We construct quasi-Monte Carlo methods to approximate the expected values of linear functionals of Petrov-Galerkin discretizations of parametric operator equations which depend on a possibly infinite sequence of parameters. Such problems arise in the numerical solution of differential and integral equations with random field inputs. We analyze the regularity of the solutions with respect to the parameters in terms of the rate of decay of the fluctuations of the input field. If p ∈ (0, 1] denotes the "summability exponent" corresponding to the fluctuations in affine-parametric families of operators, then we prove that deterministic "interlaced polynomial lattice rules" of order α = 1/p +1 in s dimensions with N points can be constructed using a fast component-by-component algorithm, in O(α s N log N + α 2 s 2 N ) operations, to achieve a convergence rate of O(N −1/p ), with the implied constant independent of s. This dimension-independent convergence rate is superior to the rate O(N −1/p+1/2 ) for 2/3 ≤ p ≤ 1, which was recently established for randomly shifted lattice rules under comparable assumptions. In our analysis we use a non-standard Banach space setting and introduce "smoothness-driven product and order dependent (SPOD)" weights for which we develop a new fast CBC construction.

We develop a convergence analysis of a multi-level algorithm combining higher order quasi-Monte Carlo (QMC) quadratures with general Petrov-Galerkin discretizations of countably affine parametric operator equations of elliptic and parabolic type, extending both the multi-level first order analysis in [F.Y. Kuo, Ch. Schwab, and I.H. Sloan, Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficient (Found. Comp. Math., 2015)] and the single level higher order analysis in [J. Dick, F.Y. Kuo, Q.T. Le Gia, D. Nuyens, and Ch. Schwab, Higher order QMC Galerkin discretization for parametric operator equations (SIAM J. Numer. Anal., 2014)]. We cover, in particular, both definite as well as indefinite, strongly elliptic systems of partial differential equations (PDEs) in non-smooth domains, and discuss in detail the impact of higher order derivatives of Karhunen-Loève eigenfunctions in the parametrization of random PDE inputs on the convergence results. Based on our a-priori error bounds, concrete choices of algorithm parameters are proposed in order to achieve a prescribed accuracy under minimal computational work. Problem classes and sufficient conditions on data are identified where multi-level higher order QMC Petrov-Galerkin algorithms outperform the corresponding single level versions of these algorithms. Numerical experiments confirm the theoretical results.Key words. Quasi-Monte Carlo methods, multi-level methods, interlaced polynomial lattice rules, higher order digital nets, affine parametric operator equations, infinite dimensional quadrature, Petrov-Galerkin discretization. AMS subject classifications. 65D30, 65D32, 65N301. Introduction. The efficient numerical computation of statistical quantities for solutions of partial differential and of integral equations with random inputs is a key task in uncertainly quantification and in the sciences. In this paper, we combine the use of higher order quasi-Monte Carlo (QMC) quadrature with Petrov-Galerkin discretization in a multi-level algorithm to estimate a quantity of interest which has been expressed as an infinite dimensional integral. This paper applies the new QMC theory developed in [8] (for a single level algorithm) to the QMC Finite Element multi-level algorithm introduced in [23], to yield a potentially reduced exponent a in the cost bound of O(ε −a ), subject to a fixed error threshold ε > 0, with the constant implied in O(·) being independent of the dimension of the integration domain.The multi-level algorithm has first been introduced in [17] in the context of integral equations and was independently rediscovered in [12] in the context of simulation of stochastic differential equations. A combination of the multi-level approach with the Monte Carlo method has recently been developed for elliptic problems with random input data in [1,3,2,16,33,5].Let y := (y j ) j≥1 denote the possibly countable set of parameters from a domain U ⊆ R N , and let A(y) denote a y-parametric bounded line...

We analyze the convergence of higher order Quasi-Monte Carlo (QMC) quadratures of solution-functionals to countably-parametric, nonlinear operator equations with distributed uncertain parameters taking values in a separable Banach space X admitting an unconditional Schauder basis.Such equations arise in numerical uncertainty quantification with random field inputs. Unconditional bases of X render the random inputs and the solutions of the forward problem countably parametric, deterministic. We show that these parametric solutions belong to a class of weighted Bochner spaces of functions of countably many variables, with a particular structure of the QMC quadrature weights: up to a (problem-dependent, and possibly large) finite dimension, product weights can be used, and beyond this dimension, weighted spaces with so-called SPOD weights recently introduced in [F.Y. Kuo, Ch. Schwab, I.H. Sloan, Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal., 50, 3351-3374, 2012.] can be used to describe the solution regularity. The regularity results in the present paper extend those in [J. Dick, F.Y. Kuo, Q.T. Le Gia, D. Nuyens, Ch. Schwab, Higher order QMC (Petrov-)Galerkin discretization for parametric operator equations. SIAM J. Numer. Anal., 52, 2676Anal., 52, -2702Anal., 52, , 2014 established for affine parametric, linear operator families; they imply, in particular, efficient constructions of (sequences of) QMC quadrature methods there, which are applicable to these problem classes. We present a hybridized version of the fast component-by-component (CBC for short) construction of a certain type of higher order digital net. We prove that this construction exploits the product nature of the QMC weights with linear scaling with respect to the integration dimension up to a possibly large, problem dependent finite dimension, and the SPOD structure of the weights with quadratic scaling with respect to the weights beyond this dimension.

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We propose and analyze deterministic multilevel approximations for Bayesian inversion of operator equations with uncertain distributed parameters, subject to additive gaussian measurement data. The algorithms use a multilevel (ML) approach based on deterministic, higher order quasi-Monte Carlo (HoQMC) quadrature for approximating the high-dimensional expectations, which arise in the Bayesian estimators, and a Petrov-Galerkin (PG) method for approximating the solution to the underlying partial differential equation (PDE). This extends the previous single-level approach from [J. Dick, R. N. Gantner, Q. T. Le Gia and Ch. Schwab, Higher order Quasi-Monte Carlo integration for Bayesian Estimation. Report 2016-13, Seminar for Applied Mathematics, ETH Zürich (in review)].Compared to the single-level approach, the present convergence analysis of the multilevel method requires stronger assumptions on holomorphy and regularity of the countably-parametric uncertaintyto-observation maps of the forward problem. As in the single-level case and in the affine-parametric case analyzed in [ J. Dick, F.Y. Kuo, Q. T. Le Gia and Ch. Schwab, Multi-level higher order QMC Galerkin discretization for affine parametric operator equations. Accepted for publication in SIAM J. Numer. Anal., 2016], we obtain sufficient conditions which allow us to achieve arbitrarily high, algebraic convergence rates in terms of work, which are independent of the dimension of the parameter space. The convergence rates are limited only by the spatial regularity of the forward problem, the discretization order achieved by the Petrov Galerkin discretization, and by the sparsity of the uncertainty parametrization.We provide detailed numerical experiments for linear elliptic problems in two space dimensions, with s = 1024 parameters characterizing the uncertain input, confirming the theory and showing that the ML HoQMC algorithms outperform, in terms of error vs. computational work, both multilevel Monte Carlo (MLMC) methods and single-level (SL) HoQMC methods.