Computing Reduced Order Models via Inner-Outer Krylov Recycling in Diffuse Optical Tomography

摘要

In nonlinear imaging problems whose forward model is described by a partialdifferential equation (PDE), the main computational bottleneck in solving theinverse problem is the need to solve many large-scale discretized PDEs at eachstep of the optimization process. In the context of absorption imaging indiffuse optical tomography, one approach to addressing this bottleneck proposedrecently (de Sturler, et al, 2015) reformulates the viewing of the forwardproblem as a differential algebraic system, and then employs model orderreduction (MOR). However, the construction of the reduced model requires thesolution of several full order problems (i.e. the full discretized PDE formultiple right-hand sides) to generate a candidate global basis. This step isthen followed by a rank-revealing factorization of the matrix containing thecandidate basis in order to compress the basis to a size suitable forconstructing the reduced transfer function. The present paper addresses thecosts associated with the global basis approximation in two ways. First, we usethe structure of the matrix to rewrite the full order transfer function, andcorresponding derivatives, such that the full order systems to be solved aresymmetric (positive definite in the zero frequency case). Then we apply MOR tothe new formulation of the problem. Second, we give an approach to computingthe global basis approximation dynamically as the full order systems aresolved. In this phase, only the incrementally new, relevant information isadded to the existing global basis, and redundant information is not computed.This new approach is achieved by an inner-outer Krylov recycling approach whichhas potential use in other applications as well. We show the value of the newapproach to approximate global basis computation on two DOT absorption imagereconstruction problems.