Computing Reduced Order Models via InnerOuter Krylov Recycling in Diffuse Optical Tomography
Abstract
In nonlinear imaging problems whose forward model is described by a partial differential equation (PDE), the main computational bottleneck in solving the inverse problem is the need to solve many largescale discretized PDEs at each step of the optimization process. In the context of absorption imaging in diffuse optical tomography, one approach to addressing this bottleneck proposed recently (de Sturler, et al, 2015) reformulates the viewing of the forward problem as a differential algebraic system, and then employs model order reduction (MOR). However, the construction of the reduced model requires the solution of several full order problems (i.e. the full discretized PDE for multiple righthand sides) to generate a candidate global basis. This step is then followed by a rankrevealing factorization of the matrix containing the candidate basis in order to compress the basis to a size suitable for constructing the reduced transfer function. The present paper addresses the costs associated with the global basis approximation in two ways. First, we use the structure of the matrix to rewrite the full order transfer function, and corresponding derivatives, such that the full order systems to be solved are symmetric (positive definite in the zero frequency case). Then we apply MOR to the new formulation of the problem. Second, we give an approach to computing the global basis approximation dynamically as the full order systems are solved. In this phase, only the incrementally new, relevant information is added to the existing global basis, and redundant information is not computed. This new approach is achieved by an innerouter Krylov recycling approach which has potential use in other applications as well. We show the value of the new approach to approximate global basis computation on two DOT absorption image reconstruction problems.
 Publication:

arXiv eprints
 Pub Date:
 January 2016
 arXiv:
 arXiv:1602.00138
 Bibcode:
 2016arXiv160200138O
 Keywords:

 Mathematics  Numerical Analysis;
 65F10;
 65F22