Stochastic Galerkin reduced basis methods for parametrized random elliptic PDEs

Mapping Intimacies ◽

10.14293/p2199-8442.1.sop-math.ffvlpm.v1 ◽

2018 ◽

Author(s):

Jens Lang ◽

Sebastian Ullmann

Keyword(s):

Elliptic Pdes ◽

Dimensional System ◽

Reduced Basis ◽

Reduced Basis Methods ◽

Stochastic Galerkin ◽

Basis Method ◽

Low Dimensional ◽

The Cost ◽

Parameter Dependent ◽

Multiple Samples

We consider the estimation of parameter-dependent statistical outputs for parametrized elliptic PDE problems with random data. We propose a stochastic Galerkin reduced basis method, which provides the expected output for a given parameter value at the cost of solving a single low-dimensional system of equations. This is substantially faster than usual Monte Carlo reduced basis methods, which require multiple samples of the reduced solution.

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An Improved Discrete Least-Squares/Reduced-Basis Method for Parameterized Elliptic PDEs

Journal of Scientific Computing ◽

10.1007/s10915-018-0661-6 ◽

2018 ◽

Vol 81 (1) ◽

pp. 76-91

Author(s):

Max Gunzburger ◽

Michael Schneier ◽

Clayton Webster ◽

Guannan Zhang

Keyword(s):

Least Squares ◽

Reduced Basis Method ◽

Elliptic Pdes ◽

Reduced Basis ◽

Basis Method

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Analysis of Transition for a Flow in a Channel via Reduced Basis Methods

Fluids ◽

10.3390/fluids4040202 ◽

2019 ◽

Vol 4 (4) ◽

pp. 202

Author(s):

Gaetano Pascarella ◽

Ioannis Kokkinakis ◽

Marco Fossati

Keyword(s):

Temporal Dynamics ◽

Dynamic Mode Decomposition ◽

Flow Structures ◽

Dynamic Mode ◽

Planar Channel ◽

Reduced Basis ◽

Reduced Basis Methods ◽

Mode Decomposition ◽

Flow Mechanisms ◽

Basis Method

The study of the flow mechanisms leading to transition in a planar channel flow is investigated by means of a reduced basis method known as Dynamic Mode Decomposition (DMD). The problem of identification of the most relevant DMD modes is addressed in terms of the ability to (i) provide a fairly accurate reconstruction of the flow field, and (ii) match the most relevant flow structures at the beginning of the transition region. A comparative study between a natural method of selection based on the energetic content of the modes and a new one based on the temporal dynamics of the modes is here presented.

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Reduced basis method for multi-parameter-dependent steady Navier–Stokes equations: Applications to natural convection in a cavity

Journal of Computational Physics ◽

10.1016/j.jcp.2009.03.008 ◽

2009 ◽

Vol 228 (12) ◽

pp. 4359-4378 ◽

Cited By ~ 40

Author(s):

Simone Deparis ◽

Gianluigi Rozza

Keyword(s):

Natural Convection ◽

Stokes Equations ◽

Reduced Basis Method ◽

Navier Stokes ◽

Reduced Basis ◽

Navier Stokes Equations ◽

Basis Method ◽

Parameter Dependent

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Solving Parameter-Dependent Lyapunov Equations Using the Reduced Basis Method with Application to Parametric Model Order Reduction

SIAM Journal on Matrix Analysis and Applications ◽

10.1137/15m1027097 ◽

2017 ◽

Vol 38 (2) ◽

pp. 478-504 ◽

Cited By ~ 4

Author(s):

Nguyen Thanh Son ◽

Tatjana Stykel

Keyword(s):

Model Order Reduction ◽

Order Reduction ◽

Parametric Model ◽

Reduced Basis Method ◽

Lyapunov Equations ◽

Reduced Basis ◽

Model Order ◽

Parametric Model Order Reduction ◽

Basis Method ◽

Parameter Dependent

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Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection–Diffusion–Reaction Equations

Fluids ◽

10.3390/fluids6080263 ◽

2021 ◽

Vol 6 (8) ◽

pp. 263

Author(s):

Sebastian Ullmann ◽

Christopher Müller ◽

Jens Lang

Keyword(s):

