Forskarutbildningskurs 2009/2010
Numerical Methods for solution of Coefficient Inverse Problems
During this course we will consider the mathematical treatment and
recently developed new methods for numerical solution of Coefficient
Inverse Problems (CIPs) with the lateral data. Such problems are
occurred in many real-world situations including medical optical and
ultrasound imaging, geological profiling and shape reconstruction.
Usually, inverse problem consists in recovering the coefficients of
Partial Differential Equations (PDEs) from the boundary data. We will
consider physical formulations leading to ill- and well-posed
problems, regularization of CIP. Particularly, we will
focus on CIP for the wave equation, adaptive FEM method and globally
convergent numerical methods for solution of this CIP.
Time and Place
Department of Mathematical Sciences,
Thursday, MVL-14, 13.15-15.00
from 19.11.2009 - 17.12.2009
and from 21.01.2010 - 18.02.2010.
Course Plan
Introduction to well- and ill-posed problems.
Lecture 1
Tikhonov scheme for solution of ill-posed problems. The concept of overdetemination.
Lecture 2
Tikhonov scheme for operator F(x)=f. A brief overview of uniqueness results.
Lecture 3
Convexity of Tikhonov functional for linear operator. Local strict convexity for nonlinear operator.
Lecture 4
Adaptivity with relaxation for ill-posed problems and global convergence for a coefficient inverse problem.
PREPRINT
Adaptive FEM for solution of an inverse problem for the wave
equation.
Lecture 5
Hybrid FEM/FDM method for solution of the time-dependent wave equation
PREPRINT
Rigorous derivation of Frechet derivative for Tikhonov functional for coefficient inverse problem.
Lecture 6
Verifying the accuracy of the solution of the wave equation. Computational implementation of the adaptive FEM/FDM method for solution of coefficient inverse problem for hyperbolic equation.
Lecture 7
Globally convergent numerical method for solution of Inverse
Problems.
Lecture 8
Synthesis of global convergence and adaptivity for solution of Inverse
problems.
Lecture 9
Picosecond scale experimental verification of a globally convergent algorithm for a coefficient inverse problem
Lecture 10
References
1. A.B. Bakushinsky and M.Yu. Kokurin, Iterative Methods
for Approximate Solution of Inverse Problems, Springer, 2004.
2. M.M.Lavrentjev, V.G.Romanov, S.P.Shishatskij, Ill-posed problems of mathematical Physics and Analysis, American Mathematical Society, translations of mathematical monographs, V.64.
3. H.W.Engl, M.Hanke, A.Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Boston, 2000.
4. L.Beilina, Adaptive hybrid FEM/FDM method for
time-dependent inverse scattering problems, Chalmers University o
Technology, PhD thesis, 2003.
5. L.Beilina and M.V.Klibanov, A globally convergent numerical
method for a coefficient inverse problem, SIAM J. Scientific Computing
, V.31, N.1, 478-509, 2008.
6. L.Beilina and M.V.Klibanov, Synthesis of global convergence
and adaptivity for a hyperbolic coefficient inverse problem, Chalmers Preprint Series, 2009:11.
7. L.Beilina, M.V.Klibanov and M.Y.Kokurin, Adaptive FEM
with relaxation for some ill-posed problems and global convergence for
a coefficient inverse problem, to appear.
Background: spatial distribution of function q in globally convergent method
of [5].