Showing papers presented at "International Conference on Numerical Analysis and Its Applications in 1996"

High-Order Stiff ODE Solvers via Automatic Differentiation and Rational Prediction

Abstract: A class of higher order methods is investigated which can be viewed as implicit Taylor series methods based on Hermite quadratures. Improved automatic differentiation techniques for the claculation of the Taylor-coefficients and their Jacobians are used. A new rational predictor is used which can allow for larger step sizes on stiff problems.

Mosaic Ranks and Skeletons

Abstract: The fact that nonsingular coefficient matrices can be covered by blocks close to low-rank matrices is well known probably for years. It was used in some cost-effective matrix-vector multiplication algorithms. However, it has been never paid a proper attention in the matrix theory. To fill in this gap we propose a notion of mosaic ranks of a matrix which reduces a description of block matrices with low-rank blocks to a single number. A general algebraic framework is presented that allows one to obtain some theoretical estimates on the mosaic ranks. An algorithm for computing upper estimates of the mosaic ranks is given with some illustrations of its efficiency on model problems.

Applications of Steklov-Type Eigenvalue Problems to Convergence of Difference Schemes for Parabolic and Hyperbolic Equations with Dynamical Boundary Conditions

Abstract: Parabolic and hyperbolic equations with dynamical boundary conditions, i.e which involve first and second order time derivatives respectively, are considered. Convergence and stability of weighted difference schemes for such problems are discussed. Norms arising from Steklov-type eigenvalues problems are used, while in previously investigations, norms corresponding to Neumann's or Robin's boundary conditions are used. More exact stability conditions are obtained for the difference schemes parameters.

Exploiting the Natural Partitioning in the Numerical Solution of ODE Systems Arising in Atmospheric Chemistry

Abstract: Large air pollution models are commonly used to study transboundary transport of air pollutants Such models are described mathematically by systems of partial differential equations (the number of equations being equal to the number of pollutants involved in the model) The use of appropriate splitting procedures leads to several sub-models If the model is discretized on a (96×96×10) grid and if the number of pollutants is 35, then a system of ODE's containing 3225600 equations is to be treated in each sub-model The ODE system of the chemical sub-model can be decoupled to (96×96×10) small systems, each of them containing 35 equations The number of time-steps, needed in each submodel, is typically several thousand

Iterative Monte Carlo Algorithms for Linear Algebra Problems

Abstract: A common Monte Carlo approach for linear algebra problems is presented. The considered problems are inverting a matrix B, solving systems of linear algebraic equations of the form Bu=b and calculating eigenvalues of symmetric matrices. Several algorithms using the same Markov chains with different random variables are described.

Boundary Value Methods for the Numerical Approximation of Ordinary Differential Equations

Abstract: Many numerical methods for the approximation of ordinary differential equations (ODEs) are obtained by using Linear Multistep Formulae (LMF). Such methods, however, in their usual implementation suffer of heavy theoretical limitations, summarized by the two well known Dahlquist barriers. For this reason, Runge-Kutta schemes have become more popular than LMF, in the last twenty years. This situation has recently changed, with the introduction of Boundary Value Methods (BVMs), which are methods still based on LMF. Their main feature consists in approximating a given continuous initial value problem (IVP) by means of a discrete boundary value problem (BVP). Such use allows to avoid order barriers for stable methods. Moreover, BVMs provide several families of methods, which make them very flexible and computationally efficient. In particular, we shall see that they allow a natural implementation of efficient mesh selection strategies.

Convergence of a Crank-Nicolson Difference Scheme for Heat Equations with Interface in the Heat Flow and Concentrated Heat Capacity

Abstract: In this paper we consider initial value problems for heat equation with discontinuous heat flow and concentrated heat capacity in interior points or at the boundary. Convergence of the Crank-Nicolson scheme is analyzed via the concept of elliptic projection. Namely, second order convergence is proved for the corresponding elliptic problems in special norms. Then, splitting the error of the heat problem into two errors we prove second order estimates in space and time in modified L2 norm.

