Embedded Multilevel Monte Carlo for Uncertainty Quantification in Random Domains

Badia, Santiago; Hampton, Jerrad; Principe, Javier

doi:10.1615/int.j.uncertaintyquantification.2021032984

Cited by 17 publications

(12 citation statements)

References 53 publications

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“…This construction permits the integration of the weak form of the problem and a judicious choice of the degrees of freedom can obtain a well posed-problem. The AgFEM method was introduced in [6], implemented in parallel in [50] and exploited to perform UQ in random domains in [3]. The reader is referred to these publications for further details.…”

Section: Model Problemmentioning

confidence: 99%

“…, 𝑀 3 Set 𝐿 = 𝐿 0 , ℓ = 𝑀, 𝑄 𝐿 , 𝑄 𝐿 = 0, 𝑛𝑟 = 0 4 while 𝑒 < 𝜖 and 𝐿 < 𝑀 do FEMPAR library, e.g., AgFEM methods. These embedded techniques have been recently exploited in [3] to perform UQ in random geometries using the MLMC implementation described herein. The code here is open source, under the GPLv3 license, and available at the FEMPAR gitlab repository.…”

Section: N Ementioning

confidence: 99%

“…As a second example, we consider a 3D Poisson problem in Section 5.2, with a random diffusion term that is described by a Karhunen-Loève Expansion (KLE), and fixed Dirichlet boundary conditions of 𝑥 1 + 𝑥 2 on the boundary of a randomized popcorn geometry similar to that described in [3]. This Poisson model is discretized using AMG, and executed in parallel with differing core counts on each level.…”

Section: N Ementioning

confidence: 99%

“…(for 𝑙 = 0, ..., 𝐿). This formula represents a theoretically ideal sample ratio when considering the computational costs and accuracies for computing integrated quantities from sufficiently regular problems discretized spatially and solved via the conjugate gradient method, as validated in [3]. To compute the scalability of the scheduling algorithm, we tune the overall computational cost of the experiments by a sample multiplier, denoted by 𝐶, and sample each level according to…”

Section: N Ementioning

confidence: 99%

“…Here we assume the domain to be a stochastic "popcorn" shape as defined in [3], which provides a complex geometry that changes for each realized sample. The boundary conditions are given by a sin |x|, Moreover, the diffusion term 𝜅(x, 𝜔), is a KLE where every point is log-normally distributed, and the associated normal distributed random variables are correlated by an exponential covariance function with correlation length 0.25.…”

Section: D Poisson In a Random Domainmentioning

confidence: 99%

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A Massively Parallel Implementation of Multilevel Monte Carlo for Finite Element Models

Badia¹,

Hampton²,

Principe³

2021

Preprint

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A. The Multilevel Monte Carlo (MLMC) method has proven to be an effective variance-reduction statistical method for Uncertainty Quantification (UQ) in Partial Differential Equation (PDE) models, combining model computations at different levels to create an accurate estimate. Still, the computational complexity of the resulting method is extremely high, particularly for 3D models, which requires advanced algorithms for the efficient exploitation of High Performance Computing (HPC). In this article we present a new implementation of the MLMC in massively parallel computer architectures, exploiting parallelism within and between each level of the hierarchy. The numerical approximation of the PDE is performed using the finite element method but the algorithm is quite general and could be applied to other discretization methods as well, although the focus is on parallel sampling. The two key ingredients of an efficient parallel implementation are a good processor partition scheme together with a good scheduling algorithm to assign work to different processors. We introduce a multiple partition of the set of processors that permits the simultaneous execution of different levels and we develop a dynamic scheduling algorithm to exploit it. The problem of finding the optimal scheduling of distributed tasks in a parallel computer is an NP-complete problem. We propose and analyze a new greedy scheduling algorithm to assign samples and we show that it is a 2-approximation, which is the best that may be expected under general assumptions. On top of this result we design a distributed memory implementation using the Message Passing Interface (MPI) standard. Finally we present a set of numerical experiments illustrating its scalability properties.

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Section: Model Problemmentioning

confidence: 99%

Section: N Ementioning

confidence: 99%

Section: N Ementioning

confidence: 99%

Section: N Ementioning

confidence: 99%

Section: D Poisson In a Random Domainmentioning

confidence: 99%

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A Massively Parallel Implementation of Multilevel Monte Carlo for Finite Element Models

Badia¹,

Hampton²,

Principe³

2021

Preprint

Self Cite

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A nonlinear stochastic finite element method for solving elastoplastic problems with uncertainties

Zheng

Nackenhorst

2023

Numerical Meth Engineering

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This article presents an efficient nonlinear stochastic finite element method to solve stochastic elastoplastic problems. Similar to deterministic elastoplastic problems, we describe history‐dependent stochastic elastoplastic behavior utilizing a series of (pseudo) time steps and go further to solve the corresponding stochastic solutions. For each time step, the original stochastic elastoplastic problem is considered as a time‐independent nonlinear stochastic problem with initial values given by stochastic displacements, stochastic strains, and internal variables of the previous time step. To solve the stochastic solution at each time step, the corresponding nonlinear stochastic problem is transformed into a set of linearized stochastic finite element equations by means of finite element discretization and a stochastic Newton linearization, while the stochastic solution at each time step is approximated by a sum of the products of random variables and deterministic vectors. Each couple of the random variable and the deterministic vector is also used to approximate the stochastic solution of the corresponding linearized stochastic finite element equation that can be solved via a weakly intrusive method. In this method, the deterministic vector is computed by solving deterministic linear finite element equations, and corresponding random variables are solved by a non‐intrusive method. Further, the proposed method avoids the curse of dimensionality successfully since its computational effort does not increase dramatically as the stochastic dimensionality increases. Four numerical cases are used to demonstrate the good performance of the proposed method.

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Random geometries for optimal control PDE problems based on fictitious domain FEMs and cut elements

Aretaki

Karatzas

2022

Journal of Computational and Applied Mathematics

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Embedded Multilevel Monte Carlo for Uncertainty Quantification in Random Domains

Cited by 17 publications

References 53 publications

A Massively Parallel Implementation of Multilevel Monte Carlo for Finite Element Models

A Massively Parallel Implementation of Multilevel Monte Carlo for Finite Element Models

A nonlinear stochastic finite element method for solving elastoplastic problems with uncertainties

Random geometries for optimal control PDE problems based on fictitious domain FEMs and cut elements

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