed to be
known in advance. The methods are proved to have a cubical rate of convergence.
Numerical examples are given.
|
math
|
2,175 |
Eigenfunctions on a Stadium Associated with Avoided Crossings of Energy Levels
|
math.NA
|
The authors examine graphical properties of eigenfunctions with stadium
boundaries associated with avoided crossings of energy levels.
|
math
|
2,176 |
A new algorithm for the volume of a convex polytope
|
math.NA
|
We provide two algorithms for computing the volume of a convex polytope with
half-space representation {x>=0; Ax <=b} for some (m,n) matrix A and some
m-vector b. Both algorithms have a O(n^m) computational complexity which makes
them especially attractive for large n and relatively small m when the other
methods with O(m^n) complexity fail. The methodology which differs from
previous existing methods uses a Laplace transform technique that is
well-suited to the half-space representation of the polytope.
|
math
|
2,177 |
Derive boundary conditions for holistic discretisations of Burgers' equation
|
math.NA
|
I previously used Burgers' equation to introduce a new method of numerical
discretisation of \pde{}s. The analysis is based upon centre manifold theory so
we are assured that the discretisation accurately models all the processes and
their subgrid scale interactions. Here I show how boundaries to the physical
domain may be naturally incorporated into the numerical modelling of Burgers'
equation. We investigate Neumann and Dirichlet boundary conditions. As well as
modelling the nonlinear advection, the method naturally derives symmetric
matrices with constant bandwidth to correspond to the self-adjoint diffusion
operator. The techniques developed here may be used to accurately model the
nonlinear evolution of quite general spatio-temporal dynamical systems on
bounded domains.
|
math
|
2,178 |
Solving the difference initial-boundary value problems by the operator exponential method
|
math.NA
|
We suggest a modification of the operator exponential method for the
numerical solving the difference linear initial boundary value problems. The
scheme is based on the representation of the difference operator for given
boundary conditions as the perturbation of the same operator for periodic ones.
We analyze the error, stability and efficiency of the scheme for a model
example of the one-dimensional operator of second difference.
|
math
|
2,179 |
A Priori Estimates for the Global Error Committed by Runge-Kutta Methods for a Nonlinear Oscillator
|
math.NA
|
The Alekseev-Gr{\"o}bner lemma is combined with the theory of modified
equations to obtain an \emph{a priori} estimate for the global error of
numerical integrators. This estimate is correct up to a remainder term of order
$h^{2p}$, where $h$ denotes the step size and $p$ the order of the method. It
is applied to a class of nonautonomous linear oscillatory equations, which
includes the Airy equation, thereby improving prior work which only gave the
$h^p$ term.
Next, nonlinear oscillators whose behaviour is described by the Emden-Fowler
equation $y'' + t^\nu y^n = 0$ are considered, and global errors committed by
Runge-Kutta methods are calculated. Numerical experiments show that the
resulting estimates are generally accurate. The main conclusion is that we need
to do a full calculation to obtain good estimates: the behaviour is different
from the linear case, it is not sufficient to look only at the leading term,
and merely considering the local error does not provide an accurate picture
either.
|
math
|
2,180 |
Numerical Computations of Viscous, Incompressible Flow Problems Using a Two-Level Finite Element Method
|
math.NA
|
We consider two-level finite element discretization methods for the stream
function formulation of the Navier-Stokes equations. The two-level method
consists of solving a small nonlinear system on the coarse mesh, then solving a
linear system on the fine mesh. The basic result states that the errors between
the coarse and fine meshes are related superlinearly. This paper demonstrates
that the two-level method can be implemented to approximate efficiently
solutions to the Navier-Stokes equations. Two fluid flow calculations are
considered to test problems which have a known solution and the driven cavity
problem. Stream function contours are displayed showing the main features of
the flow.
