arXiv subjects

Ivan G. Graham

Publications and source records attributed to Ivan G. Graham.

29 records · Page 2Linked to original sources

arXiv2020

Full Error Analysis and Uncertainty Quantification for the Heterogeneous Transport Equation in Slab Geometry

We present an analysis of multilevel Monte Carlo techniques for the forward problem of uncertainty quantification for the radiative transport equation, when the coefficients ({\em cross-sections}) are heterogenous random fields. To do this, we first give a new error analysis for the combined spatial and angular discretisation in the deterministic case, with error estimates which are explicit in the coefficients (and allow for very low regularity and jumps). This detailed error analysis is done for the 1D space - 1D angle slab geometry case with classical diamond differencing. Under reasonable assumptions on the statistics of the coefficients, we then prove an error estimate for the random problem in a suitable Bochner space. Because the problem is not self-adjoint, stability can only be proved under a path-dependent mesh resolution condition. This means that, while the Bochner space error estimate is of order $\mathcal{O}(h^η)$ for some $η$, where $h$ is a (deterministically chosen) mesh diameter, smaller mesh sizes might be needed for some realisations. Under reasonable assumptions we show that the expected cost for computing a typical quantity of interest remains of the same order as for a single sample. This leads to rigorous complexity estimates for Monte Carlo and multilevel Monte Carlo: For particular linear solvers, the multilevel version gives up to two orders of magnitude improvement over Monte Carlo. We provide numerical results supporting the theory.

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arXiv2019

Analysis of quasi-Monte Carlo methods for elliptic eigenvalue problems with stochastic coefficients

We consider the forward problem of uncertainty quantification for the generalised Dirichlet eigenvalue problem for a coercive second order partial differential operator with random coefficients, motivated by problems in structural mechanics, photonic crystals and neutron diffusion. The PDE coefficients are assumed to be uniformly bounded random fields, represented as infinite series parametrised by uniformly distributed i.i.d. random variables. The expectation of the fundamental eigenvalue of this problem is computed by (a) truncating the infinite series which define the coefficients; (b) approximating the resulting truncated problem using lowest order conforming finite elements and a sparse matrix eigenvalue solver; and (c) approximating the resulting finite (but high dimensional) integral by a randomly shifted quasi-Monte Carlo lattice rule, with specially chosen generating vector. We prove error estimates for the combined error, which depend on the truncation dimension $s$, the finite element mesh diameter $h$, and the number of quasi-Monte Carlo samples $N$. Under suitable regularity assumptions, our bounds are of the particular form $\mathcal{O}(h^2+N^{-1+δ})$, where $δ>0$ is arbitrary and the hidden constant is independent of the truncation dimension, which needs to grow as $h\to 0$ and $N\to\infty$. Although the eigenvalue problem is nonlinear, which means it is generally considered harder than the analogous source problem, in almost all cases we obtain error bounds that converge at the same rate as the corresponding rate for the source problem. The proof involves a detailed study of the regularity of the fundamental eigenvalue as a function of the random parameters. As a key intermediate result in the analysis, we prove that the spectral gap (between the fundamental and the second eigenvalues) is uniformly positive over all realisations of the random problem.

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arXiv2019

Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption

This paper rigorously analyses preconditioners for the time-harmonic Maxwell equations with absorption, where the PDE is discretised using curl-conforming finite-element methods of fixed, arbitrary order and the preconditioner is constructed using Additive Schwarz domain decomposition methods. The theory developed here shows that if the absorption is large enough, and if the subdomain and coarse mesh diameters and overlap are chosen appropriately, then the classical two-level overlapping Additive Schwarz preconditioner (with PEC boundary conditions on the subdomains) performs optimally -- in the sense that GMRES converges in a wavenumber-independent number of iterations -- for the problem with absorption. An important feature of the theory is that it allows the coarse space to be built from low-order elements even if the PDE is discretised using high-order elements. It also shows that additive methods with minimal overlap can be robust. Numerical experiments are given that illustrate the theory and its dependence on various parameters. These experiments motivate some extensions of the preconditioners which have better robustness for problems with less absorption, including the propagative case. At the end of the paper we illustrate the performance of these on two substantial applications; the first (a problem with absorption arising from medical imaging) shows the empirical robustness of the preconditioner against heterogeneity, and the second (scattering by a COBRA cavity) shows good scalability of the preconditioner with up to 3,000 processors.