Monte Carlo ◽

Monte Carlo Sampling ◽

Diffusion Reaction ◽

Reduced Basis ◽

Weak Formulation ◽

Linear System Of Equations ◽

Convection Diffusion ◽

Stochastic Galerkin ◽

Parameter Dependent ◽

Reaction Equations

We consider the estimation of parameter-dependent statistics of functional outputs of steady-state convection–diffusion–reaction equations with parametrized random and deterministic inputs in the framework of linear elliptic partial differential equations. For a given value of the deterministic parameter, a stochastic Galerkin finite element (SGFE) method can estimate the statistical moments of interest of a linear output at the cost of solving a single, large, block-structured linear system of equations. We propose a stochastic Galerkin reduced basis (SGRB) method as a means to lower the computational burden when statistical outputs are required for a large number of deterministic parameter queries. Our working assumption is that we have access to the computational resources necessary to set up such a reduced-order model for a spatial-stochastic weak formulation of the parameter-dependent model equations. In this scenario, the complexity of evaluating the SGRB model for a new value of the deterministic parameter only depends on the reduced dimension. To derive an SGRB model, we project the spatial-stochastic weak solution of a parameter-dependent SGFE model onto a reduced basis generated by a proper orthogonal decomposition (POD) of snapshots of SGFE solutions at representative values of the parameter. We propose residual-corrected estimates of the parameter-dependent expectation and variance of linear functional outputs and provide respective computable error bounds. We test the SGRB method numerically for a convection–diffusion–reaction problem, choosing the convective velocity as a deterministic parameter and the parametrized reactivity or diffusivity field as a random input. Compared to a standard reduced basis model embedded in a Monte Carlo sampling procedure, the SGRB model requires a similar number of reduced basis functions to meet a given tolerance requirement. However, only a single run of the SGRB model suffices to estimate a statistical output for a new deterministic parameter value, while the standard reduced basis model must be solved for each Monte Carlo sample.

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Error estimate of the non-intrusive reduced basis method with finite volume schemes

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2021044 ◽

2021 ◽

Vol 55 (5) ◽

pp. 1941-1961

Author(s):

Elise Grosjean ◽

Yvon Maday

Keyword(s):

Finite Volume ◽

Degrees Of Freedom ◽

Good Alternative ◽

High Fidelity ◽

Finite Volume Schemes ◽

Reduced Basis ◽

Reduced Basis Methods ◽

Flux Approximation ◽

Mimetic Finite Difference Method ◽

Parameter Dependent

The context of this paper is the simulation of parameter-dependent partial differential equations (PDEs). When the aim is to solve such PDEs for a large number of parameter values, Reduced Basis Methods (RBM) are often used to reduce computational costs of a classical high fidelity code based on Finite Element Method (FEM), Finite Volume (FVM) or Spectral methods. The efficient implementation of most of these RBM requires to modify this high fidelity code, which cannot be done, for example in an industrial context if the high fidelity code is only accessible as a "black-box" solver. The Non-Intrusive Reduced Basis (NIRB) method has been introduced in the context of finite elements as a good alternative to reduce the implementation costs of these parameter-dependent problems. The method is efficient in other contexts than the FEM one, like with finite volume schemes, which are more often used in an industrial environment. In this case, some adaptations need to be done as the degrees of freedom in FV methods have different meanings. At this time, error estimates have only been studied with FEM solvers. In this paper, we present a generalisation of the NIRB method to Finite Volume schemes and we show that estimates established for FEM solvers also hold in the FVM setting. We first prove our results for the hybrid-Mimetic Finite Difference method (hMFD), which is part the Hybrid Mixed Mimetic methods (HMM) family. Then, we explain how these results apply more generally to other FV schemes. Some of them are specified, such as the Two Point Flux Approximation (TPFA).

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Reduced Basis Methods for Quasilinear Elliptic PDEs with Applications to Permanent Magnet Synchronous Motors

10.1007/978-3-030-72983-7_14 ◽

2021 ◽

pp. 307-326

Author(s):

Michael Hinze ◽

Denis Korolev

Keyword(s):

Permanent Magnet ◽

Elliptic Pdes ◽

Reduced Basis ◽

Permanent Magnet Synchronous Motors ◽

Synchronous Motors ◽

Reduced Basis Methods ◽

Quasilinear Elliptic

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ROM-Based Inexact Subdivision Methods for PDE-Constrained Multiobjective Optimization

Mathematical and Computational Applications ◽

10.3390/mca26020032 ◽

2021 ◽

Vol 26 (2) ◽

pp. 32

Author(s):

Stefan Banholzer ◽

Bennet Gebken ◽

Lena Reichle ◽

Stefan Volkwein

Keyword(s):

Multiobjective Optimization ◽

Elliptic Partial Differential Equation ◽

Computational Effort ◽

Reduced Basis ◽

Solution Approach ◽

Gradient Based ◽

Descent Condition ◽

Basis Method ◽

Parameter Dependent ◽

Constrained Multiobjective Optimization

The goal in multiobjective optimization is to determine the so-called Pareto set. Our optimization problem is governed by a parameter-dependent semi-linear elliptic partial differential equation (PDE). To solve it, we use a gradient-based set-oriented numerical method. The numerical solution of the PDE by standard discretization methods usually leads to high computational effort. To overcome this difficulty, reduced-order modeling (ROM) is developed utilizing the reduced basis method. These model simplifications cause inexactness in the gradients. For that reason, an additional descent condition is proposed. Applying a modified subdivision algorithm, numerical experiments illustrate the efficiency of our solution approach.

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