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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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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 Sub-Quadratic Actions and Their Scaling Limits
Room 200, Graduate Student Centre
Mon 12 Dec 2016, 12:30pm-2:30pm

Details

My research concerns models of gradient type with sub-quadratic actions and their scaling limits. The model of gradient type is the density of a collection of real-valued random variables ϕ’s given by Z^{-1}e^({-ΣV(ϕ_j-ϕ_k)}). We focus our study on the case that V(t) = [1+t^2]^a with 0 < a < 1/2, which is a non-convex 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 log-concave 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 translation-invariant, 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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Sat 7 Jan 2017, 9:00am SPECIAL
MATH 126
Qualifying Exams - Analysis
MATH 126
Sat 7 Jan 2017, 9:00am-12:00pm

Details

For more information on Qualifying Exams, please visit http://www.math.ubc.ca/Grad/QualifyingExams/index.shtml
Lunch will be provided in  MATX 1101 for students writing the Analysis exam.
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Sat 7 Jan 2017, 1:00pm SPECIAL
MATH 126
Qualifying Exams - Algebra
MATH 126
Sat 7 Jan 2017, 1:00pm-4:00pm

Details

For more info, please visit http://www.math.ubc.ca/Grad/QualifyingExams/index.shtml
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Sat 7 Jan 2017, 1:00pm SPECIAL
MATH 126
Qualifying Exams - Differential Equations
MATH 126
Sat 7 Jan 2017, 1:00pm-4:00pm

Details

For more info, please visit http://www.math.ubc.ca/Grad/QualifyingExams/index.shtml
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