On the boundary-value problem for a mixed-type equation: Asymptotic analysis and an effective numerical algorithm

A singularly perturbed boundary-value problem for an equation of mixed ellipticparabolic type is considered. The first part of this work is devoted to an asymptotic study of the solution to the problem. Modification of the boundary function method for mixed-type equations with small parameters at the highest derivatives is used. The second part is devoted to creating a numerical method that considers the structure of the solution for small parameter values. The idea of an approximate factorization of an elliptic operator into the product of two parabolic operators is employed. An effective numerical algorithm based on an asymptotic representation of the solution is developed.     

Rational interpolation and approximate solution of integral equations

In the space of continuous periodic functions, we construct interpolation rational operators, use them to obtain quadrature formulas with positive coefficients which are exact on rational trigonometric functions of order 2n, and suggest an algorithm for an approximate solution of integral equations of the second kind. We estimate the accuracy of the approximate solution via the best trigonometric rational approximations to the kernel and the right-hand side of the integral equation.     

A regularizing algorithm for the solution of the inverse problem for an inhomogeneous string

We consider uniqueness of the solution of the inverse problem of determining the coefficient of the one-dimensional wave equation on the real halfline. Necessary conditions of existence of a unique solution of this inverse problem are obtained. A Tikhonov regularizing algorithm is constructed for approximate solution of the inverse problem. The algorithm has an efficient numerical implementation.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 55–66, 1985.     

Degeneration and regularization of the operator in the Cauchy problem for the Schrödinger equation

The Cauchy problem for the Schrödinger equation whose operator degenerates on a half-line is studied. In order to approximate a solution to the problem with degeneracy by solutions to well-posed problems, the notion of regularization for an operator with degeneracy is introduced; an approximative solution to a problem with degeneracy is defined as the limit of a sequence of regularized problems. The dependence of the approximative solution on the choice of the class of admissible regularizations is studied. The weak compactness of sequences of states determined by sequences of solutions to regularized problems in the topologies determined by the space of all bounded linear operators and by subspaces of mutually commuting bounded linear operators is investigated.     

On dynamics of quantum states generated by the Cauchy problem for the Schrödinger equation with degeneration on the half-line

The paper considers the Cauchy problem for the Schrödinger equation with operator degenerate on the semiaxis and the family of regularized Cauchy problems with uniformly elliptic operators whose solutions approximate the solution of the degenerate problem. The author studies the strong and weak convergences of the regularized problems and the convergence of values of quadratic forms of bounded operators on solutions of the regularized problems when the regularization parameter tends to zero.     

A penalty method for a finite-dimensional obstacle problem with derivative constraints

We propose a power penalty method for an obstacle problem arising from the discretization of an infinite-dimensional optimization problem involving differential operators in both its objective function and constraints. In this method we approximate the mixed nonlinear complementarity problem (NCP) arising from the KKT conditions of the discretized problem by a nonlinear penalty equation. We then show the solution to the penalty equation converges exponentially to that of the mixed NCP. Numerical results will be presented to demonstrate the theoretical convergence rates of the method.     

Evolutionary Algorithms for Approximate Optimal Control of the Heat Equation with Thermal Sources

In this paper, optimal control problem (OCP) governed by the heat equation with thermal sources is considered. The aim is to find an optimal control which puts the system in a finite time T, into a stationary regime and to minimize a general objective function. To obtain an approximate solution of this problem, a partition of the time-control space is considered and the discrete form of the problem is converted to a quasi assignment problem. Then by using an evolutionary algorithm, an approximate optimal control function is obtained as a piecewise linear function. Numerical examples are given to show the proficiency of the presented algorithm.     

Numerical solution by the quasi-reversibility method of unstable problems for the first-order evolution equation

Prior bounds are derived on the solution of the perturbed problem in different versions of the quasi-reversibility method used for approximate solution of unstable problems for first-order evolution equations. An example of such a problem is provided by the problem backward in time for the equation of heat conduction. Approximate solution of perturbed problems by difference methods is considered. The investigation of the difference schemes of the quasi-reversibility method relies on the general theory of p-stability of difference schemes. Specific features of solution of problems with non-self-adjoint operators are considered. Efficient difference schemes are constructed for multidimensional problems.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach. Matematicheskoi Fiziki, pp. 93–124, 1993.     

