# Tensor-Galerkin POD for UQ in PDEs with Multivariate Random Parameters

virtuelle GAMM 202021 an der Uni Kassel

# Introduction

## Example Problem Setup

• The heat equation with uncertainty in the coefficient $\kappa$: $-\kappa(\alpha) \Delta y = f, \quad \text{in }\Omega,$ where $\alpha$ is a random variable.

• Then, the solution $y$ is a random variable depending on $\alpha$.

• Of interest: $\mathbb E_\alpha y, \quad \text{in }\Omega$ – the expected value of the solution $y$.

## Monte Carlo (MC)

• slow, but highly parallelizable

• many improvements like Multi Level MC, Markov Chain MC

## Galerkin/Collocation methods

• e.g., Polynomial Chaos Expansion (PCE)

• good convergence, effort grows exponentially with the dimensions

• model reduction needed: PCA, sparse grids, PGD, low-rank tensor formats

• general Hilbert space theory as in (Soize and Ghanem 2004)

• for applications in PDE approximations see (Babuska, Tempone, and Zouraris 2004)

## This talk:

• tensor representation of PCE

• reduction through multidimensional POD

# Multidimensional Galerkin POD

## Setup

$\DeclareMathOperator{\spann}{span} \def\yijk{\mathbf y^{i\,j\,k}} \def\Vec{\mathop{\mathrm {vec}}\nolimits} \def\Ltt{L^2((0,T))} \def\Lto{L^2(\Omega)} \def\Ltg{L^2(\Gamma;d\mathbb P_\alpha)} \def\by{\mathbf y}$ Consider a multivariable function $y(t,x;\alpha)$: $y\colon (0,T) \times \Omega \times \Gamma \to \mathbb R$

and the separated spaces for time, space, and uncertainty: $\Ltt,\quad \Lto, \quad\text{and}\quad\Ltg.$

## Time-Space-PCE Galerkin Discretization

Finite dimensional “subspaces”:

• $S = \spann\{\psi_1, \dotsc, \psi_s\} \subset \Ltt$,
• $X = \spann\{\phi_1, \dotsc, \phi_r\} \subset \Lto$,
• $W = \spann\{\eta_1, \dotsc, \eta_p\}~\subset\mspace{-4mu}~ \Ltg$,

and the Galerkin ansatz in $\by \in S\otimes X \otimes W$: $\by = \sum_{i=1}^s\sum_{j=1}^r\sum_{k=1}^p \yijk \psi_i \phi_j \eta_k.$

## A Modest Example

• $s=100$ – “time steps”

• $r=1000$ – “nodes in the mesh”

• $p=10$ – “features in the uncertainty”

• gives $s\cdot r \cdot p = 10^6$ – number of unknowns

## Time-Space-PCE Galerkin POD

Goal: Dimension Reduction

Idea: Find a subspace $\hat S \subset S$ and projection $\Pi_{\hat S}$ such that $\|\Pi_{\hat S} \by - \by\|_{S\otimes X \otimes W}$ is minimal…

minimal in the sense that if there exists $\hat{\hat S}$ such that $\|\Pi_{\hat{\hat S}} \by - \by\|_{S\otimes X \otimes W}$ is smaller, than the dimension of $\hat{\hat S}$ is larger than that of $\hat S$.

## Solution: HOSVD – higher order SVD

Recall: $\by = \sum_{i=1}^s\sum_{j=1}^r\sum_{k=1}^p \yijk \psi_i \phi_j \eta_k$ that is, with $\mathbf Y = [\yijk]$, the discrete function

$y \in S\otimes X \otimes W \longleftrightarrow \mathbf Y \in \mathbb R^{s \times r \times p}$

can be interpreted and reduced as a tensor $\mathbf Y$.

Vice versa:

Theorem: The $\hat s$-dimensional subspace $\hat S\subset S$ that optimally parametrizes $y\in S\otimes X \otimes W$ in $\hat S \otimes X \otimes W$ is defined by the $\hat s$ leading mode-(1) singular vectors of $\mathbf Y \in \mathbb R^{s \times r \times p}$.

Notes:

• The reduced spaces define a reduced Galerkin discretization.
• This works for any dimension in a product space $V = \prod_{\ell=1}^NV_i$.

# Application Example

A generic convection-diffusion problem $b\cdot \nabla y- \nabla\cdot ( \kappa_\alpha \nabla y) = f,$ where we assume that the diffusivity coefficient depends on a random vector $\alpha=(\alpha_1, \alpha_2, \alpha_3, \alpha_4)$.

## Ansatz

Locate the solution $y$ (depending on space $x$ and the random variable $\alpha$) in $\Lto \cdot L^2(\Gamma _ {1};\mathsf{d} \mathbb P _ {1}) \cdot L^2(\Gamma _ {2};\mathsf{d} \mathbb P _ {2}) \cdot L^2(\Gamma _ {3};\mathsf{d} \mathbb P _ {3}) \cdot L^2(\Gamma _ {4};\mathsf{d} \mathbb P _ {4})$

and use

• a standard FEM space $X$ to discretize $\Lto$,
• and Polynomial Chaos Expansions (PCE), e.g.,
• Lagrange polynomials with
• weights and nodes chosen according to the distribution of $\alpha_i$
• to define $W_i$, $i=1,2,3,4$.

## Approach

1. Compute the discrete solution $y\in X\otimes \bar W_1 \otimes \bar W_2 \otimes \bar W_3 \otimes \bar W_4$ for a low-dimensional PCE discretizations.

2. Reduce the spatial discretization $X \leftarrow \hat X$.

3. Compute the discrete solution $y\in \hat X\otimes W_1 \otimes W_2 \otimes W_3 \otimes W_4$ for a high-dimensional PCE discretizations.

4. Compare to $y\in X\otimes W_1 \otimes W_2 \otimes W_3 \otimes W_4$.

## Result

• Error in expected value over space.

• Full solve: $5^4 \times 90'000$ (PCE x FEM)

• POD + training: $2^4 \times 90'000$ + $5^4 \times 12$

• Error level $\approx 10^{-6}$

• Speed up factor $\approx 16$

• Memory savings: $\approx 97$%

• Monte Carlo: No convergence after $10^6 \times 90'000$.

# Conclusion

## … and Outlook

• Multidimensional Galerkin POD applies naturally for FEM/PCE discretizations.

• Significant savings of computation time and memory.

• Outlook: Optimal control and time dependent problems.

Thank You!

## Resources

• Submitted to Int. J. Numerical Methods in Engineering

• Preprint: arxiv:2009.01055

Babuska, Ivo, Raúl Tempone, and Georgios E. Zouraris. 2004. “Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations.” SIAM J. Numer. Anal. 42 (2): 800–825. https://doi.org/10.1137/S0036142902418680.

Soize, Christian, and Roger G. Ghanem. 2004. “Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure.” SIAM J. Sci. Comput. 26 (2): 395–410. https://doi.org/10.1137/S1064827503424505.

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