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arXiv2019

Bounding the spectral gap for an elliptic eigenvalue problem with uniformly bounded stochastic coefficients

A key quantity that occurs in the error analysis of several numerical methods for eigenvalue problems is the distance between the eigenvalue of interest and the next nearest eigenvalue. When we are interested in the smallest or fundamental eigenvalue, we call this the spectral or fundamental gap. In a recent manuscript [Gilbert et al., arXiv:1808.02639], the current authors, together with Frances Kuo, studied an elliptic eigenvalue problem with homogeneous Dirichlet boundary conditions, and with coefficients that depend on an infinite number of uniformly distributed stochastic parameters. In this setting, the eigenvalues, and in turn the eigenvalue gap, also depend on the stochastic parameters. Hence, for a robust error analysis one needs to be able to bound the gap over all possible realisations of the parameters, and because the gap depends on infinitely-many random parameters, this is not trivial. This short note presents, in a simplified setting, an important result that was shown in the paper above. Namely, that, under certain decay assumptions on the coefficient, the spectral gap of such a random elliptic eigenvalue problem can be bounded away from 0, uniformly over the entire infinite-dimensional parameter space.

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arXiv2018

The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances

We consider the exterior Dirichlet problem for the heterogeneous Helmholtz equation, i.e. the equation $\nabla\cdot(A \nabla u ) + k^2 n u =-f$ where both $A$ and $n$ are functions of position. We prove new a priori bounds on the solution under conditions on $A$, $n$, and the domain that ensure nontrapping of rays; the novelty is that these bounds are explicit in $k$, $A$, $n$, and geometric parameters of the domain. We then show that these a priori bounds hold when $A$ and $n$ are $L^\infty$ and satisfy certain monotonicity conditions, and thereby obtain new results both about the well-posedness of such problems and about the resonances of acoustic transmission problems (i.e. $A$ and $n$ discontinuous) where the transmission interfaces are only assumed to be $C^0$ and star-shaped; the novelty of this latter result is that until recently the only known results about resonances of acoustic transmission problems were for $C^\infty$ convex interfaces with strictly positive curvature.

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arXiv2018

Circulant embedding with QMC -- analysis for elliptic PDE with lognormal coefficients

In a previous paper (J. Comp. Phys. 230 (2011), 3668--3694), the authors proposed a new practical method for computing expected values of functionals of solutions for certain classes of elliptic partial differential equations with random coefficients. This method was based on combining quasi-Monte Carlo (QMC) methods for computing the expected values with circulant embedding methods for sampling the random field on a regular grid. It was found capable of handling fluid flow problems in random heterogeneous media with high stochastic dimension, but a convergence theory was missing. This paper provides a convergence analysis for the method in the case when the QMC method is a specially designed randomly shifted lattice rule. The convergence result depends on the eigenvalues of the underlying nested block circulant matrix and can be independent of the number of stochastic variables under certain assumptions. In fact the QMC analysis applies to general factorisations of the covariance matrix to sample the random field. The error analysis for the underlying fully discrete finite element method allows for locally refined meshes (via interpolation from a regular sampling grid of the random field). Numerical results on a non-regular domain with corner singularities in two spatial dimensions and on a regular domain in three spatial dimensions are included.

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arXiv2018

Analysis of circulant embedding methods for sampling stationary random fields

In this paper we prove, under mild conditions, that the positive definiteness of the circulant matrix appearing in the circulant embedding method is always guaranteed, provided the enclosing cube is sufficiently large. We examine in detail the case of the Matérn covariance, and prove (for fixed correlation length) that, as $h_0\rightarrow 0$, positive definiteness is guaranteed when the random field is sampled on a cube of size order $(1 + ν^{1/2} \log h_0^{-1})$ times larger than the size of the physical domain. (Here $h_0$ is the mesh spacing of the regular grid and $ν$ the Matérn smoothness parameter.) We show that the sampling cube can become smaller as the correlation length decreases when $h_0$ and $ν$ are fixed. Our results are confirmed by numerical experiments. We prove several results about the decay of the eigenvalues of the circulant matrix. These lead to the conjecture, verified by numerical experiment, that they decay with the same rate as the Karhunen--Loève eigenvalues of the covariance operator.

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arXiv2017

Modern Monte Carlo Variants for Uncertainty Quantification in Neutron Transport

We describe modern variants of Monte Carlo methods for Uncertainty Quantification (UQ) of the Neutron Transport Equation, when it is approximated by the discrete ordinates method with diamond differencing. We focus on the mono-energetic 1D slab geometry problem, with isotropic scattering, where the cross-sections are log-normal correlated random fields of possibly low regularity. The paper includes an outline of novel theoretical results on the convergence of the discrete scheme, in the cases of both spatially variable and random cross-sections. We also describe the theory and practice of algorithms for quantifying the uncertainty of a linear functional of the scalar flux, using Monte Carlo and quasi-Monte Carlo methods, and their multilevel variants. A hybrid iterative/direct solver for computing each realisation of the functional is also presented. Numerical experiments show the effectiveness of the hybrid solver and the gains that are possible through quasi-Monte Carlo sampling and multilevel variance reduction. For the multilevel quasi-Monte Carlo method, we observe gains in the computational $\varepsilon$-cost of up to 2 orders of magnitude over the standard Monte Carlo method, and we explain this theoretically. Experiments on problems with up to several thousand stochastic dimensions are included.

