Numerical Solution of a Cauchy Problem for a Parabolic Equation in Two or more Space Dimensions by the Arnoldi Method
2010 (English)Report (Other academic)
We consider the numerical solution of a Cauchy problem for a parabolic equation in multi-dimensional space with cylindrical domain in one spatial space direction. It is desired to find the lower boundary values from the Cauchy data on the upper boundary. This problem is severely ill-posed. The formal solution is written as a hyperbolic cosine function in terms of a multidimensional parabolic (unbounded) operator. We compute an approximate solution by projecting onto a smaller subspace generated via the Arnoldi algorithm applied on the discretized inverse of the operator. Further we regularize the projected problem. The hyperbolic cosine is evaluated explicitly on a low-dimensional subspace. In each iteration step of the Arnoldi method a well-posed parabolic problem is solved. Numerical examples are given to illustrate the performance of the method.
Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press , 2010. , 23 p.
LiTH-MAT-R, ISSN 0348-2960 ; 2010:4
Cauchy problem, inverse problem, ill-posed, iterative method, Arnoldi method, Schur decomposition, parabolic PDE
IdentifiersURN: urn:nbn:se:liu:diva-54299OAI: oai:DiVA.org:liu-54299DiVA: diva2:302604