PDE-betinga optimering : prekondisjonerarar og metodar for diffuse domene

This thesis is mainly concerned with the efficient numerical solution of optimization problems subject to linear PDE-constraints, with particular focus on robust preconditioners and diffuse domain methods. Associated with such constrained optimization problems are the famous first-order KarushKuhn-Tucker (KKT) conditions. For certain minimization problems, the functions satisfying the KKT conditions are also optimal solutions of the original optimization problem, implying that we can solve the KKT system to obtain the optimum; the so-called “all-at-once” approach. We propose and analyze preconditioners for the different KKT systems we derive in this thesis.Denne avhandlinga ser i hovudsak på effektive numeriske løysingar av PDE-betinga optimeringsproblem, med eit særskilt fokus på robuste prekondisjonerar og “diffuse domain”-metodar. Assosiert med slike optimeringsproblem er dei velkjende Karush-Kuhn-Tucker (KKT)-føresetnadane. For mange betinga optimeringsproblem, vil funksjonar som tilfredstillar KKT-vilkåra samstundes vere ei optimal løysing på det opprinnelege optimeringsproblemet. Dette impliserar at vi kan løyse KKT-likningane for å finne optimum. Vi konstruerar og analyserar prekondisjonerar for dei forskjellige KKT-systema vi utleiar i denne avhandlinga

Sparsity regularization for inverse problems with non-trivial nullspaces

We study a weighted $\ell^1$-regularization technique for solving inverse problems when the forward operator has a significant nullspace. In particular, we prove that a sparse source can be exactly recovered as the regularization parameter $\alpha$ tends to zero. Furthermore, for positive values of $\alpha$, we show that the regularized inverse solution equals the true source multiplied by a scalar $\gamma$, where $\gamma = 1 - c\alpha$. Our analysis is supported by numerical experiments for cases with one and several local sources. This investigation is motivated by PDE-constrained optimization problems arising in connection with ECG and EEG recordings, but the theory is developed in terms of Euclidean spaces. Our results can therefore be applied to many problems

Identifying the source term in the potential equation with weighted sparsity regularization

We explore the possibility for using boundary measurements to recover a sparse source term f(x) in the potential equation. Employing weighted sparsity regularization and standard results for subgradients, we derive simple-to-check criteria which assure that a number of sinks (f(x) 0) can be identified. Furthermore, we present two cases for which these criteria always are fulfilled: a) well-separated sources and sinks, and b) many sources or sinks located at the boundary plus one interior source/sink. Our approach is such that the linearity of the associated forward operator is preserved in the discrete formulation. The theory is therefore conveniently developed in terms of Euclidean spaces, and it can be applied to a wide range of problems. In particular, it can be applied to both isotropic and anisotropic cases. We present a series of numerical experiments. This work is motivated by the observation that standard methods typically suggest that internal sinks and sources are located close to the boundary

The inverse problem of electrocardiography : increasing the stability by using a simplified geometry for the ischemic region

We aim to obtain stability when using ECG recordings, the bidomain model and lab measurements to locate an ischemic region in the heart. Historically, this has proven to be a difficult task when using a general geometry, so our approach is to assume a priori that we can approximate the ischemic region as a ball. The approximation has been viewed from both a theoretical stand as well as more practically with numerical simulations. First, the theoretical continuity properties of the system with the simplified geometry were explored, followed by several numerical simulations to illuminate the practical behavior with this geometry. We did find a theoretical stability, as well as promising numerical results. From a small compact domain of the heart, we have proved a continuous inverse. In addition, we found a necessary demand for uniqueness of the inverse solution throughout the entire heart – which then also guarantee continuity. Numerically, we were able to retrieve the ischemic region without noise as well as with a proper amount of noise on our synthetic forward data. These findings are interesting to pursuit further. Since we worked on synthetic data in this thesis, it will be of great importance to further work with true patient data to see how well we can approximate the ischemic region in a real case. Also, the key point of this thesis was to gain more stability. In theory we found this to be stable, but we do not know how stable it is when it comes to numerical simulations. It might therefore be interesting to try to determine how sensitive the numerics can be to noise

Preconditioners for PDE-constrained optimization problems with box constraints: Towards high resolution inverse ECG images

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