A Hierarchical Multilevel Markov Chain Monte Carlo Algorithm with Applications to Uncertainty Quantification in Subsurface Flow

Abstract

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).

Correction of coarse-graining errors by a two-level method: Application to the Asakura-Oosawa model

“…where we used that the denominator of (20) converges for large N f to the expectation of (19) with respect to p c , which is (Ξ f /Ξ c ). [The approximate equality in (21) is accurate up to a small correction associated with the difference between the denominator of (20) and its average, see Sec.…”

“…where I p (C) is the value of (24) obtained on the pth iteration of the computation. Finally, using (25) in (20) gives the weight factorsŵ i .…”

“…Our method is inspired by recent mathematical studies, particularly the multilevel approach, [16][17][18] which has been developed recently for applications in Bayesian statistics and uncertainty quantification. [19][20][21] (We note that the term "multi-level" has a specific mathematical meaning and should not be confused with the "multi-scale" terminology used in physics and chemistry. 10,11 ) Our method corresponds to a multi-level implementation with only two levels, so we call it a two-level (TL) method.…”

Regression-based sparse polynomial chaos for uncertainty quantification of subsurface flow models

“…Therefore, UQ for a Quantity of Interest (QoI) is usually performed numerically through multiple evaluations of expensive reservoir simulations [12]. This corresponds to significant computational resources especially when dealing with a high number of uncertain parameters [13] and for the cases where model high resolution is requirement [14]. Several lines of research have been pursued to address this challenge.…”

A practical guide to pseudo-marginal methods for computational inference in systems biology

“…For practical illustrative purposes, we focus on the fundamental method of particle marginal Metropolis-Hastings (Andrieu et al, 2010) using the bootstrap particle filter (Gordon et al, 1993) for likelihood estimation. There are many other variants to this classic approach, such as particle Gibbs sampling (Andrieu et al, 2010;Doucet et al, 2015), coupled Markov chains (Dodwell et al, 2015(Dodwell et al, , 2019, and more advanced particle filters (Doucet and Johanson, 2011) and proposal mechanisms (Botha et al, 2019;Cotter et al, 2013). It is also important to note that the pseudo-marginal approach is equally valid for Bayesian sampling strategies based on sequential Monte Carlo (Del Moral et al, 2006;Li et al, 2019;Sisson et al, 2007).…”