Mohammad Motamed
MATH 579, Introduction to Uncertainty Quantification of PDEs

General description

The behavior of many physical systems can be mathematically modeled by deterministic partial differential equations (PDEs). These models however differ from reality for two main reasons: (1) our inability to accurately characterize all parameters of the mathematical model; and/or (2) an intrinsic variability of the physical system. Uncertainty is therefore a fundamental feature of physical systems and needs to be takin into account when building a mathematical model. Examples appear in the description of flows in porous media, behavior of living tissues, combustion problems, deformation of composite materials, earthquake motions, etc. In order to understand the uncertainties inherent in the model, and often represented in a probabilistic setting, we need a process called uncertainty quantification (UQ).

In this course we consider different types of PDEs (elliptic, parabolic, and hyperbolic) with stochastic input parameters (coefficients, forcing terms, boundary conditions, shape of the physical domain, etc.) and study varios numerical techniques for solving the stochastic problems.

Office hours

W 14:30-16:00 SMLC 216

Syllabus

Questionnaire

Assignments

Assignment 1

Final Project

Projects schedule and report due: Schedule

Lecture notes

  • Introduction: Lecture 1, Lecture 2A

  • Basics of Probability Theory: Lecture 2B, Lecture 3, Lecture 4, Lecture 5, Lecture 6

  • Forward UQ (Numerical Mathods for Stochastic PDEs):
    • Abstract Framework: Lecture 7, Lecture 8A
    • Monte Carlo Smapling Mathods: Lecture 8B, Lecture 9, Lecture 10, Lecture 11, Lecture 12, Lecture 13
    • Polynomial Approimation:
      • Introduction and Background: Lecture 14, Lecture 15
      • Univariate and Multivariate Polynomials: Lecture 16, Lecture 17, Lecture 18, Lecture 19
      • Stochastic Regularity of Elliptic Problems: Lecture 20
      • Stochastic Galerkin for Elliptic Problems: Lecture 21
      • Stochastic Collocation for Elliptic Problems: Lecture22, Lecture23, Lecture24, Lecture25, Lecture 26, Lecture 27
      • Stochastic Least Squares for Elliptic Problems: Lecture 28, Lecture 29
      • Time-dependent Problems: Lecture 30   Lecture 31   Lecture 32   Lecture 33
      • Optimizing Polynomial Spaces: Lecture 34

    motamed@math.unm.edu
    Last updated: April 2014