arXiv subjects

Alexander D. Gilbert

Publications and source records attributed to Alexander D. Gilbert.

18 recordsLinked to original sources

arXiv2026

Multilevel lattice-based kernel approximation for elliptic PDEs with random coefficients

This paper introduces a multilevel kernel-based approximation method to estimate efficiently solutions to elliptic partial differential equations (PDEs) with periodic random coefficients. Building upon the work of Kaarnioja, Kazashi, Kuo, Nobile, Sloan (Numer. Math., 2022) on kernel interpolation with quasi-Monte Carlo (QMC) lattice point sets, we leverage multilevel techniques to enhance computational efficiency while maintaining a given level of accuracy. In the function space setting with product-type weight parameters, the single-level approximation can achieve an accuracy of $\varepsilon>0$ with cost $\mathcal{O}(\varepsilon^{-η-ν-θ})$ for positive constants $η, ν, θ$ depending on the rates of convergence associated with dimension truncation, kernel approximation, and finite element approximation, respectively. Our multilevel approximation can achieve the same $\varepsilon$ accuracy at a reduced cost $\mathcal{O}(\varepsilon^{-η-\max(ν,θ)})$. Full regularity theory and error analysis are provided, followed by numerical experiments that validate the efficacy of the proposed multilevel approximation in comparison to the single-level approach.

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arXiv2026

Quasi-Monte Carlo methods for uncertainty quantification of tumor growth modeled by a parametric semi-linear parabolic reaction-diffusion equation

We study the application of a quasi-Monte Carlo (QMC) method to a class of semi-linear parabolic reaction-diffusion partial differential equations used to model tumor growth. Mathematical models of tumor growth are largely phenomenological in nature, capturing infiltration of the tumor into surrounding healthy tissue, proliferation of the existing tumor, and patient response to therapies, such as chemotherapy and radiotherapy. Considerable inter-patient variability, inherent heterogeneity of the disease, sparse and noisy data collection, and model inadequacy all contribute to significant uncertainty in the model parameters. It is crucial that these uncertainties can be efficiently propagated through the model to compute quantities of interest (QoIs), which in turn may be used to inform clinical decisions. We show that QMC methods can be successful in computing expectations of meaningful QoIs. Well-posedness results are developed for the model and used to show a theoretical error bound for the case of uniform random fields. The theoretical linear error rate, which is superior to that of standard Monte Carlo, is verified numerically. Encouraging computational results are also provided for lognormal random fields, prompting further theoretical development.

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arXiv2026

Minimal Subsampled Rank-1 Lattices for Multivariate Approximation with Optimal Convergence Rate

In this paper we show error bounds for randomly subsampled rank-1 lattices. We pay particular attention to the ratio of the size of the subset to the size of the initial lattice, which is decisive for the computational complexity. In the special case of Korobov spaces, we achieve the optimal polynomial sampling complexity whilst having the smallest initial lattice possible. We further characterize the frequency index set for which a given lattice is reconstructing by using the reciprocal of the worst-case error achieved using the lattice in question. This connects existing approaches used in proving error bounds for lattices. We make detailed comments on the implementation and test different algorithms using the subsampled lattice in numerical experiments.

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arXiv2025

Finite element analysis of density estimation using preintegration for elliptic PDE with random input

This paper analyses the finite element component of the error when using preintegration to approximate the cdf and pdf for uncertainty quantification (UQ) problems involving elliptic PDEs with random inputs. It is a follow up to Gilbert, Kuo, Srikumar, SIAM J. Numer. Anal. 63 (2025), pp. 1025-1054, which introduced a method of density estimation for a class of UQ problems, based on computing the integral formulations of the cdf and pdf by performing an initial smoothing preintegration step and then applying a quasi-Monte Carlo quadrature rule to approximate the remaining high-dimensional integral. That paper focussed on the quadrature aspect of the method, whereas this paper studies the spatial discretisation of the PDE using finite element methods. First, it is shown that the finite element approximation satisfies the required assumptions for the preintegration theory, including the important monotonicity condition. Then the finite element error is analysed and finally, the combined finite element and quasi-Monte Carlo error is bounded. It is shown that under similar assumptions, the cdf and pdf can be approximated with the same rate of convergence as the much simpler problem of computing expected values.