|
math
|
2,181 |
Direct linearization method for nonlinear PDE's and the related kernel RBFs
|
math.NA
|
The standard methodology handling nonlinear PDE's involves the two steps:
numerical discretization to get a set of nonlinear algebraic equations, and
then the application of the Newton iterative linearization or its variants to
solve the nonlinear algebraic systems. Here we present an alternative strategy
called direct linearization method (DLM). The DLM discretization algebraic
equations of nonlinear PDE's is simply linear rather than nonlinear. The basic
idea behind the DLM is that we see a nonlinear term as a new independent
systematic variable and transfer a nonlinear PDE into a linear PDE with more
than one independent variable. It is stressed that the DLM strategy can be
applied combining any existing numerical discretization techniques. The
resulting linear discretization equations can be either over-posed or
well-posed. In particular, we also discuss how to create proper radial basis
functions in conjunction with the DLM.
|
math
|
2,182 |
Shock-capturing with natural high frequency oscillations
|
math.NA
|
This paper explores the potential of a newly developed conjugate filter
oscillation reduction (CFOR) scheme for shock-capturing under the influence of
natural high-frequency oscillations. The conjugate low-pass and high-pass
filters are constructed based on the principle of the discrete singular
convolution. Two Euler systems, the advection of an isentropy vortex flow and
the interaction of shock-entropy wave are considered to demonstrate the utility
of the CFOR scheme. Computational accuracy and order of approximation are
examined and compared with the literature. Some of the best numerical results
are obtained for the shock-entropy wave interaction. Numerical experiments
indicate that the proposed scheme is stable, conservative and reliable for the
numerical simulation of hyperbolic conservation laws.
|
math
|
2,183 |
Holistically discretise the Swift-Hohenberg equation on a scale larger than its spatial pattern
|
math.NA
|
I introduce an innovative methodology for deriving numerical models of
systems of partial differential equations which exhibit the evolution of
spatial patterns. The new approach directly produces a discretisation for the
evolution of the pattern amplitude, has the rigorous support of centre manifold
theory at finite grid size $h$, and naturally incorporates physical boundaries.
The results presented here for the Swift-Hohenberg equation suggest the
approach will form a powerful method in computationally exploring pattern
selection in general. With the aid of computer algebra, the techniques may be
applied to a wide variety of equations to derive numerical models that
accurately and stably capture the dynamics including the influence of possibly
forced boundaries.
|
math
|
2,184 |
A semi-numerical computation for the added mass coefficients of an oscillating hemi-sphere at very low and very high frequencies
|
math.NA
|
A floating hemisphere under forced harmonic oscillation at very high and very
low frequencies is considered. The problem is reduced to an elliptic one, that
is, the Laplace operator in the exterior domain with standard Dirichlet and
Neumann boundary conditions, so the flow problem is simplified to standard
ones, with well known analytic solutions in some cases. The general procedure
is based in the use of spherical harmonics and its derivation is based on a
physics insight. The results can be used to test the accuracy achieved by
numerical codes as, for example, by finite elements or boundary elements.
|
math
|
2,185 |
Phase retrieval by iterated projections
|
math.NA
|
Several strategies in phase retrieval are unified by an iterative "difference
map" constructed from a pair of elementary projections and a single real
parameter $\beta$. For the standard application in optics, where the two
projections implement Fourier modulus and object support constraints
respectively, the difference map reproduces the "hybrid" form of Fienup's
input-output map for $\beta = 1$. Other values of $\beta$ are equally effective
in retrieving phases but have no input-output counterparts. The geometric
construction of the difference map illuminates the distinction between its
fixed points and the recovered object, as well as the mechanism whereby
stagnation is avoided. When support constraints are replaced by object
histogram or atomicity constraints, the difference map lends itself to
crystallographic phase retrieval. Numerical experiments with synthetic data
suggest that structures with hundreds of atoms can be solved.
|
math
|
2,186 |
Bayesian Blocks in Two or More Dimensions: Image Segmentation and Cluster Analysis
|
math.NA
|
This paper describes an extension, to higher dimensions, of the Bayesian
Blocks algorithm for estimating signals in noisy time series data (Scargle
1998, 2000). The mathematical problem is to find the partition of the data
space with the maximum posterior probability for a model consisting of a
homogeneous Poisson process for each partition element. For model M_{n},
attributing the data within region n of the data space to a Poisson process
with a fixed event rate lambda_{n}, the global posterior is:
P(M_{n}) = Phi(N,V) = Gamma(N+1)Gamma(V-N+1) / Gamma(V+2) = N!(V-N)!/(V+1)! .
