Figure in P. Pettersson, J. Nordström, A. Doostan,* A Well-posed and Stable Stochastic Galerkin Formulation of the Incompressible Navier-Stokes Equations with Random Data*, Linköping University, LiTH-MAT-R, No. 2015:06, 2015.*.*

# Uncertainty quantification for partial differential equations

Numerical simulations are always prone to errors that come in many different shapes. In order to understand the relation between realworld systems and numerical simulations, knowledge of error and uncertainty propagation is essential. This course is about the propagation of uncertainty through partial differential equations with focus on methods based on the polynomial chaos framework.

## Schedule

Lecture week: Feb 22-26, 2016, Linköping univ.

## Teachers

Jan Nordström (**J**) and Per Pettersson (**P**)

## Examination:

Mandatory assignments with topics provided by the lecturers.

## DAY 1: Basic Concepts [J & P]

*(Estimated time: 3 × 45 minutes)*

Introduction: Why uncertainty quantification? General problem definition: PDE with

stochastic IC, BC or parameters

### • [P] Representation of random fields via spectral expansions:

- Karhunen-Loeve decomposition based on covariance statistics, optimal representation
- Generalized polynomials chaos, orthogonal polynomials, L2 convergence
- Localized representations for non-smooth functions
- Multiple stochastic dimensions: stochastic tensor grids, sparse grids, Smolyak

quadrature rules

### • [J] PDE Theory

- Elliptic/hyperbolic/parabolic PDEs
- Well-posedness and boundary conditions
- Linear vs nonlinear PDEs

### • [P] Example: gPC formulation for simple problem

- Stochastic Galerkin: coupled system to be solved once – new problem
- Stochastic Collocation: Few samples of original problem
- Least-squares/regression: Samples dependent on order of gPC

Reading Material: TPME11.

## DAY 2: Elliptic Problems (Pressure Equation) [P]

*(Estimated time: 2 × 45 minutes)*

- Demonstration of very efficient gPC representation e.g. in comparison to standard

Monte Carlo - Sparsity of gPC basis for elliptic problems
- Multiple stochastic dimensions (if not covered during day 1)

Reading Material: XK02

## DAY 3: Hyperbolic-Parabolic Problems [J]

*(Estimated time: 2 × 45 minutes)*

- The advection-diffusion equation: ut + vux = εuxx

– What happens when ε → 0?

– What happens when v → 0?

– What are the requirements for well-posedness ?

Reading material: PIN15 Ch.5, GX08

## DAY 4: Nonlinear Problems (Burgers’ equation) [P (J)]

*(Estimated time: 2 × 45 minutes)*

- Nonlinear analysis for stochastic problem guided by deterministic analysis (e.g. wellposedness)
- Analysis of the exact solution of the stochastic viscous Burgers’ equation
- Multiple discontinuities in stochastic Galerkin systems
- Robust solvers for the discrete problem (if time permits)

Reading material: PIN15 Ch. 6

## DAY 5: Advanced topics [J & P]

*(Estimated time: 2/3 × 45 minutes)*

- Many stochastic dimensions
- Alternative gPC basis functions: wavelets, spatially adaptive gPC
- A topic to be decided

Reading material: To be distributed at the course start.

Day 5 will give perspective on the course material. Three important topics will be presented

briefly. These topics are input in the assignments in the examination.

## Literature

**GX08**Gottlieb, Xiu, Galerkin Method for Wave Equations with Uncertain Coefficients, Commun. Comput. Phys., Vol. 3, No. 2, pp. 505-518, 2008.**PIN15**Pettersson, Iaccarino, Nordström, Polynomial Chaos Methods for Hyperbolic Partial

Differential Equations, Springer, 2015.**TPME11**Tuminaro, Phipps, Miller, Elman, Assessment of Collocation and Galerkin Approaches

to Linear Diffusion Equations with Random Data, International Journal for Uncertainty Quantification, Vol. 1, No. 1, pp. 19-33, 2011.**XK02**Xiu, Karniadakis, Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos, CMAME, Vol. 191, pp. 49274948, 2002.