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arXiv2024

A complex-projected Rayleigh quotient iteration for targeting interior eigenvalues

We introduce a new Projected Rayleigh Quotient Iteration aimed at improving the convergence behaviour of classic Rayleigh Quotient iteration (RQI) by incorporating approximate information about the target eigenvector at each step. While classic RQI exhibits local cubic convergence for Hermitian matrices, its global behaviour can be unpredictable, whereby it may converge to an eigenvalue far away from the target, even when started with accurate initial conditions. This problem is exacerbated when the eigenvalues are closely spaced. The key idea of the new algorithm is at each step to add a complex-valued projection to the original matrix (that depends on the current eigenvector approximation), such that the unwanted eigenvalues are lifted into the complex plane while the target stays close to the real line, thereby increasing the spacing between the target eigenvalue and the rest of the spectrum. Making better use of the eigenvector approximation leads to more robust convergence behaviour and the new method converges reliably to the correct target eigenpair for a significantly wider range of initial vectors than does classic RQI. We prove that the method converges locally cubically and we present several numerical examples demonstrating the improved global convergence behaviour. In particular, we apply it to compute eigenvalues in a band-gap spectrum of a Sturm-Liouville operator used to model photonic crystal fibres, where the target and unwanted eigenvalues are closely spaced. The examples show that the new method converges to the desired eigenpair even when the eigenvalue spacing is very small, often succeeding when classic RQI fails.

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arXiv2024

Density estimation for elliptic PDE with random input by preintegration and quasi-Monte Carlo methods

In this paper, we apply quasi-Monte Carlo (QMC) methods with an initial preintegration step to estimate cumulative distribution functions and probability density functions in uncertainty quantification (UQ). The distribution and density functions correspond to a quantity of interest involving the solution to an elliptic partial differential equation (PDE) with a lognormally distributed coefficient and a normally distributed source term. There is extensive previous work on using QMC to compute expected values in UQ, which have proven very successful in tackling a range of different PDE problems. However, the use of QMC for density estimation applied to UQ problems will be explored here for the first time. Density estimation presents a more difficult challenge compared to computing the expected value due to discontinuities present in the integral formulations of both the distribution and density. Our strategy is to use preintegration to eliminate the discontinuity by integrating out a carefully selected random parameter, so that QMC can be used to approximate the remaining integral. First, we establish regularity results for the PDE quantity of interest that are required for smoothing by preintegration to be effective. We then show that an $N$-point lattice rule can be constructed for the integrands corresponding to the distribution and density, such that after preintegration the QMC error is of order $\mathcal{O}(N^{-1+ε})$ for arbitrarily small $ε>0$. This is the same rate achieved for computing the expected value of the quantity of interest. Numerical results are presented to reaffirm our theory.

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arXiv2024

Multilevel Monte Carlo methods for stochastic convection-diffusion eigenvalue problems

We develop new multilevel Monte Carlo (MLMC) methods to estimate the expectation of the smallest eigenvalue of a stochastic convection-diffusion operator with random coefficients. The MLMC method is based on a sequence of finite element (FE) discretizations of the eigenvalue problem on a hierarchy of increasingly finer meshes. For the discretized, algebraic eigenproblems we use both the Rayleigh quotient (RQ) iteration and implicitly restarted Arnoldi (IRA), providing an analysis of the cost in each case. By studying the variance on each level and adapting classical FE error bounds to the stochastic setting, we are able to bound the total error of our MLMC estimator and provide a complexity analysis. As expected, the complexity bound for our MLMC estimator is superior to plain Monte Carlo. To improve the efficiency of the MLMC further, we exploit the hierarchy of meshes and use coarser approximations as starting values for the eigensolvers on finer ones. To improve the stability of the MLMC method for convection-dominated problems, we employ two additional strategies. First, we consider the streamline upwind Petrov--Galerkin formulation of the discrete eigenvalue problem, which allows us to start the MLMC method on coarser meshes than is possible with standard FEs. Second, we apply a homotopy method to add stability to the eigensolver for each sample. Finally, we present a multilevel quasi-Monte Carlo method that replaces Monte Carlo with a quasi-Monte Carlo (QMC) rule on each level. Due to the faster convergence of QMC, this improves the overall complexity. We provide detailed numerical results comparing our different strategies to demonstrate the practical feasibility of the MLMC method in different use cases. The results support our complexity analysis and further demonstrate the superiority over plain Monte Carlo in all cases.

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arXiv2023

Theory and construction of Quasi-Monte Carlo rules for option pricing and density estimation

In this paper we propose and analyse a method for estimating three quantities related to an Asian option: the fair price, the cumulative distribution function, and the probability density. The method involves preintegration with respect to one well chosen integration variable to obtain a smooth function of the remaining variables, followed by the application of a tailored lattice Quasi-Monte Carlo rule to integrate over the remaining variables.

