Browsing by Author "O'Connell, Meghan"

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    Computing Reduced Order Models via Inner-Outer Krylov Recycling in Diffuse Optical Tomography
    O'Connell, Meghan; Kilmer, Misha E.; de Sturler, Eric; Gugercin, Serkan (Siam Publications, 2017-01-01)
    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.
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    Nonlinear Parametric Inversion Using Interpolatory Model Reduction
    de Sturler, Eric; Gugercin, Serkan; Kilmer, Misha E.; Chaturantabut, Saifon; Beattie, Christopher A.; O'Connell, Meghan (Siam Publications, 2015-01-01)
    Nonlinear parametric inverse problems appear in several prominent applications; one such application is Diffuse Optical Tomography (DOT) in medical image reconstruction. Such inverse problems present huge computational challenges, mostly due to the need for solving a sequence of large-scale discretized, parametrized, partial diferential equations (PDEs) in the forward model. In this paper, we show how interpolatory parametric model reduction can significantly reduce the cost of the inversion process in DOT by drastically reducing the computational cost of solving the forward problems. The key observation is that function evaluations for the underlying optimization problem may be viewed as transfer function evaluations along the imaginary axis; a similar observation holds for Jacobian evaluations as well. This motivates the use of system-theoretic model order reduction methods. We discuss the construction and use of interpolatory parametric reduced models as surrogates for the full forward model. Within the DOT setting, these surrogate models can approximate both the cost functional and the associated Jacobian with very little loss of accuracy while significantly reducing the cost of the overall inversion process. Four numerical examples illustrate the effciency of the proposed approach. Although we focus on DOT in this paper, we believe that our approach is applicable much more generally.