PRECONDITIONERS FOR PDE-CONSTRAINED OPTIMIZATION PROBLEMS WITH BOX CONSTRAINTS: TOWARDS HIGH RESOLUTION INVERSE ECG IMAGES

摘要

By combining the Minimal Residual Method and the Primal-Dual Active Set algorithm, we derive an efficient scheme for solving a class of PDE-constrained optimization problems with inequality constraints. The approach studied in this paper addresses box constrains on the control function, and leads to an iterative scheme in which linear optimality systems must be solved in each iteration. We prove that the spectra of the associate saddle point operators, appearing in each iteration, are well behaved: Almost all the eigenvalues are contained in three bounded intervals, not containing zero. In fact, for severely ill-posed problems, the number of eigenvalues outside these three intervals are of order O(ln(alpha(-1))) as alpha -> 0, where a is the parameter employed in the Tikhonov regularization. Krylov subspace methods are well known to handle such systems of algebraic equations very well, and we thus obtain a fast method for PDE-constrained optimization problems with box constraints. In contrast to previous papers, our investigation is not targeted at analyzing a specific model, but instead covers a rather large class of problems. Our theoretical findings are illuminated by several numerical experiments. An example covered by our theoretical findings, as well as cases not fulfilling all the assumptions needed in the analysis, are presented. Also, in addition to computations only involving synthetic data, we briefly explore whether these new techniques can be applied to real world problems. More specifically, the algorithm is tested on a medical imaging problem with clinical patient data. These tests suggest that the method is fast and reliable.