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arXiv2016

Domain Decomposition preconditioning for high-frequency Helmholtz problems with absorption

In this paper we give new results on domain decomposition preconditioners for GMRES when computing piecewise-linear finite-element approximations of the Helmholtz equation $-Δu - (k^2+ {\rm i} \varepsilon)u = f$, with absorption parameter $\varepsilon \in \mathbb{R}$. Multigrid approximations of this equation with $\varepsilon \not= 0$ are commonly used as preconditioners for the pure Helmholtz case ($\varepsilon = 0$). However a rigorous theory for such (so-called "shifted Laplace") preconditioners, either for the pure Helmholtz equation, or even the absorptive equation ($\varepsilon \not=0$), is still missing. We present a new theory for the absorptive equation that provides rates of convergence for (left- or right-) preconditioned GMRES, via estimates of the norm and field of values of the preconditioned matrix. This theory uses a $k$- and $\varepsilon$-explicit coercivity result for the underlying sesquilinear form and shows, for example, that if $|\varepsilon|\sim k^2$, then classical overlapping additive Schwarz will perform optimally for the absorptive problem, provided the subdomain and coarse mesh diameters are carefully chosen. Extensive numerical experiments are given that support the theoretical results. The theory for the absorptive case gives insight into how its domain decomposition approximations perform as preconditioners for the pure Helmholtz case $\varepsilon = 0$. At the end of the paper we propose a (scalable) multilevel preconditioner for the pure Helmholtz problem that has an empirical computation time complexity of about $\mathcal{O}(n^{4/3})$ for solving finite element systems of size $n=\mathcal{O}(k^3)$, where we have chosen the mesh diameter $h \sim k^{-3/2}$ to avoid the pollution effect. Experiments on problems with $h\sim k^{-1}$, i.e. a fixed number of grid points per wavelength, are also given.

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arXiv2013

Mixed Finite Element Analysis of Lognormal Diffusion and Multilevel Monte Carlo Methods

This work is motivated by the need to develop efficient tools for uncertainty quantification in subsurface flows associated with radioactive waste disposal studies. We consider single phase flow problems in random porous media described by correlated lognormal distributions. We are interested in the error introduced by a finite element discretisation of these problems. In contrast to several recent works on the analysis of standard nodal finite element discretisations, we consider here mass-conservative lowest order Raviart-Thomas mixed finite elements. This is very important since local mass conservation is highly desirable in realistic groundwater flow problems. Due to the limited spatial regularity and the lack of uniform ellipticity and boundedness of the operator the analysis is non-trivial in the presence of lognormal random fields. We establish finite element error bounds for Darcy velocity and pressure, as well as for a more accurate recovered pressure approximation. We then apply the error bounds to prove convergence of the multilevel Monte Carlo algorithm for estimating statistics of these quantities. Moreover, we prove convergence for a class of bounded, linear functionals of the Darcy velocity. An important special case is the approximation of the effective permeability in a 2D flow cell. We perform numerical experiments to confirm the convergence results.

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arXiv2010

Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation

We consider the classical coupled, combined-field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle. In recent work, we have proved lower and upper bounds on the $L^2$ condition numbers for these formulations, and also on the norms of the classical acoustic single- and double-layer potential operators. These bounds to some extent make explicit the dependence of condition numbers on the wave number $k$, the geometry of the scatterer, and the coupling parameter. For example, with the usual choice of coupling parameter they show that, while the condition number grows like $k^{1/3}$ as $k\to\infty$, when the scatterer is a circle or sphere, it can grow as fast as $k^{7/5}$ for a class of `trapping' obstacles. In this paper we prove further bounds, sharpening and extending our previous results. In particular we show that there exist trapping obstacles for which the condition numbers grow as fast as $\exp(γk)$, for some $γ>0$, as $k\to\infty$ through some sequence. This result depends on exponential localisation bounds on Laplace eigenfunctions in an ellipse that we prove in the appendix. We also clarify the correct choice of coupling parameter in 2D for low $k$. In the second part of the paper we focus on the boundary element discretisation of these operators. We discuss the extent to which the bounds on the continuous operators are also satisfied by their discrete counterparts and, via numerical experiments, we provide supporting evidence for some of the theoretical results, both quantitative and asymptotic, indicating further which of the upper and lower bounds may be sharper.

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