Regularization of the Solution of the Cauchy Problem: The Quasi-Reversibility Method

Some regularization algorithm is proposed related to the problem of continuation of the wave field from the planar boundary into the half-plane. We consider a hyperbolic equation whose main part coincideswith the wave operator, whereas the lowest term contains a coefficient depending on the two spatial variables. The regularization algorithm is based on the quasi-reversibility method proposed by Lattes and Lions. We consider the solution of an auxiliary regularizing equation with a small parameter; the existence, the uniqueness, and the stability of the solution in the Cauchy data are proved. The convergence is substantiated of this solution to the exact solution as the small parameter vanishes. A solution of an auxiliary problem is constructed with the Cauchy data having some error. It is proved that, for a suitable choice of a small parameter, the approximate solution converges to the exact solution.     

Approximate solution to a boundary value problem for the Lyapunov differential equation with a parameter

A special boundary value problem is studied for the Lyapunov differential equation which is used for investigation of the asymptotic properties of solutions to systems of periodic differential equations with a parameter. An algorithm is proposed for constructing an approximate solution to this boundary value problem, and conditions on the parameter are found under which the zero solution to the system is asymptotically stable.     

On the perturbation algorithm for the semidiscrete scheme for the evolution equation and estimation of the approximate solution error using semigroups

In a Banach space, for the approximate solution of the Cauchy problem for the evolution equation with an operator generating an analytic semigroup, a purely implicit three-level semidiscrete scheme that can be reduced to two-level schemes is considered. Using these schemes, an approximate solution to the original problem is constructed. Explicit bounds on the approximate solution error are proved using properties of semigroups under minimal assumptions about the smoothness of the data of the problem. An intermediate step in this proof is the derivation of an explicit estimate for the semidiscrete Crank–Nicolson scheme. To demonstrate the generality of the perturbation algorithm as applied to difference schemes, a four-level scheme that is also reduced to two-level schemes is considered.     

Set-valued mappings specified by regularization of the Schrödinger equation with degeneration

The Cauchy problem for the Schrödinger equation with an operator degenerating on a half-line and a family of regularized Cauchy problems with uniformly elliptic operators, whose solutions approximate the solution to the degenerate problem, are considered. A set-valued mapping is investigated that takes a bounded operator to a set of partial limits of values of its quadratic form on solutions of the regularized problems when the regularization parameter tends to zero. The dynamics of quantum states are determined by applying an averaging procedure to the set-valued mapping.     

Modification of upwind finite difference fractional step methods by the transient state of the semiconductor device

The mathematical model of the three‐dimensional semiconductor devices of heat conduction is described by a system of four quasi‐linear partial differential equations for initial boundary value problem. One equation of elliptic form is for the electric potential; two equations of convection‐dominated diffusion type are for the electron and hole concentration; and one heat conduction equation is for temperature. Upwind finite difference fractional step methods are put forward. Some techniques, such as calculus of variations, energy method multiplicative commutation rule of difference operators, decomposition of high order difference operators, and the theory of prior estimates and techniques are adopted. Optimal order estimates in L² norm are derived to determine the error in the approximate solution.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

The approximate solution of integro‐ differential equations

We consider a class of partial integro‐differential equations with appropriate conditions and its corresponding equation in the field of Mikusiński operators. As is usual in numerical analysis, we construct the corresponding difference equation, determine its solution, analyze its character and treat it as the approximate solution of the considered problem. We also estimate the error of approximation.     

