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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Ph.D. Candidate: Zichun Ye
Mathematics, UBC

Mon 12 Dec 2016, 12:30pm
SPECIAL
Room 200, Graduate Student Centre

Doctoral Exam: Models of Gradient Type with SubQuadratic Actions and Their Scaling Limits

Room 200, Graduate Student Centre
Mon 12 Dec 2016, 12:30pm2:30pm
Details
My research concerns models of gradient type with subquadratic actions and their scaling limits. The model of gradient type is the density of a collection of realvalued random variables ϕ’s given by Z^{1}e^({ΣV(ϕ_jϕ_k)}). We focus our study on the case that V(t) = [1+t^2]^a with 0 < a < 1/2, which is a nonconvex potential.
The first result concerns the thermodynamic limits of the model of gradient type. We introduce an auxiliary field t for each edge and represent the model as the marginal of a model with logconcave density. Based on this method, we prove that finite moments of the fields are bounded uniformly in the volume for the finite volume measure. This bound leads to the existence of infinite volume measures.
The second result is the random walk representation and the scaling limit of the translationinvariant, ergodic gradient infinite volume Gibbs measure. We represent every infinite volume Gibbs measure as a mixture over Gaussian gradient measures with a random coupling constant ω for each edge. With such representation, we give estimations on the decay of the two point correlation function. Then by the quenched functional central limit theorem in random conductance model, we prove that every ergodic, infinite volume Gibbs measure with mean zero for the potential V above scales to a Gaussian free field.
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Seminar Information Pages

Note for Attendees
Latecomers will not be admitted.