Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection–Diffusion–Reaction Equations
Abstract
:1. Introduction
- The Monte Carlo reduced basis (MCRB) method: The underlying equations are formulated weakly regarding the physical space, that is, the problem depends on both the deterministic and stochastic parameters. Monte Carlo sampling is used to estimate the parameter-dependent expected value and variance of a functional output of interest. An MCRB method for linear elliptic problems with error bounds for the expectation and variance of a linear functional output is derived in [5]. Improved error bounds are provided by [6]. Further advances are the introduction of a weighted error estimator [7] and the embedding in a multi-level procedure [8]. MCRB methods have also been applied to parabolic problems [9], saddle point problems [10], Bayesian inverse problems [11,12,13], and the assessment of rare events [14].
- Stochastic Galerkin reduced basis (SGRB) method: The underlying equations are formulated weakly regarding the spatial and stochastic dimensions, so that the problem depends on the deterministic parameters only. Parameter-dependent estimates of the expected value and variance of a functional output are obtained by direct integration of the reduced solution. The principle of SGRB methods is introduced in [15] for stochastic time-dependent incompressible Navier–Stokes problems, formulated weakly regarding the spatial and stochastic dimensions, with time acting as a parameter. Applications to linear dynamical systems are studied in [16,17]. SGRB methods can be related to space-time reduced basis methods [18,19], which rely on a weak formulation with respect to space and time. The idea of using SGRB methods to estimate parameter-dependent expected values is discussed in ([20], Section 8.2.1).
2. Materials and Methods
2.1. Monte Carlo Reduced Basis Method
2.1.1. Monte Carlo Finite Element (MCFE) Model
2.1.2. Monte Carlo Reduced Basis Model
2.1.3. Output Statistics and Error Estimates
2.2. Stochastic Galerkin Reduced Basis Method
2.2.1. Stochastic Galerkin Finite Element Model
2.2.2. Stochastic Galerkin Reduced Basis Model
2.2.3. Output Statistics and Error Estimates
2.3. Reduced Spaces
Algorithm 1 Proper orthogonal decomposition. |
|
2.3.1. Spatial POD
2.3.2. Spatial-Stochastic POD
3. Results
3.1. Convection–Diffusion–Reaction Model
3.2. Discretization
3.2.1. Finite Element Method
3.2.2. Monte Carlo Method
3.2.3. Stochastic Galerkin Method
3.2.4. Reduced Basis
3.3. Computational Costs
3.4. Numerical Results
4. Discussion
Author Contributions
Funding
Conflicts of Interest
References
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Symbol | Value | Description |
---|---|---|
expected value of reactivity | ||
1 | expected value of diffusivity | |
200 | standard deviation of reactivity | |
standard deviation of diffusivity | ||
L | 1 | correlation length |
K | 5 | Karhunen-Loève truncation index |
spatial domain with boundary | ||
deterministic parameter domain | ||
random parameter domain |
Symbol | Default | Reference | Description |
---|---|---|---|
225 | 961 | number of FE degrees of freedom | |
1024 | 16384 | number of MC samples of | |
d | 2 | 3 | degree of SG polynomials |
243 | 1024 | number of SG degrees of freedom: |
# | Dimension | Problem to Solve | Output |
---|---|---|---|
eigenvalue problem | coercivity constant | ||
system of equations | snapshots | ||
5 | singular value decomposition | reduced basis |
# | Dimension | Problem to Solve | Output |
---|---|---|---|
1 | eigenvalue problem | coercivity constant | |
system of equations | snapshots | ||
4 | singular value decomposition | reduced basis |
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Ullmann, S.; Müller, C.; Lang, J. Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection–Diffusion–Reaction Equations. Fluids 2021, 6, 263. https://doi.org/10.3390/fluids6080263
Ullmann S, Müller C, Lang J. Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection–Diffusion–Reaction Equations. Fluids. 2021; 6(8):263. https://doi.org/10.3390/fluids6080263
Chicago/Turabian StyleUllmann, Sebastian, Christopher Müller, and Jens Lang. 2021. "Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection–Diffusion–Reaction Equations" Fluids 6, no. 8: 263. https://doi.org/10.3390/fluids6080263