Teaching

Fundamentals of Uncertainty Quantification in Computational Science and Engineering

Course Description: Computing the statistical properties of nonlinear random systems is of fundamental importance in many areas of science and engineering. The primary objective of the course is to introduce students to state-of-the-art methods for uncertainty propagation and quantification in model-based computations, focusing on the computational and algorithmic features of these methods most useful in dealing with systems specified in terms of ordinary and partial differential equations.

The course will focus mainly on the so-called forward UQ problem, in which uncertainties in input parameters such as initial conditions, boundary conditions, geometry or forcing terms are propagated through the equations of motion of the system into the solution. The course will also discuss cutting edge topics on data-driven modeling, deep learning with stochastic neural networks, control of nonlinear systems under uncertainty, and uncertainty propagation in high-dimensional systems via tensor methods.

 

Instructor: Prof. Daniele Venturi

Course Notes

  • Chapter 1: Review of probability theory (PDF) 
  • Chapter 2: PDF equations for dynamical systems and PDEs (PDF)
  • Chapter 3: Deep learning with stochastic neural networks (PDF)
  • Chapter 4: Polynomial chaos (PDF)
  • Chapter 5: Sampling methods (PDF)
  • Chapter 6: Tensor methods (PDF)

Tentative Syllabus 

  • Week 1: Introduction to uncertainty quantification, objectives, model equations and probabilistic framework. Review of probability theory: random variables.
  • Week 2: Review of probability theory: random vectors, stochastic processes, random fields and their representation.
  • Week 3: Applications of probabilistic tools to nonlinear dynamical systems: system evolving form random initial states and system driven by random forcing terms: Liouville equation, Fokker Planck equation. Application to PDEs (Hopf's equations).
  • Week 4: Reduced-order PDF equations, data-driven closures. Deep learning with stochastic neural network models.
  • Week 5: Sampling methods (Monte-Carlo and quasi-MC). Convergence analysis.
  • Week 6: Control of nonlinear dynamical systems under uncertainty.
  • Week 7: Polynomial chaos (orthogonal polynomials and approximation theory), generalized polynomial. chaos (gPC); Multi-element generalized polynomial chaos (ME-gPC); Stochastic Galerkin method.
  • Week 8: Probabilistic collocation method; multi-element probabilistic collocation; tensor product collocation.
  • Week 9: Sparse grid collocation, Smolyak algorithm, stochastic Galerkin versus probabilistic collocation.
  • Week 10: Numerical tensor methods for uncertainty propagation: DO/BO field equation methods, tensor methods for high-dimensional PDF equations.

 Grading Policy

Students will be evaluated on the basis of:

  1. Homework assignments
  2. Final research project related to the course material

The final project can be an original research paper on an in-depth report on an existing published paper. In the latter case, the students are required to reproduce all analytical and numerical results in the paper. The final projects have two components: a piece of work (paper) handed in before the last day of class, and an oral presentation which will take place during finals week.

Reading List

  1. Lecture notes and research papers discussed in class.
  2. D. Xiu, ``Numerical Methods for Stochastic Computations: A Spectral Method Approach'', Princeton University Press (2010)
  3. O. Le Maitre and O. M. Knio, ``Spectral methods for uncertainty quantification: with applications to computational fluid dynamics'', Springer (2010)
  4. R. G. Ghanem and P. D. Spanos, ``Stochastic finite elements: a spectral approach'', Springer (2012)
  5. J. Garcke and M. Griebel (Ed.), ``Sparse grids and applications'', Springer (2013)
  6. M. P. Pettersson, G. Iaccarino and J. Nordstrom, ``Polynomial chaos methods for hyperbolic partial differential equations: numerical techniques for fluid dynamics problems in the presence of uncertainties'', Springer (2015)
  7. J. Dick, F. Y. Kuo, G. W. Peters and I. Sloan (Ed.), ``Monte Carlo and Quasi-Monte Carlo methods 2012'', Springer (2013)
  8. F. Moss and P. V. E. McClintock (Ed.), ``Noise in nonlinear dynamical systems: Vol 1 (theory of continuous Fokker-Planck systems)'', Cambridge (1989)