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arXiv2022

Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems I: Regularity and error analysis

Stochastic PDE eigenvalue problems are useful models for quantifying the uncertainty in several applications from the physical sciences and engineering, e.g., structural vibration analysis, the criticality of a nuclear reactor or photonic crystal structures. In this paper we present a multilevel quasi-Monte Carlo (MLQMC) method for approximating the expectation of the minimal eigenvalue of an elliptic eigenvalue problem with coefficients that are given as a series expansion of countably-many stochastic parameters. The MLQMC algorithm is based on a hierarchy of discretisations of the spatial domain and truncations of the dimension of the stochastic parameter domain. To approximate the expectations, randomly shifted lattice rules are employed. This paper is primarily dedicated to giving a rigorous analysis of the error of this algorithm. A key step in the error analysis requires bounds on the mixed derivatives of the eigenfunction with respect to both the stochastic and spatial variables simultaneously. Under stronger smoothness assumptions on the parametric dependence, our analysis also extends to multilevel higher-order quasi-Monte Carlo rules. An accompanying paper [Gilbert and Scheichl, 2022], focusses on practical extensions of the MLQMC algorithm to improve efficiency, and presents numerical results.

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arXiv2022

Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems II: Efficient algorithms and numerical results

Stochastic PDE eigenvalue problems often arise in the field of uncertainty quantification, whereby one seeks to quantify the uncertainty in an eigenvalue, or its eigenfunction. In this paper we present an efficient multilevel quasi-Monte Carlo (MLQMC) algorithm for computing the expectation of the smallest eigenvalue of an elliptic eigenvalue problem with stochastic coefficients. Each sample evaluation requires the solution of a PDE eigenvalue problem, and so tackling this problem in practice is notoriously computationally difficult. We speed up the approximation of this expectation in four ways: we use a multilevel variance reduction scheme to spread the work over a hierarchy of FE meshes and truncation dimensions; we use QMC methods to efficiently compute the expectations on each level; we exploit the smoothness in parameter space and reuse the eigenvector from a nearby QMC point to reduce the number of iterations of the eigensolver; and we utilise a two-grid discretisation scheme to obtain the eigenvalue on the fine mesh with a single linear solve. The full error analysis of a basic MLQMC algorithm is given in the companion paper [Gilbert and Scheichl, 2022], and so in this paper we focus on how to further improve the efficiency and provide theoretical justification for using nearby QMC points and two-grid methods. Numerical results are presented that show the efficiency of our algorithm, and also show that the four strategies we employ are complementary.

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arXiv2022

Analysis of preintegration followed by quasi-Monte Carlo integration for distribution functions and densities

In this paper, we analyse a method for approximating the distribution function and density of a random variable that depends in a non-trivial way on a possibly high number of independent random variables, each with support on the whole real line. Starting with the integral formulations of the distribution and density, the method involves smoothing the original integrand by preintegration with respect to one suitably chosen variable, and then applying a suitable quasi-Monte Carlo (QMC) method to compute the integral of the resulting smoother function. Interpolation is then used to reconstruct the distribution or density on an interval. The preintegration technique is a special case of conditional sampling, a method that has previously been applied to a wide range of problems in statistics and computational finance. In particular, the pointwise approximation studied in this work is a specific case of the conditional density estimator previously considered in L'Ecuyer et al., arXiv:1906.04607. Our theory provides a rigorous regularity analysis of the preintegrated function, which is then used to show that the errors of the pointwise and interpolated estimators can both achieve nearly first-order convergence. Numerical results support the theory.

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arXiv2021

Preintegration is not smoothing when monotonicity fails

Preintegration is a technique for high-dimensional integration over $d$-dimensional Euclidean space, which is designed to reduce an integral whose integrand contains kinks or jumps to a $(d-1)$-dimensional integral of a smooth function. The resulting smoothness allows efficient evaluation of the $(d-1)$-dimensional integral by a Quasi-Monte Carlo or Sparse Grid method. The technique is similar to conditional sampling in statistical contexts, but the intention is different: in conditional sampling the aim is to reduce the variance, rather than to achieve smoothness. Preintegration involves an initial integration with respect to one well chosen real-valued variable. Griebel, Kuo, Sloan [Math. Comp. 82 (2013), 383--400] and Griewank, Kuo, Leövey, Sloan [J. Comput. Appl. Maths. 344 (2018), 259--274] showed that the resulting $(d-1)$-dimensional integrand is indeed smooth under appropriate conditions, including a key assumption -- the integrand of the smooth function underlying the kink or jump is strictly monotone with respect to the chosen special variable when all other variables are held fixed. The question addressed in this paper is whether this monotonicity property with respect to one well chosen variable is necessary. We show here that the answer is essentially yes, in the sense that without this property the resulting $(d-1)$-dimensional integrand is generally not smooth, having square-root or other singularities.

