Centre for Computational Science and Engineering

CCSE Student Seminar

Date & Time: February 3rd 2023, 12:10pm-1:30pm

Location: UTIAS Lecture Hall and Zoom

Zoom registration: https://utoronto.zoom.us/meeting/register/tZ0rdOquqTgtHNNodhK3VoMkdNxzWnoQNSBv

Talk 1:

Title: Intro to Gradient-Enhanced Gaussian Processes and Overcoming their Ill-Conditioning Problem

Speaker: André Marchildon

Abstract: This talk will introduce Gaussian processes as a method of constructing probabilistic surrogates that can be used for various applications including classification, uncertainty quantification, and optimization. Having a probabilistic surrogate is beneficial since it quantifies the uncertainty of the surrogate’s approximation to a function of interest. Furthermore, a Gaussian process can be used to construct a surrogate even when the evaluations of the function of interest and of its gradients are noisy.

Gradient-enhanced Gaussian processes utilize the gradient of a function of interest to construct a more accurate surrogate. Unfortunately, a major challenge in their use is the ill-conditioning of the gradient-enhanced covariance matrix. Existing approaches to address this challenge include drawbacks such as having to reduce the number of data points used to construct the surrogate and also limiting how close the data points can be to each other in the parameter space. A new method has been developed that allows all of the data points to be kept and it provides a provable upper bound on the condition number of the gradient-enhanced covariance matrix. Optimization results will be presented that demonstrate how this method overcomes the ill-conditioning problem and thus allows for deep convergence to be achieved.

Talk 2:

Title: An overview of Summation By Parts Methods for CFD: how to make your CFD discretizations more attractive to both engineers and mathematicians

Speaker: Alex Bercik

Abstract: The three main mesh-based discretization techniques for Computational Fluid Dynamics (CFD), Finite Difference, Finite Volume, and Finite Element, are traditionally thought to be distinct from one another. However, some may be aware that the overlaps between them can be significant. This is especially apparent with the use of Summation By Parts (SBP) methods, in which by constructing these numerical schemes carefully, we can ensure they possess certain mathematical properties that not only give us several computational advantages, but also provide certain analytical tools that allow us to analyze and interpret the schemes in a new light. In this talk, an SBP scheme for the linear advection equation will be presented that can be equivalently interpreted using any of the main three discretization techniques. We will use this test problem to demonstrate the powerful analytical framework allowed by the SBP property, allowing us to prove desirable mathematical properties of the scheme, such as arbitrary design-order, conservation, and notably, stability. Several numerical advantages provided by these schemes will then be discussed, such as straightforward parallelization, flexible element-type blocking for complex geometries, and facilitated treatment of special points through discontinuous or multi-valued solutions. Finally, extensions to the Euler and Navier-Stokes equations will briefly be explored, accompanied by a high-level presentation of non-linearly “entropy-stable” schemes, as well as a general overview of current research being done in Prof. Zingg’s lab on SBP methods.