virtual USNCCM 2021
Jan Heiland (MPI Magdeburg)
Peter Benner (MPI Magdeburg)
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 \(y\).
slow, but highly parallelizable
many improvements like Multi Level MC
little overhead for additional dimensions
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)
tensor representation of PCE
reduction through multidimensional POD
\[ \DeclareMathOperator{\spann}{span} \def\yijk{\mathbf y^{i\,j\,k}} \def\Vec{\mathop{\mathrm {vec}}\nolimits} \def\Ltt{L^2((0,T))} \def\Lto{L^2(\Omega)} \def\Ltg{L^2(\Gamma;d\mathbb P_\alpha)} \def\by{\mathbf y} \] 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: \[ \Ltt,\quad \Lto, \quad\text{and}\quad\Ltg. \]
Finite dimensional “subspaces”:
and the Galerkin ansatz in \(\by \in S\otimes X \otimes W\): \[ \by = \sum_{i=1}^s\sum_{j=1}^r\sum_{k=1}^p \yijk \psi_i \phi_j \eta_k. \]
\(s=100\) – “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
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 \(\hat S\).
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 \(\mathbf Y\).
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 \(\mathbf Y \in \mathbb R^{s \times r \times p}\).
Note that:
The reduced spaces define a reduced Galerkin discretization.
This works for any dimension in a product space \(V = \prod_{\ell=1}^NV_i\).
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 \(\alpha=(\alpha_1, \alpha_2, \alpha_3, \alpha_4)\).
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
Compute the discrete solution \[ \bar y\in X\otimes \bar W_1 \otimes \bar W_2 \otimes \bar W_3 \otimes \bar W_4 \] for a low-dimensional PCE discretization.
Use the resulting tensor data \[ \bar y \longleftrightarrow \bar {\mathbf Y} \in \mathbb R^{d_x \times \bar d_1 \times \dotsm \times \bar d_4} \] to compute an optimized low-dimensional approximation \[\hat X\] for the FEM space \(X\).
Compute the discrete solution \[ \hat y\in \hat X\otimes W_1 \otimes W_2 \otimes W_3 \otimes W_4 \] for high-dimensional PCE discretizations and compare to
\[y\in X\otimes W_1 \otimes W_2 \otimes W_3 \otimes W_4,\]
i.e. the full-order solution.
Error in expected value over space.
Full solve: \(5^4 \times 90'000\) (PCE x FEM)
Training: \(2^4 \times 90'000\)
ROM solve: \(5^4 \times 12\)
Error level \(\approx 10^{-6}\)
Speed up factor \(\approx 16\)
Memory savings: \(\approx 97\)%
Monte Carlo: No convergence after \(10^6 \times 90'000\).
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!
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.