1 research outputs found
Computing Reduced Order Models via Inner-Outer Krylov Recycling in Diffuse Optical Tomography
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 large-scale 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 right-hand sides) to generate a candidate global basis. This step is
then followed by a rank-revealing 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 inner-outer 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