The Cauchy problem for the Helmholtz equation appears in applications related to acoustic or electromagnetic wave phenomena. The problem is ill–posed in the sense that the solution does not depend on the data in a stable way. In this paper we give a detailed study of the problem. Specifically we investigate how the ill–posedness depends on the shape of the computational domain and also on the wave number. Furthermore, we give an overview over standard techniques for dealing with ill–posed problems and apply them to the problem.
The Cauchy problem for the Helmholtz equation appears in various applications. The problem is severely ill-posed and regularization is needed to obtain accurate solutions. We start from a formulation of this problem as an operator equation on the boundary of the domain and consider the equation in (H-1/2)* spaces. By introducing an artificial boundary in the interior of the domain we obtain an inner product for this Hilbert space in terms of a quadratic form associated with the Helmholtz equation; perturbed by an integral over the artificial boundary. The perturbation guarantees positivity property of the quadratic form. This inner product allows an efficient evaluation of the adjoint operator in terms of solution of a well-posed boundary value problem for the Helmholtz equation with transmission boundary conditions on the artificial boundary. In an earlier paper we showed how to take advantage of this framework to implement the conjugate gradient method for solving the Cauchy problem. In this work we instead use the Conjugate gradient method for minimizing a Tikhonov functional. The added penalty term regularizes the problem and gives us a regularization parameter that can be used to easily control the stability of the numerical solution with respect to measurement errors in the data. Numerical tests show that the proposed algorithm works well. (C) 2016 Elsevier Ltd. All rights reserved.
In this paper we study the Cauchy problem for the Helmholtz equation. This problem appears in various applications and is severely ill–posed. The modified alternating procedure has been proposed by the authors for solving this problem but the convergence has been rather slow. We demonstrate how to instead use conjugate gradient methods for accelerating the convergence. The main idea is to introduce an artificial boundary in the interior of the domain. This addition of the interior boundary allows us to derive an inner product that is natural for the application and that gives us a proper framework for implementing the steps of the conjugate gradient methods. The numerical results performed using the finite difference method show that the conjugate gradient based methods converge considerably faster than the modified alternating iterative procedure studied previously.
We present a modification of the alternating iterative method, which was introduced by V.A. Kozlov and V. Maz’ya in for solving the Cauchy problem for the Helmholtz equation in a Lipschitz domain. The method is implemented numerically using the finite difference method.
The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Maz’ya does not converge for large wavenumbers in the Helmholtz equation. We prove here that if we alternate Robin and Dirichlet boundary conditions instead of Neumann and Dirichlet boundary conditions, then the algorithm will converge. We present also another algorithm based on the same idea, which converges for large wavenumbers. Numerical implementations obtained using the finite difference method are presented. Numerical results illustrate that the algorithms suggested in this paper, produce a convergent iterative sequences.
The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Mazya does not converge for large wavenumbers k in the Helmholtz equation. Here, we present some simple modifications of the algorithm which may restore the convergence. They consist of the replacement of the Neumann-Dirichlet iterations by the Robin-Dirichlet ones which repairs the convergence for less than the first Dirichlet-Laplacian eigenvalue. In order to treat large wavenumbers, we present an algorithm based on iterative solution of Robin-Dirichlet boundary value problems in a sufficiently narrow border strip. Numerical implementations obtained using the finite difference method are presented. The numerical results illustrate that the algorithms suggested in this paper, produce convergent iterative sequences.
We study the impact of age-structure and temporal environmental variability on the persistence of populations. We use a linear age-structured model with time-dependent vital rates. It is the same as the one presented by Chipot in (Arch. Ration. Mech. Anal. 82(1):13-25, 1983), but the assumptions on the vital rates are slightly different. Our main interest is in describing the large-time behavior of a population provided that we know its initial distribution and transient vital rates. Using upper and lower solutions for the characteristic equation, we define time-dependent upper and lower boundaries for a solution in a constant environment. Moreover, we estimate solutions for the general time-dependent case and also for a special case when the environment is changing periodically.
For a locally convex space , where the topology is given by a familyof seminorms, we study the existence and uniqueness of fixed points for a mapping defined on some set . We require that there exists a linear and positive operator , acting on functions defined on the index set , such that for every
Under some additional assumptions, one of which is the existence of a fixed point for the operator, we prove that there exists a fixed point of . For a class of elements satisfying as , we show that fixed points are unique. This class includes, in particular, the class for which we prove the existence of fixed points.We consider several applications by proving existence and uniqueness of solutions to first and second order nonlinear differential equations in Banach spaces. We also consider pseudo-differential equations with nonlinear terms.
This paper considers to the equation integral(S) U(Q)/vertical bar P - Q vertical bar(N-1) dS(Q) = F(P), P is an element of S, where the surface S is the graph of a Lipschitz function phi on R-N, which has a small Lipschitz constant. The integral on the left-hand side is the single layer potential corresponding to the Laplacian in RN+1. Let Lambda(r) be the Lipschitz constant of phi on the ball centered at the origin with radius 2r. Our analysis is carried out in local L-p-spaces and local Sobolev spaces, where 1 less than p less than infinity, and results are presented in terms of Lambda. Estimates of solutions to the equation are provided, which can be used to obtain knowledge about the behavior of the solutions near a point on the surface. These estimates are given in terms of seminorms. Solutions are also shown to be unique if they are subject to certain growth conditions. Local estimates are provided and some applications are supplied.
We consider the following equation for the Riesz potential of order one:
Uniqueness of solutions is proved in the class of solutions for which the integral is absolutely convergent for almost every x. We also prove anexistence result and derive an asymptotic formula for solutions near the origin.Our analysis is carried out in local L^{p}-spaces and Sobolev spaces, which allows us to obtain optimal results concerning the class of right-hand sides and solutions. We also apply our results to weighted L^{p}-spaces and homogenous Sobolev spaces.
The book systematically develops nonlinear potential theory and the Sobolev space theory covers results concerning approximation, extension, and interpolation, Sobolev and Poincare inequalities, Maz'ya type embedding theorems, and isoperimetric inequalities