736 research outputs found
Multilevel Monte Carlo Finite Volume Methods for Random Conservation Laws with Discontinuous Flux
We consider conservation laws with discontinuous flux where the initial
datum, the flux function, and the discontinuous spatial dependency coefficient
are subject to randomness. We establish a notion of random adapted entropy
solutions to these equations and prove well-posedness provided that the spatial
dependency coefficient is piecewise constant with finitely many
discontinuities. In particular, the setting under consideration allows the flux
to change across finitely many points in space whose positions are uncertain.
We propose a single- and multilevel Monte Carlo method based on a finite volume
approximation for each sample. Our analysis includes convergence rate estimates
of the resulting Monte Carlo and multilevel Monte Carlo finite volume methods
as well as error versus work rates showing that the multilevel variant
outperforms the single-level method in terms of efficiency. We present
numerical experiments motivated by two-phase reservoir simulations for
reservoirs with varying geological properties.Comment: 25 pages, 9 figures, 4 tables, major revision
Multilevel Monte Carlo for Random Degenerate Scalar Convection Diffusion Equation
We consider the numerical solution of scalar, nonlinear degenerate
convection-diffusion problems with random diffusion coefficient and with random
flux functions. Building on recent results on the existence, uniqueness and
continuous dependence of weak solutions on data in the deterministic case, we
develop a definition of random entropy solution. We establish existence,
uniqueness, measurability and integrability results for these random entropy
solutions, generalizing \cite{Mishr478,MishSch10a} to possibly degenerate
hyperbolic-parabolic problems with random data. We next address the numerical
approximation of random entropy solutions, specifically the approximation of
the deterministic first and second order statistics. To this end, we consider
explicit and implicit time discretization and Finite Difference methods in
space, and single as well as Multi-Level Monte-Carlo methods to sample the
statistics. We establish convergence rate estimates with respect to the
discretization parameters, as well as with respect to the overall work,
indicating substantial gains in efficiency are afforded under realistic
regularity assumptions by the use of the Multi-Level Monte-Carlo method.
Numerical experiments are presented which confirm the theoretical convergence
estimates.Comment: 24 Page
Adaptive Uncertainty Quantification for Stochastic Hyperbolic Conservation Laws
We propose a predictor-corrector adaptive method for the study of hyperbolic
partial differential equations (PDEs) under uncertainty. Constructed around the
framework of stochastic finite volume (SFV) methods, our approach circumvents
sampling schemes or simulation ensembles while also preserving fundamental
properties, in particular hyperbolicity of the resulting systems and
conservation of the discrete solutions. Furthermore, we augment the existing
SFV theory with a priori convergence results for statistical quantities, in
particular push-forward densities, which we demonstrate through numerical
experiments. By linking refinement indicators to regions of the physical and
stochastic spaces, we drive anisotropic refinements of the discretizations,
introducing new degrees of freedom (DoFs) where deemed profitable. To
illustrate our proposed method, we consider a series of numerical examples for
non-linear hyperbolic PDEs based on Burgers' and Euler's equations
A Fully Parallelized and Budgeted Multi-level Monte Carlo Framework for Partial Differential Equations: From Mathematical Theory to Automated Large-Scale Computations
All collected data on any physical, technical or economical process is subject to uncertainty. By incorporating this uncertainty in the model and propagating it through the system, this data error can be controlled. This makes the predictions of the system more trustworthy and reliable. The multi-level Monte Carlo (MLMC) method has proven to be an effective uncertainty quantification tool, requiring little knowledge about the problem while being highly performant.
In this doctoral thesis we analyse, implement, develop and apply the MLMC method to partial differential equations (PDEs) subject to high-dimensional random input data. We set up a unified framework based on the software M++ to approximate solutions to elliptic and hyperbolic PDEs with a large selection of finite element methods. We combine this setup with a new variant of the MLMC method. In particular, we propose a budgeted MLMC (BMLMC) method which is capable to optimally invest reserved computing resources in order to minimize the model error while exhausting a given computational budget. This is achieved by developing a new parallelism based on a single distributed data structure, employing ideas of the continuation MLMC method and utilizing dynamic programming techniques. The final method is theoretically motivated, analyzed, and numerically well-tested in an automated benchmarking workflow for highly challenging problems like the approximation of wave equations in randomized media
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