Advanced Techniques in the Computation of Reduced Order Models and Krylov Recycling for Diffuse Optical Tomography - O'Connell, Meghan. (2017)
Abstract: Nonlinear parametric inverse problems whose forward model is described by a partial differential equation (PDE) arise in many applications, such as diffuse optical tomography (DOT). The main computational bottleneck in solving these types of inverse problems is the need to repeatedly solve the forward model, which requires solves of large-scale discretized parametrized PDEs. The main focus of this thesis is developing methods to reduce this cost. In the context of absorption imaging in DOT, interpolatory model reduction can be employed to reduce the computational cost associated with the forward model solves. We use surrogate models to approximate both the function evaluations and the Jacobian evaluations, which significantly reduces the cost while maintaining accuracy. We consider two methods for construction of the global basis required for the reduced model. Both methods require several full order model solves. The first method solves the fully discretized PDE for multiple right-hand sides and then uses a rank-revealing factorization to compress the basis. The second method reduces the cost of the construction of the global basis in two ways. First, we show how we exploit the structure of the matrix to rewrite the full order transfer function and corresponding derivatives in terms of a symmetric matrix. We then apply model order reduction to the new symmetric formulation of the problem. Second, we give an inner-outer Krylov approach to dynamically build the global basis while the full order systems are solved. This means that we only update the global basis with the incrementally new, relevant information eliminating the need to do an expensive rank-revealing factorization. Next, we extend the inner-outer Krylov recycling approach to solving sequences of shifted linear systems. We show the value of the above approaches with 2-dimensional and 3-dimensional examples from DOT, however, we believe our methods have the potential to be useful for other applications as well. In the final chapter, we explore different approaches to constructing the recycle spaces for shifted systems. We show how the use of generalized eigenvectors has the potential to be extremely useful for large shifts.; Thesis (Ph.D.)--Tufts University, 2016.; Submitted to the Dept. of Mathematics.; Advisor: Misha Kilmer.; Committee: Eric de Sturler, James Adler, and Xiaozhe Hu.; Keywords: Mathematics, and Applied mathematics.