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].