Goal-oriented adaptive finite element multilevel Monte Carlo with convergence rates

Beck, Joakim; Liu, Yang; Schwerin, Erik von; Tempone, Raúl

doi:10.1016/j.cma.2022.115582

Cited by 6 publications

(4 citation statements)

References 65 publications

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“…Sparse grid methods to integrate high-dimensional integrals are considered in [74,10,32,40,56,27,57,17,61]. An idea to generate goal-oriented adaptive spatial grids and use them in the multilevel MC (MLMC) framework was presented in [22,6].…”

Section: √ 𝑁mentioning

confidence: 99%

Uncertainty Quantification in the Henry Problem Using the Multilevel Monte Carlo Method

Litvinenko

Logashenko²,

Tempone

et al. 2023

Preprint

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Section: √ 𝑁mentioning

confidence: 99%

Uncertainty Quantification in the Henry Problem Using the Multilevel Monte Carlo Method

Litvinenko

Logashenko²,

Tempone

et al. 2023

Preprint

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“…Sparse grid methods to integrate high-dimensional integrals are considered in [68,9,31,38,51,26,52,15,56]. An idea to generate goal-oriented adaptive spatial grids and use them in the multilevel MC (MLMC) framework was presented in [20,5].…”

Section: Introductionmentioning

confidence: 99%

Uncertainty Quantification in Coastal Aquifers Using the Multilevel Monte Carlo Method

Litvinenko¹,

Logashenko²,

Tempone³

et al. 2023

Preprint

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We consider a class of density-driven flow problems. We are particularly interested in the problem of the salinization of coastal aquifers. We consider the Henry saltwater intrusion problem with uncertain porosity, permeability, and recharge parameters as a test case. The reason for the presence of uncertainties is the lack of knowledge, inaccurate measurements, and inability to measure parameters at each spatial or time location. This problem is nonlinear and time-dependent. The solution is the salt mass fraction, which is uncertain and changes in time. Uncertainties in porosity, permeability, recharge, and mass fraction are modeled using random fields. This work investigates the applicability of the well-known multilevel Monte Carlo (MLMC) method for such problems. The MLMC method can reduce the total computational and storage costs. Moreover, the MLMC method runs multiple scenarios on different spatial and time meshes and then estimates the mean value of the mass fraction. The parallelization is performed in both the physical space and stochastic space. To solve every deterministic scenario, we run the parallel multigrid solver ug4 in a black-box fashion. We use the solution obtained from the quasi-Monte Carlo method as a reference solution.

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“…Extended multiple sampling by significance is used for realistic rendering of materials such as leather, marble, wax, etc. The Monte Carlo method is often used to simulate light transport [1,2]. However, the Monte Carlo method is characterized by slow convergence.…”

Section: Introductionmentioning

confidence: 99%

The method of multiple sampling by significance for the visualization of functionally defined scenes

Vyatkin

Dolgovesov²

2023

E3S Web of Conf.

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In ray tracing methods, the key point is to choose the direction for the rays. If many rays are needed not everywhere, but only in some parts of the scene, it is reasonable to increase the selection in these places. As a result, computing resources are not wasted where there is no such need. That is, you need to make selections by significance. In this paper, the visualization of functionally defined scenes is considered. A method of multiple sampling by significance is proposed. The method uses weight functions for multiple sampling by significance. The weighting functions minimize the variance of the multiple sample estimation by significance. Weights can be negative, which reduces the variance. In addition, weights allow you to have additional flexibility when developing a sampling method that accelerates calculations. As a result, acceptable weights were obtained when modeling light transfer. The variance was reduced by using weights in the sample. The dependence of the mean square error on the number of samples is given. Highly realistic functionally defined scenes are visualized. The method is implemented using CPU and GPU. Diagrams of the method's performance are given.

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Goal-oriented adaptive finite element multilevel Monte Carlo with convergence rates

Cited by 6 publications

References 65 publications

Uncertainty Quantification in the Henry Problem Using the Multilevel Monte Carlo Method

Uncertainty Quantification in the Henry Problem Using the Multilevel Monte Carlo Method

Uncertainty Quantification in Coastal Aquifers Using the Multilevel Monte Carlo Method

The method of multiple sampling by significance for the visualization of functionally defined scenes

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