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arXiv2021

Equivalence between Sobolev spaces of first-order dominating mixed smoothness and unanchored ANOVA spaces on $\mathbb{R}^d$

We prove that a variant of the classical Sobolev space of first-order dominating mixed smoothness is equivalent (under a certain condition) to the unanchored ANOVA space on $\mathbb{R}^d$, for $d \geq 1$. Both spaces are Hilbert spaces involving weight functions, which determine the behaviour as different variables tend to $\pm \infty$, and weight parameters, which represent the influence of different subsets of variables. The unanchored ANOVA space on $\mathbb{R}^d$ was initially introduced by Nichols & Kuo in 2014 to analyse the error of quasi-Monte Carlo (QMC) approximations for integrals on unbounded domains; whereas the classical Sobolev space of dominating mixed smoothness was used as the setting in a series of papers by Griebel, Kuo & Sloan on the smoothing effect of integration, in an effort to develop a rigorous theory on why QMC methods work so well for certain non-smooth integrands with kinks or jumps coming from option pricing problems. In this same setting, Griewank, Kuo, Leövey & Sloan in 2018 subsequently extended these ideas by developing a practical smoothing by preintegration technique to approximate integrals of such functions with kinks or jumps. We first prove the equivalence in one dimension (itself a non-trivial task), before following a similar, but more complicated, strategy to prove the equivalence for general dimensions. As a consequence of this equivalence, we analyse applying QMC combined with a preintegration step to approximate the fair price of an Asian option, and prove that the error of such an approximation using $N$ points converges at a rate close to $1/N$.

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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

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

Hiding the weights -- CBC black box algorithms with a guaranteed error bound

The component-by-component (CBC) algorithm is a method for constructing good generating vectors for lattice rules for the efficient computation of high-dimensional integrals in the "weighted" function space setting introduced by Sloan and Woźniakowski. The "weights" that define such spaces are needed as inputs into the CBC algorithm, and so a natural question is, for a given problem how does one choose the weights? This paper introduces two new CBC algorithms which, given bounds on the mixed first derivatives of the integrand, produce a randomly shifted lattice rule with a guaranteed bound on the root-mean-square error. This alleviates the need for the user to specify the weights. We deal with "product weights" and "product and order dependent (POD) weights". Numerical tables compare the two algorithms under various assumed bounds on the mixed first derivatives, and provide rigorous upper bounds on the root-mean-square integration error.

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arXiv2018

Efficient implementations of the Multivariate Decomposition Method for approximating infinite-variate integrals

In this paper we focus on efficient implementations of the Multivariate Decomposition Method (MDM) for approximating integrals of $\infty$-variate functions. Such $\infty$-variate integrals occur for example as expectations in uncertainty quantification. Starting with the anchored decomposition $f = \sum_{\mathfrak{u}\subset\mathbb{N}} f_\mathfrak{u}$, where the sum is over all finite subsets of $\mathbb{N}$ and each $f_\mathfrak{u}$ depends only on the variables $x_j$ with $j\in\mathfrak{u}$, our MDM algorithm approximates the integral of $f$ by first truncating the sum to some `active set' and then approximating the integral of the remaining functions $f_\mathfrak{u}$ term-by-term using Smolyak or (randomized) quasi-Monte Carlo (QMC) quadratures. The anchored decomposition allows us to compute $f_\mathfrak{u}$ explicitly by function evaluations of $f$. Given the specification of the active set and theoretically derived parameters of the quadrature rules, we exploit structures in both the formula for computing $f_\mathfrak{u}$ and the quadrature rules to develop computationally efficient strategies to implement the MDM in various scenarios. In particular, we avoid repeated function evaluations at the same point. We provide numerical results for a test function to demonstrate the effectiveness of the algorithm.

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arXiv2017

Small Superposition Dimension and Active Set Construction for Multivariate Integration Under Modest Error Demand

Constructing active sets is a key part of the Multivariate Decomposition Method. An algorithm for constructing optimal or quasi-optimal active sets is proposed in the paper. By numerical experiments, it is shown that the new method can provide sets that are significantly smaller than the sets constructed by the already existing method. The experiments also show that the superposition dimension could surprisingly be very small, at most 3, when the error demand is not smaller than $10^{-3}$ and the weights decay sufficiently fast.

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