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 HighOrder TimeStepping Methods

Room 5104, Earth Sciences Building (ESB)
Mon 5 Dec 2016, 9:00am11:00am
Details
Abstract:
This thesis examines the derivation and implementation of the discrete adjoint method for several timestepping methods. Our results are important for gradientbased numerical optimization in the context of largescale parameter estimation problems that are constrained by nonlinear timedependent 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 RungeKutta (RK) methods, and several variants of these are discussed. Regular LM and RK methods can be used for ODE systems arising from the semidiscretization of general nonlinear timedependent PDEs, whereas implicitexplicit and staggered variants can be applied when the PDE has a more specialized form. Exponential timedifferencing RK methods are also discussed. Our motivation is the application of the discrete adjoint method to highorder timestepping methods, but the approach taken here does not exclude lowerorder methods. Within each class, each timestepping method has an associated adjoint method and we give details on its implementation.
All of the algorithms have been implemented in MATLAB using an objectoriented 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 SwiftHohenberg model.
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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 )

Mulitto onedimensional optimal transport

ESB Room 4127 (PIMS Videoconferencing Room )
Thu 8 Dec 2016, 4:00pm5:00pm
Abstract
I will discuss joint work with PierreAndre Chiappori and Robert McCann on the MongeKantorovich 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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Seminar Information Pages

Note for Attendees
Latecomers will not be admitted.