Adaptive Discretizations for the Regularization of Inverse Problems

• Applicant: Professor Dr. Boris Vexler
• Subject Area: Mathematics
• Term: from 2009 to 2014
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 119705116

Many complex processes in the field of natural sciences, medicine and engineering are described by mathematical models with partial differential equations (PDEs). The mentioned systems of PDEs mostly contain unknown data, e.g. space-dependent coefficient functions, source terms, initial and boundary data, whose determination leads to high-dimensional inverse problems.The numerical effort for solving inverse problems with PDEs is usually much higher than for the numerical simulation of the underlying process with a given data set. Moreover the inherent instability of inverse problems requires the use of appropriate regularization techniques. Great potential for the construction of efficient algorithms for the solution of such inverse problems lies in adaptive discretizations. While the use of adaptive concepts for the choice of the discretization for numerical simulation has become prevalent in the last years, adaptivity in the context of inverse problems presents a new and highly relevant topic.The goal of the project consists in finding generally applicable and analytically justified methods for the adaptive discretization of inverse problems. In this process, the main focus is on the efficiency of the constructed algorithms on the one hand and on the rigorous convergence analysis in the context of regularization methods on the other hand.

• DFG Programme: Research Grants
• International Connection: Austria
• Participating Person: Professorin Dr. Barbara Kaltenbacher