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.     

On the solution of a Schrödinger type equation by a projection-difference method with an implicit Euler scheme with respect to time

A linear nonstationary Schrödinger type problem in a separable Hilbert space is approximately solved by a projection-difference method. The problem is discretized in space by the Galerkin method using finite-dimensional subspaces of finite-element type, and an implicit Euler scheme is used with respect to time. We establish error estimates uniform with respect to the time grid for the approximate solutions; as to the spatial variables, the estimates are given in the norm of the original space as well as in the energy norm. The estimates considered here not only permit one to prove the convergence of approximate solutions to the exact solution but also give a numerical characterization of the convergence rate.     

Solution of elliptic optimal control problem with pointwise and non-local state constraints

We study an optimal control problem of a system governed by a linear elliptic equation, with pointwise control constraints and pointwise and non-local (integral) state constraints. We construct a finite-difference approximation of the problem, we prove the existence and the convergence of the approximate solutions to the exact solution. We construct and study mesh saddle point problem and its iterative solution method and analyze the results of numerical experiments.     

Solution of a singular integro-differential equation with the use of asymptotic polynomials

We suggest a new method for solving a singular equation of elasticity theory, which is based on the use of asymptotic polynomials constructed on the basis of Chebyshev polynomials of the second kind. Under certain conditions imposed on the functions occurring in the operator equation, the approximate solution tends as n → ∞ to the best uniform approximation polynomial, which converges to the exact solution as n increases. The method permits one to express the remainder term of the approximate solution in the form of an infinite sum via linear functionals. If the originally chosen degree of the polynomial does not provide the desired accuracy, then one can find the corresponding term of the remainder starting from which the desired accuracy is attained and compute the polynomial of the corresponding degree. The proof of the convergence is presented for the case in which the variable ranges in a closed interval.     

一类球型区域上变系数反向热传导问题

考虑了一类球型区域上变系数反向热传导问题.这个问题是不适定的,即问题的解(若存在)并不连续依赖于测量数据.构造了投影迭代正则化方法,得到了该反问题的正则近似解,同时给出了在先验和后验参数选取规则下精确解与正则近似解之间的收敛性误差估计.最后,通过数值结果验证了该方法的有效性.     

Wavelet method for solving second-order quasilinear parabolic equations with a conservative principal part

A method based on wavelet transforms is proposed for finding classical solutions to initial-boundary value problems for second-order quasilinear parabolic equations. For smooth data, the convergence of the method is proved and the convergence rate of an approximate weak solution to a classical one is estimated in the space of wavelet coefficients. An approximate weak solution of the problem is found by solving a nonlinear system of equations with the help of gradient-type iterative methods with projection onto a fixed subspace of basis wavelet functions.     

Solution of boundary-value problems for elliptic equations in the space of distributions

We extend the well-known approach to solution of generalized boundary-value problems for second-order elliptic and parabolic equations and for second-order strongly elliptic systems of variational type to the case of a general normal boundary-value problem for an elliptic equation of order2m. The representation of a distribution from (C ^∞ (S))’ is established and is usedfor the proof of convergence of an approximate method of solution of a normal elliptic boundary-value problem in unnormed spaces of distributions.     

一类集值非线性混合变分包含问题的逼近解

在Hilbert空间中讨论了一类集值非线性混合变分包含问题逼近解的存在性,建立了变分包含问题与其预解方程的等价性,获得了3个迭代算法并研究了算法的收敛性．该结果推广统一了近期一些学者关于变分包含问题的相关结果．     

Solution of the Cauchy problem for a system of integrodifferential equations of viscoelasticity

Existence and uniqueness of the solution of the Cauchy problem is proved for a system of integrodifferential equations of the hereditary theory of viscoelasticity. A method of constructing an approximate solution is proposed. An estimate of the error of the approximate solution is presented.     

Solving a variational parabolic equation with the periodic condition by a projection-difference method with the Crank–Nicolson scheme in time

A solution to a smoothly solvable linear variational parabolic equation with the periodic condition is sought in a separable Hilbert space by an approximate projection-difference method using an arbitrary finite-dimensional subspace in space variables and the Crank–Nicolson scheme in time. Solvability, uniqueness, and effective error estimates for approximate solutions are proven. We establish the convergence of approximate solutions to a solution as well as the convergence rate sharp in space variables and time.     

Euler Discretization and Inexact Restoration for Optimal Control

A computational technique for unconstrained optimal control problems is presented. First, an Euler discretization is carried out to obtain a finite-dimensional approximation of the continuous-time (infinite-dimensional) problem. Then, an inexact restoration (IR) method due to Birgin and Martínez is applied to the discretized problem to find an approximate solution. Convergence of the technique to a solution of the continuous-time problem is facilitated by the convergence of the IR method and the convergence of the discrete (approximate) solution as finer subdivisions are taken. The technique is numerically demonstrated by means of a problem involving the van der Pol system; comprehensive comparisons are made with the Newton and projected Newton methods.     