Note that lambda_{n} does not appear, since it has been marginalized, using a
flat, improper prior. Other priors yield similar formulas. This expression is
valid for a data space of any dimension. It depends on only N, the number of
data points within the region, and V, the volume of the region. No information
about the actual locations of the points enters this expression.
Suppose two such regions, described by N_{1},V_{1} and N_{2},V_{2}, are
candidates for being merged into one. From the above equation, construct a
Bayes merge factor, giving the ratio of posteriors for the two regions merged
and not merged, respectively:
P(Merge) = Phi(N_{1}+N_{2},V_{1}+V_{2}) / Phi(N_{1},V_{1}) Phi(N_{2},V_{2}) .
Then collect data points into blocks with a greedy cell coalescence
algorithm.
|
math
|
2,187 |
New RBF collocation schemes and their applications
|
math.NA
|
The purpose of this study is to apply some new RBF collocation schemes and
recently-developed kernel RBFs to various types of partial differential
equation systems. By analogy with the Fasshauer's Hermite interpolation, we
recently developed the symmetric BKM and boundary particle methods (BPM), where
the latter is based on the multiple reciprocity principle. The resulting
interpolation matrix of them is always symmetric irrespective of boundary
geometry and conditions. Furthermore, the proposed direct BKM and BPM apply the
practical physical variables rather than expansion coefficients and become very
competitive alternative to the boundary element method. On the other hand, by
using the Green integral we derive a new domain-type symmetrical RBF scheme
called as the modified Kansa method (MKM), which differs from the Fasshaure's
scheme in that the MKM discretizes both governing equation and boundary
conditions on the same boundary nodes. Therefore, the MKM significantly reduces
calculation errors at nodes adjacent to boundary with explicit mathematical
basis. Experimenting these novel RBF schemes with 2D and 3D Laplace, Helmholtz,
and convection-diffusion problems will be subject of this study. In addition,
the nonsingular high-order fundamental or general solution will be employed as
the kernel RBFs in the BKM and MKM.
|
math
|
2,188 |
Detection of Edges in Spectral Data II. Nonlinear Enhancement
|
math.NA
|
We discuss a general framework for recovering edges in piecewise smooth
functions with finitely many jump discontinuities, where $[f](x):=f(x+)-f(x-)
\neq 0$. Our approach is based on two main aspects--localization using
appropriate concentration kernels and separation of scales by nonlinear
enhancement.
To detect such edges, one employs concentration kernels, $K_\epsilon(\cdot)$,
depending on the small scale $\epsilon$. It is shown that odd kernels, properly
scaled, and admissible (in the sense of having small $W^{-1,\infty}$-moments of
order ${\cal O}(\epsilon)$) satisfy $K_\epsilon*f(x) = [f](x) +{\cal
O}(\epsilon)$, thus recovering both the location and amplitudes of all edges.As
an example we consider general concentration kernels of the form
$K^\sigma_N(t)=\sum\sigma(k/N)\sin kt$ to detect edges from the first
$1/\epsilon=N$ spectral modes of piecewise smooth f's. Here we improve in
generality and simplicity over our previous study in [A. Gelb and E. Tadmor,
Appl. Comput. Harmon. Anal., 7 (1999), pp. 101-135]. Both periodic and
nonperiodic spectral projections are considered. We identify, in particular, a
new family of exponential factors, $\sigma^{exp}(\cdot)$, with superior
localization properties.
