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UBC Theses and Dissertations

Dimensionality reduction for solving large scale inverse problems with PDE constraints through simultaneous source method Liu, Michelle


Optimization problems with PDE constraints arise in many applications of science and engineering. The PDEs often describe physical systems and are characterized by a set of parameters, such as the material properties of the system, or velocity of the wave traveling through the earth’s subsurface. Usually the parameters cannot be directly measured and can only be inferred by solving inverse problems. Solving large scale inverse problems usually require prohibitively large number of PDE solves. As a result, various dimensionality reduction methods have been proposed to reduce the number of PDE solves. This work builds on a type of dimensionality method called the Simultaneous Source (SS) method. This method relies on random sampling and the stochastic trace estimator to reduce the number of PDE solves to a more manageable size. One of the limitations of the SS method is it can only be applied to data sets with no missing entries. Unfortunately, data sets with missing entries are common in practice, such as in geophysical applications. In this thesis, we propose a new coupled optimization problem that extend the SS method to data with missing entries. The block coordinate descent method (BCDM) is used to break the coupled problem into two subproblems. The first subproblem optimizes over the data and fills the missing entries by the rank minimization technique. The second subproblem minimizes over the model parameter and uses SS method to reduce the number of PDE solves into a more manageable size. We test our method in the context of a full-waveform inversion (FWI) problem and present our numerical results in which we are able to reduce the number of PDE solves by 98% with less than 2% difference between the model using 1 PDE solve and the model using full PDE solves.

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