The V-ray Technology of Optimizing Programs to Parallel Computers

Abstract: This paper gives a brief overview of the V-Ray technology, based on the rigorous mathematical theory of analysis and transformation of programs, and intended for optimization of programs to parallel computers. This technology provides a basis for resolving the whole scope of problems related to mapping of applications onto parallel computers, starting from the commonly adopted control-flow and data-flow analysis up to the optimization of data distribution and data locality. High efficiency of the V-Ray technology is illustrated by successful optimization of the TRFD Perfect Benchmark to vector/parallel CRAY Y-MP M90/C90 as well as to massively parallel CRAY T3D supercomputers.

Basic Techniques for Numerical Linear Algebra on Bulk Synchronous Parallel Computers

Abstract: The bulk synchronous parallel (BSP) model promises scalable and portable software for a wide range of applications. A BSP computer consists of several processors, each with private memory, and a communication network that delivers access to remote memory in uniform time.

Convex Combinations of Matrices-Nonsingularity and Schur Stability

Abstract: Using the notion of a block P-matrix, introduced previously by the authors, a characterization of the nonsingularity (Schur stability, resp.) of all convex combinations of three nonsingular (Schur stable, resp.) real matrices is derived.

Spectral Portrait of Matrices by Block Diagonalization

Abstract: We first describe an algorithm that reduces a matrix A to a block diagonal form using only well conditioned transformations. The spectral properties of A are then carried out from the resulting block diagonal matrix. We show in particular that the spectral portrait of A can be obtained cheaply from that of the block diagonal matrix.

Third Order Explicit Method for the Stiff Ordinary Differential Equations

Abstract: The time-step in integration process has two restrictions. The first one is the time-step restriction due to accuracy requirement τ ac and the second one is the time-step restriction due to stability requirement τ st . The stability property of the Runge-Kutta method depend on stability region of the method. The stability function of the explicit methods is the polynomial. The stability regions of the polynomials are relatively small. The most of explicit methods have small stability regions and consequently small τ st . It obliges us to solve the ODE with the small step size τ st ≪τac. The goal of our article is to construct the third order explicit methods with enlarged stability region (with the big τ st : τ st ≥τ ac ). To achieve this aim we construct the third order polynomials: 1−z+z2/2−z3/6+∑ i=4 n d i z i with the enlarge stability regions. Then we derive the formula for the embedded Runge-Kutta third order accuracy methods with the stability functions equal to above polynomials. The methods can use only three arrays of the storage. It gives us opportunity to solve large systems of differential equations.

Bessel and Neumann Fitted Methods for the Numerical Solution of the Schrödinger Equation

Abstract: Two 2-step methods for the numerical solution of some problems of the Schrodinger equation are developed in this paper. One is of the Numerov-type and of algebraic order 4 and the other is of the Runge-Kutta type and of algebraic order 5. Each of these methods have free parameters which will be defined such that the methods are fitted to spherical Bessel and Neumann functions. Based on these methods we have obtained a variable-step method. The results produced based on the phase-shift problem of the radial Schrodinger equation indicate that this new approach is more efficient than other well known methods.

P-stable Mono-Implicit Runge-Kutta-Nyström Modifications of the Numerov Method

Abstract: We present families of fourth-order mono-implicit Runge-Kutta-Nystrom methods Each member of these families can be considered as a modification of the Numerov method Some parameters of these new methods are used to optimize the linear stability properties, ie to obtain P-stable methods with a minimal phase-lag Also we show that in some cases there exist P-stable methods with stage-order 3 Since the methods considered are mono-implicit, the computational work needed in each time-step to solve the implicit equations is reduced seriously

Operator Problems in Strengthened Sobolev Spaces and Numerical Methods for Them

Abstract: The strengthened Sobolev spaces are naturally connected, e.g., with such important (two or three-dimensional) problems of mathematical physics as those in theory of plates and shells with stiffeners or in the capillary hydrodynamics involving the surface tension. These nonstandard Hilbert spaces allow also to set variational and operator problems on composed manifolds of different dimensionality. Spectral (eigenvalue) problems can be considered as well.

