Overlapping domain decomposition and multigrid methods for inverse problems

Our emphasis is on the numerical treatment of discontinuous coefficient and efficiency of the numerical methods. It is well known that such an inverse problem is illposed. Its numerical solution often suffers from undesirable numerical oscillation and very slow convergence. When the coefficient is smooth, successful numerical methods have been developed in [5] [7]. When the coefficient has large jumps, the numerical problem is much more difficult and some techniques have been proposed in [2] and [3]. See also [9], [4] and [6] for some related numerical results in identifying some discontinuous coefficients. The two fundamental tools we use in [2] and [3] are the total variation (TV) regularization technique and the augmented Lagrangian technique. The TV regularization allows the coefficient to have large jumps and at the same time it will discourage the oscillations that normally appear in the computations. The augmented Lagrangian method enforces the equation constraint in an H−1 norm and was studied in detail in [7]. Due to the bilinear structure of the equation constraint, the augmented Lagrangian reduces the output-least-squares (OLS) minimization to a system of coupled algebraic equations. How to solve these algebraic equations is of great importance in speeding up the solution procedure. The contribution of the present work is to propose an overlapping domain decomposition (DD) and a multigrid (MG) technique to evaluate the H−1 norm and at the same time use them as a preconditioner for one of the algebraic equations. Numerical tests will be given to show the speed-up using these techniques.