Affiliation:
1. Department of Mathematics , National Nuclear Research University “MEPHI” , Kashirskoe Shosse 31, 115409 Moscow , Russia
Abstract
Abstract
We consider inverse problems of recovering a source term in initial boundary value problems for linear multidimensional partial differential equations (PDEs) of a general form.
A universal stable method suitable for solving such inverse problems is proposed.
The method allows one to obtain in the same way approximations to exact sources in different kinds of PDEs using various types of linear supplementary conditions specified with an error.
The method is suitable for both spacewise dependent and time-dependent sources.
The method consists in preliminary calculation of a special matrix introduced in the article, the matrix of the source inverse problem, and then inverting it using Tikhonov regularization.
The matrix can be obtained by solving a number of initial boundary value problems in question with sources in the form of basis functions.
Having spent some time for preliminary finding the matrix (for example, by finite element method with a sufficiently detailed grid), we can then use this matrix to quickly solve the inverse problem with various data.
The same technique can be applied to solve inverse source problems in linear steady-state PDEs.
We also propose an a posteriori error estimation method for the obtained approximate solution and give a numerical algorithm for such estimation.
In addition, a relationship is established between the posterior estimate and the lower estimate for the optimal accuracy of solving the inverse problem.
The proposed method of solving inverse source problems is illustrated by the numerical solution of model examples for one-dimensional and two-dimensional PDEs of different kinds with a posteriori error estimates.
Funder
Russian Foundation for Basic Research
National Research Nuclear University MEPhI
Reference42 articles.
1. A. Bakushinsky and A. Goncharsky,
Ill-Posed Problems: Theory and Applications,
Kluwer Academic, Dordrecht, 1994.
2. A. B. Bakushinsky and M. Y. Kokurin,
Iterative Methods for Approximate Solution of Inverse Problems,
Math. Appl. (New York) 577,
Springer, Dordrecht, 2004.
3. J. Diestel,
Geometry of Banach Spaces—Selected Topics,
Lecture Notes in Math. 485,
Springer, Berlin, 1975.
4. H. W. Engl, M. Hanke and A. Neubauer,
Regularization of Inverse Problems,
Math. Appli. 375,
Kluwer Academic, Dordrecht, 1996.
5. A. Erdem, D. Lesnic and A. Hasanov,
Identification of a spacewise dependent heat source,
Appl. Math. Model. 37 (2013), no. 24, 10231–10244.
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献