On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow

Summary. The aim of this work is to study a decoupled algorithm of a fixed point for solving a finite element (FE) problem for the approximation of viscoelastic fluid flow obeying an Oldroyd B differential model. The interest for this algorithm lies in its applications to numerical simulation and in the cost of computing. Furthermore it is easy to bring this algorithm into play. The unknowns are the viscoelastic part of the extra stress tensor, the velocity and the pressure. We suppose that the solution is sufficiently smooth and small. The approximation of stress, velocity and pressure are resp. discontinuous, continuous, continuous FE. Upwinding needed for convection of , is made by discontinuous FE. The method consists to solve alternatively a transport equation for the stress, and a Stokes like problem for velocity and pressure. Previously, results of existence of the solution for the approximate problem and error bounds have been obtained using fixed point techniques with coupled algorithm. In this paper we show that the mapping of the decoupled fixed point algorithm is locally (in a neighbourhood of ) contracting and we obtain existence, unicity (locally) of the solution of the approximate problem and error bounds. Received July 29, 1994 / Revised version received March 13, 1995     

The accurate solution of certain continuous problems using only single precision arithmetic

A typical approach for finding the approximate solution of a continuous problem is through discretization with meshsizeh such that the truncation error goes to zero withh. The discretization problem is solved in floating point arithmetic. Rounding-errors spoil the theoretical convergence and the error may even tend to infinity.In this paper we present algorithms of moderate cost which use only single precision and which compute the approximate solution of the integration and elliptic equation problems with full accuracy. These algorithms are based on the modified Gill-Møller algorithm for summation of very many terms, iterative refinement of a linear system with a special algorithm for the computation of residuals in single precision and on a property of floating point subtraction of nearby numbers.On leave of absence from Institute of Informatics, University of Warsaw, 00-901 Warsaw, Poland.This research was supported in part by the National Science Foundation under Grant MCS-7823676.     

The Sinc‐Galerkin method for solving an inverse parabolic problem with unknown source term

The inverse problem of determining an unknown source term depending on space variable in a parabolic equation is considered. A numerical algorithm is presented for recovering the unknown function and obtaining a solution of the problem. As this inverse problem is ill‐posed, Tikhonov regularization is used for finding a stable solution. For solving the direct problem, a Galerkin method with the Sinc basis functions in both the space and time domains is presented. This approximate solution displays an exponential convergence rate and is valid on the infinite time interval. Finally, some examples are presented to illustrate the ability and efficiency of this numerical method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Riemannian Newton-type methods for joint diagonalization on the Stiefel manifold with application to independent component analysis

The joint approximate diagonalization of non-commuting symmetric matrices is an important process in independent component analysis. This problem can be formulated as an optimization problem on the Stiefel manifold that can be solved using Riemannian optimization techniques. Among the available optimization techniques, this study utilizes the Riemannian Newton’s method for the joint diagonalization problem on the Stiefel manifold, which has quadratic convergence. In particular, the resultant Newton’s equation can be effectively solved by means of the Kronecker product and the vec and veck operators, which reduce the dimension of the equation to that of the Stiefel manifold. Numerical experiments are performed to show that the proposed method improves the accuracy of the approximate solution to this problem. The proposed method is also applied to independent component analysis for the image separation problem. The proposed Newton method further leads to a novel and fast Riemannian trust-region Newton method for the joint diagonalization problem.     

Constrained radial symmetry for monotone elliptic quasilinear operators

Some overdetermined problems associated to monotone elliptic quasilinear operators are investigated. A model operator is the p-Laplacian. Assuming that a solution exists, the domain of our problem is shown to be either a ball centered at the origin or an annulus centered at the origin. In the special case of the Laplace equation, a result of approximate radial symmetry is also obtained. Proofs are based on comparisons with radial solutions.     

Numerical method for a quadratic minimization problem with an ellipsoidal constraint and an a priori estimate for the solution norm

An algorithm for solving a quadratic minimization problem on an ellipsoidal set in a Hilbert space is proposed. The algorithm is stable to nonuniform perturbations of the operators. A key condition for its application is that we know an estimate for the norm of the exact solution. Applications to boundary control problems for the one-dimensional wave equation are considered. Numerical results are presented.