No abstract

We consider the application of multilevel Monte Carlo methods to elliptic PDEs with random coefficients. We focus on models of the random coefficient that lack uniform ellipticity and boundedness with respect to the random parameter, and that only have limited spatial regularity. We extend the finite element error analysis for this type of equation, carried out in [6], to more difficult problems, posed on non-smooth domains and with discontinuities in the coefficient. For this wider class of model problem, we prove convergence of the multilevel Monte Carlo algorithm for estimating any bounded, linear functional and any continuously Fréchet differentiable non-linear functional of the solution. We further improve the performance of the multilevel estimator by introducing level dependent truncations of the Karhunen-Loève expansion of the random coefficient. Numerical results complete the paper.

We consider a finite element approximation of elliptic partial differential equations with random coefficients. Such equations arise, for example, in uncertainty quantification in subsurface flow modelling. Models for random coefficients frequently used in these applications, such as log-normal random fields with exponential covariance, have only very limited spatial regularity, and lead to variational problems that lack uniform coercivity and boundedness with respect to the random parameter. In our analysis we overcome these challenges by a careful treatment of the model problem almost surely in the random parameter, which then enables us to prove uniform bounds on the finite element error in standard Bochner spaces. These new bounds can then be used to perform a rigorous analysis of the multilevel Monte Carlo method for these elliptic problems that lack full regularity and uniform coercivity and boundedness. To conclude, we give some numerical results that confirm the new bounds.

In this paper we address the problem of the prohibitively large computational cost of existing Markov chain Monte Carlo methods for large-scale applications with high dimensional parameter spaces, e.g. in uncertainty quantification in porous media flow. We propose a new multilevel Metropolis-Hastings algorithm, and give an abstract, problem dependent theorem on the cost of the new multilevel estimator based on a set of simple, verifiable assumptions. For a typical model problem in subsurface flow, we then provide a detailed analysis of these assumptions and show significant gains over the standard Metropolis-Hastings estimator. Numerical experiments confirm the analysis and demonstrate the effectiveness of the method with consistent reductions of more than an order of magnitude in the cost of the multilevel estimator over the standard Metropolis-Hastings algorithm for tolerances ε < 10 −2 . for any q < ∞ and δ > 0, where the (generic) constant C k,f,ψ,q (here and below) depends on the data k, f , ψ and on q, but is independent of any other parameters.Proof. This follows from [34, Proposition 4.1].Convergence results for functionals of the solution p can now be derived from Theorem 4.2 using a duality argument. We will here for simplicity only consider bounded, linear functionals, but the results extend to continuously Frèchet differentiable functionals (see [34, §3.2]). We make the following assumption on the functional G (cf. Assumption F1 in [34]).A2. Let G : H 1 (D) → R be linear, and suppose there exists C G ∈ R, such thatfor all δ > 0.An example of a functional which satisfies A2 is a local average of the pressure, 1 |D * | D * p dx for some D * ⊂ D. The main result on the convergence for functionals is the following. Corollary 4.3. Let the assumptions of Theorem 4.2 be satisfied, and suppose G satisfies A2. Thenfor any q < ∞ and δ > 0.Proof. This follows from [34, Corollary 4.1].Note that assumption A2 is crucial in order to get the faster convergence rates of the spatial discretisation error in Corollary 4.3. For multilevel estimators based on i.i.d. samples, it follows immediately from Corollary 4.3 that the (corresponding) assumptions M1 and M2 are satisfied, with α = 1/d + δ, α = 1/2 + δ and β = 2α, β = 2α , for any δ > 0 (see [34] for details).

Stochastic collocation methods for approximating the solution of partial differential equations with random input data (e.g., coefficients and forcing terms) suffer from the curse of dimensionality whereby increases in the stochastic dimension cause an explosion of the computational effort. We propose and analyze a multilevel version of the stochastic collocation method that, as is the case for multilevel Monte Carlo (MLMC) methods, uses hierarchies of spatial approximations to reduce the overall computational complexity. In addition, our proposed approach utilizes, for approximation in stochastic space, a sequence of multi-dimensional interpolants of increasing fidelity which can then be used for approximating statistics of the solution as well as for building highorder surrogates featuring faster convergence rates. A rigorous convergence and computational cost analysis of the new multilevel stochastic collocation method is provided, demonstrating its advantages compared to standard single-level stochastic collocation approximations as well as MLMC methods. Numerical results are provided that illustrate the theory and the effectiveness of the new multilevel method.