Existence and Stability of Traveling Discrete Shocks

Abstract: We are concerned with existence and stability questions for finite difference schemes approximating solutions of scalar conservation laws with shocks. A suitable model for the study of the artifacts created by these schemes near the shocks are the traveling discrete shock profiles; these are discrete shock profiles υ=(υk)k∈ℤ which reappear shifted, when the scheme is applied on them, according to the speed of the shock. Existence of such profiles connecting entropy admissible shocks is already established for monotone schemes, first and third order accurate schemes and the Lax-Wendroff scheme. Jennings showed existence and l1-stability of these profiles for conservative monotone schemes. Smyrlis showed existence and parametrization by the amount of excess mass and stability for stationary profiles of the Lax-Wendroff scheme. Shih Hsien Yu showed existence of traveling profiles of mild strength for the Lax-Wendroff scheme using inertial manifolds theory. Here we study traveling discrete shock profiles for Lax-Wendroff, Engquist-Osher and monotone schemes. We show existence of such profiles with small shock speed. We also show that these profiles are stable with respect to suitably weighted l2-norms.

A Finite State Stochastic Minimax Optimal Control Problem with Infinite Horizon

Abstract: We consider here a stochastic discrete minimax control problem with infinite horizon. We prove the existence of solution, we characterize it and we present iterative methods to compute it numerically.

Improved Perturbation Bounds for the Matrix Exponential

Abstract: In this paper we give asymptotic series expansions in e=∥E∥ for the bound of the perturbation ∥exp(t(A+E))−exp(tA)∥ in the matrix exponential exp(tA).

High Performance Solution of Linear Systems Arising from Conforming Spectral Approximations for Non-Conforming Domain Decompositions

Abstract: We apply a conforming spectral collocation technique to non-conforming domain decompositions. The resulting global matrices have a particular block structure. We study the performance of various direct methods of solution of the resulting linear system on a RS6000 workstation, a SGI Power Challenge and a Cray J-916 supercomputer.

Convergence in Iterative Polynominal Root-Finding

Abstract: For target α of the Nth-degree polynomial P (z), ¦δ*/δ¦≡ ¦(z* −α)/(z −α)¦=O [σδ¦q−1] 1 and ¦σδ¦ ≪ 1, regardless of ¦δ¦ itself. Even if α is not a zero but the centroid of a cluster, the recomputed multiplicity estimate m (z) could lead to a component zero. In global iterations, popular methods proved inadequate, yet for symmetric clusters the CLAM formula z*=z −(NP/P′) (1 −Q m/n )/(1 −Q), where Q=[N (1 −PP″/P′2) −1]/(N/m −1), converges in principle to an m-fold zero in one iteration, using any finite guess outside the cluster centroid. Equipped with countermeasures against rebounds caused by local clusters, the formula has never been found to fail for general polynomials, and with an initial guess based on zeros of a symmetric cluster, usually converge in a few iterations.

Stabilization and Experience with the Partitioning Method for Tridiagonal Systems

Abstract: The partitioning algorithm which is a modification of Wang's method for tridiagonal equations is stabilized to the case of arbitrary well conditioned matrix. A realization on Parallel Virtual Machine (PVM) is presented. The parallel solution is analysed under different loads: system dimension, variable numbers of virtual machines and different kind of bandwidth local area networks.

Rate of Convergence of a Numerical Procedure for Impulsive Control Problems

Abstract: We consider a deterministic impulsive control problem. We discretize the Hamilton-Jacobi-Bellman equation satisfied by the optimal cost function and we obtain discrete solutions of the problem. We give an explicit rate of convergence of the approximate solutions to the solution of the original problem. We consider the optimal switching problem as a special case of impulsive control problem and we apply the same structure of discretization to obtain also a rate of convergence in this case. We present a numerical example.

Interpolation Technique and Convergence Rate Estimates for Finite Difference Method

Abstract: In this work we expose a methodology for establishing convergence rate estimates for finite difference schemes based on the interpolation theory of Banach spaces. As a model problem we consider Dirichlet boundary value problem for second order linear elliptic equation with variable coefficients from Sobolev spaces. Using interpolation theory we construct fractional-order convergence rate estimates which are consistent with the smoothness of data.