The other aspect of our edge detection involves a nonlinear enhancement
procedure which is based on separation of scales between the edges, where
$K_\epsilon*f(x)\sim [f](x) \neq 0$, and the smooth regions where $K_\epsilon*f
= {\cal O}(\epsilon) \sim 0$. Numerical examples demonstrate that by coupling
concentration kernels with nonlinear enhancement one arrives at effective edge
detectors.
|
math
|
2,189 |
Adaptive Mollifiers for High Resolution Recovery of Piecewise Smooth Data from its Spectral Information
|
math.NA
|
We discuss the reconstruction of piecewise smooth data from its (pseudo-)
spectral information. Spectral projections enjoy superior resolution provided
the data is globally smooth, while the presence of jump discontinuities is
responsible for spurious ${\cal O}(1)$ Gibbs oscillations in the neighborhood
of edges and an overall deterioration to the unacceptable first-order
convergence rate. The purpose is to regain the superior accuracy in the
piecewise smooth case, and this is achieved by mollification.
Here we utilize a modified version of the two-parameter family of spectral
mollifiers introduced by Gottlieb & Tadmor [GoTa85]. The ubiquitous
one-parameter, finite-order mollifiers are based on dilation. In contrast, our
mollifiers achieve their high resolution by an intricate process of high-order
cancelation. To this end, we first implement a localization step using edge
detection procedure, [GeTa00a, GeTa00b]. The accurate recovery of piecewise
smooth data is then carried out in the direction of smoothness away from the
edges, and adaptivity is responsible for the high resolution. The resulting
adaptive mollifier greatly accelerates the convergence rate, recovering
piecewise analytic data within exponential accuracy while removing spurious
oscillations that remained in [GoTa85]. Thus, these adaptive mollifiers offer a
robust, general-purpose ``black box'' procedure for accurate post processing of
piecewise smooth data.
|
math
|
2,190 |
High resolution conjugate filters for the simulation of flows
|
math.NA
|
This paper proposes a Hermite-kernel realization of the conjugate filter
oscillation reduction (CFOR) scheme for the simulation of fluid flows. The
Hermite kernel is constructed by using the discrete singular convolution (DSC)
algorithm, which provides a systematic generation of low-pass filter and its
conjugate high-pass filters. The high-pass filters are utilized for
approximating spatial derivatives in solving flow equations, while the
conjugate low-pass filter is activated to eliminate spurious oscillations
accumulated during the time evolution of a flow. As both low-pass and high-pass
filters are derived from the Hermite kernel, they have similar regularity,
time-frequency localization, effective frequency band and compact support.
Fourier analysis indicates that the CFOR-Hermite scheme yields a nearly optimal
resolution and has a better approximation to the ideal low-pass filter than
previously CFOR schemes. Thus, it has better potential for resolving natural
high frequency oscillations from a shock. Extensive one- and two-dimensional
numerical examples, including both incompressible and compressible flows, with
or without shocks, are employed to explore the utility, test the resolution,
and examine the stability of the present CFOR-Hermite scheme. Extremely small
ratio of point-per-wavelength (PPW) is achieved in solving the Taylor problem,
advancing a wavepacket and resolving a shock/entropy wave interaction. The
present results for the advection of an isentropic vortex compare very
favorably to those in the literature.
|
math
|
2,191 |
Shape reconstruction in scattering media with voids using a transport model and level sets
|
math.NA
|
A two-step shape reconstruction method for diffuse optical tomography (DOT)
is presented which uses adjoint fields and level sets. The propagation of
near-infrared photons in tissue is modeled by the time-dependent linear
transport equation, of which the absorption parameter has to be reconstructed
from boundary measurements. In the shape reconstruction approach, it is assumed
that the inhomogeneous background absorption parameter and the values inside
the obstacles (which typically have a high contrast to the background) are
known, but that the number, sizes, shapes, and locations of these obstacles
have to be reconstructed from the data. An additional difficulty arises due to
the presence of so-called clear regions in the medium. The first step of the
reconstruction scheme is a transport-backtransport (TBT) method which provides
us with a low-contrast approximation to the sought objects. The second step
uses this result as an initial guess for solving the shape reconstruction
problem. A key point in this second step is the fusion of the 'level set
technique' for representing the shapes of the reconstructed obstacles, and an
'adjoint-field technique' for solving the nonlinear inverse problem. Numerical
experiments are presented which show that this novel method is able to recover
one or more objects very fast and with good accuracy.