On a finite element method for a certain class of time-dependent problems

The finite element method is applied to solve a linear initial-boundary value problem. The basic idea is to combine this method for a disretization in space variables with the Laplace transform technique for a time variable. Formulation, existence and uniqueness of a weak solution is investigated. The convergence and the rate of convergence of the proposed approximate solution is discussed     

Approximation of Solution Components for Ill-Posed Problems by the Tikhonov Method with Total Variation

An ill-posed problem in the form of a linear operator equation given on a pair of Banach spaces is considered. Its solution is representable as a sum of a smooth and a discontinuous component. A stable approximation of the solution is obtained using a modified Tikhonov method in which the stabilizer is constructed as a sum of the Lebesgue norm and total variation. Each of the functionals involved in the stabilizer depends only on one component and takes into account its properties. Theorems on the componentwise convergence of the regularization method are stated, and a general scheme for the finite-difference approximation of the regularized family of approximate solutions is substantiated in the n-dimensional case.     

kth Order convergence for a semilinear elliptic boundary value problem in the divergence form

The convergence problem of approximate solutions for a semilinear elliptic boundary value problem in the divergence form is studied. By employing the method of quasilinearization, a sequence of approximate solutions converging with the kth (k ? 2) order convergence to a weak solution for a semilinear elliptic problem is obtained via the variational approach.     

Approximate Solution of Optimization Problems for Infinite-Dimensional Singular Systems

An optimal control problem is considered for a system described by a singular equation of parabolic type. The study bases on a special regularization method. We establish existence of a solution to the regularized problem, as well as the corresponding necessary optimality conditions. The results enable us to find an approximate solution to the original problem even in the absence of solvability.     

A Hybrid Approximate Extragradient – Proximal Point Algorithm Using the Enlargement of a Maximal Monotone Operator

We propose a modification of the classical extragradient and proximal point algorithms for finding a zero of a maximal monotone operator in a Hilbert space. At each iteration of the method, an approximate extragradient-type step is performed using information obtained from an approximate solution of a proximal point subproblem. The algorithm is of a hybrid type, as it combines steps of the extragradient and proximal methods. Furthermore, the algorithm uses elements in the enlargement (proposed by Burachik, Iusem and Svaiter) of the operator defining the problem. One of the important features of our approach is that it allows significant relaxation of tolerance requirements imposed on the solution of proximal point subproblems. This yields a more practical proximal-algorithm-based framework. Weak global convergence and local linear rate of convergence are established under suitable assumptions. It is further demonstrated that the modified forward-backward splitting algorithm of Tseng falls within the presented general framework.     

On the Two-step Newton Method for the Solution of Nonlinear Operator Equations

Building on the method of Kantorovich majorants, we give convergence results and error estimates for the two-step Newton method for the approximate solution of a nonlinear operator equation.     

Finite element solution of nonlinear diffusion problems

In this paper we describe the Rothe-finite element numerical scheme to find an approximate solution of a nonlinear diffusion problem modeled as a parabolic partial differential equation of even order. This scheme is based on the Rothe’s approximation in time and on the finite element method (FEM) approximation in the spatial discretization. A proof of convergence of the approximate solution is given and error estimates are shown.     

Inexact Restoration for Euler Discretization of Box-Constrained Optimal Control Problems

The Inexact Restoration method for Euler discretization of state and control constrained optimal control problems is studied. Convergence of the discretized (finite-dimensional optimization) problem to an approximate solution using the Inexact Restoration method and convergence of the approximate solution to a continuous-time solution of the original problem are established. It is proved that a sufficient condition for convergence of the Inexact Restoration method is guaranteed to hold for the constrained optimal control problem. Numerical experiments employing the modelling language AMPL and optimization software Ipopt are carried out to illustrate the robustness of the Inexact Restoration method by means of two computationally challenging optimal control problems, one involving a container crane and the other a free-flying robot. The experiments interestingly demonstrate that one might be better-off using Ipopt as part of the Inexact Restoration method (in its subproblems) rather than using Ipopt directly on its own.     

Solution of a system of nonstrictly hyperbolic conservation laws

In this paper we study a special case of the initial value problem for a 2×2 system of nonstrictly hyperbolic conservation laws studied by Lefloch, whose solution does not belong to the class ofL ^∞ functions always but may contain δ-measures as well: Lefloch's theory leaves open the possibility of nonuniqueness for some initial data. We give here a uniqueness criteria to select the entropy solution for the Riemann problem. We write the system in a matrix form and use a finite difference scheme of Lax to the initial value problem and obtain an explicit formula for the approximate solution. Then the solution of initial value problem is obtained as the limit of this approximate solution.