Conforming Spectral Domain Decomposition Schemes

Abstract: Spectral domain decomposition schemes are presented for the numerical solution of second and fourth order problems These schemes, which are formulated in the collocation framework yield spectral approximations which are conforming along the subdomain interfaces for both conforming and non-conforming decompositions For conforming decompositions the approximations are pointwise C1 continuous across the interfaces for second order problems and C3 continuous across the interfaces for fourth order problems For non-conforming decompositions the corresponding approximations are pointwise C0 and C1 continuous for second and fourth order problems, respectively Efficient direct methods for the solution of the resulting systems are also presented

The Effectiveness of Band-Toeplitz Preconditioners: A Survey

Abstract: In many applications such as signal processing [26], differential equations [40], image restoration [20] and statistics [22] we have to solve n × n Hermitian Toeplitz linear systems An(f)x=b where the symbol f, the generating function, is an L1 function and the entries of An(f) along the k- th diagonal coincide with the k-th Fourier coefficient a k of f. When the generating function has (essential) zeros of even orders and is nonnegative, several very satisfactory band-Toeplitz preconditioners have been proposed [9, 18, 12, 32] leading to optimal and superoptimal preconditioned conjugate gradient (PCG) methods with a total cost of O(n log n) ops, even in the ill-conditioned case where other celebrated techniques fail [17]. More recently these preconditioners have been successfully extended to the block [27] and nondefinite [29] cases as well as to the case of zeros of any (also noninteger) orders [34]. The latter extension is very important because it concerns with a lot of practical situations not considered before. Therefore, the only relevant criticism to this approach continues to be the assumption that, at least, the sign of f, the position of the zeros and their order are known. In fact, in some applications, it is possible to assume the availability of this information but, in other fields, e.g. image restoration, this just results an ideal hypothesis. In the latter case, very recently, we have introduced [38] a practical and economical criterion in order to discover all the needed analytical properties of f by using as data input the coefficients ak. In such way, we are convinced that the set of all the band-Toeplitz preconditioners based strategies can become much more attractive from an applicative point of view.

Least Squares and Total Least Squares Methods in Image Restoration

Abstract: Image restoration is the process of removing or minimizing degradations (blur) in an image. Mathematically, it can be modeled as a discrete ill-posed problem Hf=g, where H is a matrix of large dimension representing the blurring phenomena, and g is a vector representing the observed image. Often H is severely ill-conditioned, and both H and g are corrupted with noise. Regularization is used to reduce the noise sensitivity of the numerical scheme. Most of these methods, however, assume that there is no noise in H, and therefore least squares techniques are used to reconstruct f. In some applications, though, H is also corrupted with noise. In this case a total least squares approach may be more appropriate. These least squares and total least squares methods will be investigated in the context of signal restoration.

Justification of Difference Schemes for Derivative Nonlinear Evolution Equations

Abstract: We consider the difference schemes applied to a derivative nonlinear system of evolution equations. For the boundary-value problem with initial conditions $$\begin{gathered}\frac{{\partial u}}{{\partial t}} = A\frac{{\partial ^2 u}}{{\partial x^2 }} + B\frac{{\partial u}}{{\partial x}} + f(x,u) + g(x,u)\frac{{\partial u}}{{\partial x}},(t,x) \in (0,T] \times (0,1), \hfill \\u(t,0) = u(t,1) = 0,t \in [0,T], \hfill \\u(0,x) = u^{(0)} (x),x \in (0,1) \hfill \\\end{gathered}$$ we use the Crank-Nicolson discretizations. A is complex and B — real diagonal matrixes; u,f and g are complex vector-functions. The analysis shows that proposed schemes are uniquely solvable, convergent and stable in a grid norm W 2 2 if all (diagonal) elements in Re(A) are positive. This is true without any restrictions on the ratio of time and space grid steps.

The Use of Discrete Sine Transform in Computations with Toeplitz Matrices

Abstract: The T algebra, related to the discrete sine transform, is an efficient tool for approximating Toeplitz matrices arising in image processing. We present two applications concerning the computation of singular values and the preconditioning of least squares problems.