|
math
|
2,192 |
Finite volume methods for incompressible flow
|
math.NA
|
Two finite volume methods are derived and applied to the solution of problems
of incompressible flow. In particular, external inviscid flows and
boundary-layer flows are examined. The firstmethod analyzed is a cell-centered
finite volume scheme. It is shown to be formally first order accurate on
equilateral triangles and used to calculate inviscid flow over an airfoil. The
second method is a vertex-centered least-squares method and is second order
accurate. It's quality is investigated for several types of inviscid flow
problems and to solve Prandtl's boundary-layer equations over a flat plate.
Future improvements and extensions of the method are discussed.
|
math
|
2,193 |
Double newtonisation of fixed point sequences
|
math.NA
|
A neutral fixed point of a real iteration map $u$ becomes a super attracting
fixed point using a suitable double newtonisation. The map $u$ is so
transformed into a map $w$ which is here called the standard accelerator of
$u$. The map $w$ provides a unifying process to deal with a large set of fixed
point sequences which are not convergent or converge slowly. Several examples
illustrate the main results obtained.
|
math
|
2,194 |
New Method to obtain Exact-Fit Polynomial and Exponential
|
math.NA
|
The existing methods to obtain an exact-fit polynomial does not give the
resulting polynomial in its standard form, and further manipulations are needed
to obtain that. The new method presented here gives the coefficients of the
polynomial in the standard form directly. It is also possible to obtain the
exact-fit exponential using a similar method. Part I of the Document explains
the method to find the Exact-Fit Polynomial and Part II explains the method to
obtain an Exact-Fit Exponential.
|
math
|
2,195 |
Algorithm to generate ideals in a Lie algebra of matrices at any particular characteristic with Mathematica
|
math.NA
|
We present in this paper a routine which construct the ideal generated by a
list of elements in a matrix Lie algebra at any particular characteristic. We
have used this algorithm to analyze the problem of the simplicity of some Lie
algebras.
|
math
|
2,196 |
Algorithm to compute the rank and a Cartan subalgebra of a matrix Lie algebra with Mathematica
|
math.NA
|
We present in this paper a set of routines constructed to compute the rank of
a matrix Lie algebra and also to determine a Cartan subalgebra from a given
list of elements
|
math
|
2,197 |
Large Eddy Simulation of Turbulent Channel Flows by the Rational LES Model
|
math.NA
|
The rational large eddy simulation (RLES) model is applied to turbulent
channel flows. This approximate deconvolution model is based on a rational
(subdiagonal Pade') approximation of the Fourier transform of the Gaussian
filter and is proposed as an alternative to the gradient (also known as the
nonlinear or tensor-diffusivity) model. We used a spectral element code to
perform large eddy simulations of incompressible channel flows at Reynolds
numbers based on the friction velocity and the channel half-width Re{sub tau} =
180 and Re{sub tau} = 395. We compared the RLES model with the gradient model.
The RLES results showed a clear improvement over those corresponding to the
gradient model, comparing well with the fine direct numerical simulation. For
comparison, we also present results corresponding to a classical subgrid-scale
eddy-viscosity model such as the standard Smagorinsky model.
|
math
|
2,198 |
The Lie algebra splitg2 with Mathematica using Zorn's matrices
|
math.NA
|
We will obtain in this paper a generic expression of any element in athe Lie
algebra of the derivations of the split octonions a over an arbitrary field.
For this purpose, we will use the Zorn's matrices. We will also compute the
multiplication table of this Lie algebra.
|
math
|
2,199 |
About Calculation of the Hankel Transform Using Preliminary Wavelet Transform
|
math.NA
|
The purpose of this paper is to present an algorithm for evaluating Hankel
transform of the null and the first kind. The result is the exact analytical
representation as the series of the Bessel and Struve functions multiplied by
the wavelet coefficients of the input function. Numerical evaluation of the
test function with known analytical Hankel transform illustrates the proposed
algorithm.
|
math
|
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