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Ph.D. Candidate: Kai Rothauge
Mathematics, UBC
Mon 5 Dec 2016, 9:00am SPECIAL
Room 5104, Earth Sciences Building (ESB)
Doctoral Exam: The Discrete Adjoint Method for High-Order Time-Stepping Methods
Room 5104, Earth Sciences Building (ESB)
Mon 5 Dec 2016, 9:00am-11:00am

Details

Abstract:
This thesis examines the derivation and implementation of the discrete adjoint method for several time-stepping methods. Our results are important for gradient-based numerical optimization in the context of large-scale parameter estimation problems that are constrained by nonlinear time-dependent PDEs. To this end, we discuss finding the gradient and the action of the Hessian of the data misfit function with respect to three sets of parameters: model parameters, source parameters and the initial condition. We also discuss the closely related topic of computing the action of the sensitivity matrix on a vector, which is required when performing a sensitivity analysis. The gradient and Hessian of the data misfit function with respect to these parameters requires the derivatives of the misfit with respect to the simulated data, and we give the procedures for computing these derivatives for several data misfit functions that are of use in seismic imaging and elsewhere.

The methods we consider can be divided into two categories, linear multistep (LM) methods and Runge-Kutta (RK) methods, and several variants of these are discussed. Regular LM and RK methods can be used for ODE systems arising from the semi-discretization of general nonlinear time-dependent PDEs, whereas implicit-explicit and staggered variants can be applied when the PDE has a more specialized form. Exponential time-differencing RK methods are also discussed. Our motivation is the application of the discrete adjoint method to high-order time-stepping methods, but the approach taken here does not exclude lower-order methods. Within each class, each time-stepping method has an associated adjoint method and we give details on its implementation.

All of the algorithms have been implemented in MATLAB using an object-oriented design and are written with extensibility in mind. It is illustrated numerically that the adjoint methods have the same order of accuracy as their corresponding forward methods, and for linear PDEs we give a simple proof that this must always be the case. The applicability of some of the methods developed here to pattern formation problems is demonstrated using the Swift-Hohenberg model.

Note for Attendees

Latecomers will not be admitted.
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University of Alberta
Thu 8 Dec 2016, 4:00pm SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
ESB Room 4127 (PIMS Videoconferencing Room )
Mulit-to one-dimensional optimal transport
ESB Room 4127 (PIMS Videoconferencing Room )
Thu 8 Dec 2016, 4:00pm-5:00pm

Abstract

I will discuss joint work with Pierre-Andre Chiappori and Robert McCann on the Monge-Kantorovich problem of transporting a probability measure on \mathbb{R}^n to another on the real line. We introduce a nestededness criterion relating the cost to the marginals, under which it is possible to solve this problem uniquely (and essentially explicitly), by constructing an optimal map one level set at a time. I plan to discuss examples for which the nestedness condition holds, as well as some for which it fails; some of these examples arise from a matching problem in economics which originally motivated our work. If time permits, I will also briefly discuss how level set dynamics can be used to develop a local regularity theory in the nested case
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