Ph.D. Candidate: Curt Da Silva
Mathematics, UBC

Mon 21 Aug 2017, 12:30pm
SPECIAL
Room 202, Anthropology and Sociology Bldg., 6303 NW Marine Drive, UBC

Oral Examination: Largescale optimization algorithms for missing data completion and inverse problems

Room 202, Anthropology and Sociology Bldg., 6303 NW Marine Drive, UBC
Mon 21 Aug 2017, 12:30pm2:30pm
Details
ABSTRACT: Inverse problems are an important class of problems found in many areas of science and engineering. In these problems, one aims to estimate unknown parameters of a physical system through indirect multiexperiment measurements. Inverse problems arise in a number of fields including seismology, medical imaging, and astronomy, among others.
An important aspect of inverse problems is the quality of the acquired data itself. Realworld data acquisition restrictions, such as time and budget constraints, often result in measured data with missing entries. Many inversion algorithms assume that the input data is fully sampled and relatively noise free and produce poor results when these assumptions are violated. Given the multidimensional nature of realworld data, we propose a new lowrank optimization method on the smooth manifold of Hierarchical Tucker tensors. Tensors that exhibit this lowrank structure can be recovered from solving this nonconvex program in an efficient manner. We successfully interpolate realistically sized seismic data volumes using this approach.
If our lowrank tensor is corrupted with nonGaussian noise, the resulting optimization program can be formulated as a convexcomposite problem. This class of problems involves minimizing a nonsmooth but convex objective composed with a nonlinear smooth mapping. In this thesis, we develop a level set method for solving compositeconvex problems and prove that the resulting subproblems converge linearly. We demonstrate that this method is competitive when applied to examples in noisy tensor completion, analysisbased compressed sensing, audio declipping, totalvariation deblurring and denoising, and onebit compressed sensing.
With respect to solving the inverse problem itself, we introduce a new software design framework that manages the cognitive complexity of the various components involved. Our framework is modular by design, which enables us to easily integrate and replace components such as linear solvers, finite difference stencils, preconditioners, and parallelization schemes. As a result, a researcher using this framework can formulate her algorithms with respect to highlevel components such as objective functions and hessian operators. We showcase the ease with which one can prototype such algorithms in a 2D test problem and, with little code modification, apply the same method to largescale 3D problems.
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Latecomers will not be admitted.