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).
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 .
Multilevel Monte Carlo (MLMC) is a recently proposed variation of Monte Carlo (MC) simulation that achieves variance reduction by simulating the governing equations on a series of spatial (or temporal) grids with increasing resolution. Instead of directly employing the fine grid solutions, MLMC estimates the expectation of the quantity of interest from the coarsest grid solutions as well as differences between each two consecutive grid solutions. When the differences corresponding to finer grids become smaller, hence less variable, fewer MC realizations of finer grid solutions are needed to compute the difference expectations, thus leading to a reduction in the overall work. This paper presents an extension of MLMC, referred to as multilevel control variates (MLCV), where a low-rank approximation to the solution on each grid, obtained primarily based on coarser grid solutions, is used as a control variate for estimating the expectations involved in MLMC. Cost estimates as well as numerical examples are presented to demonstrate the advantage of this new MLCV approach over the standard MLMC when the solution of interest admits a low-rank approximation and the cost of simulating finer grids grows fast.
SUMMARY Bootstrap algebraic multigrid (BAMG) is a multigrid‐based solver for matrix equations of the form Ax = b. Its aim is to automatically determine the interpolation weights used in algebraic multigrid by locally fitting a set of test vectors that have been relaxed as solutions to the corresponding homogeneous equation, Ax = 0. This paper studies an improved form of BAMG, called relaxation‐corrected bootstrap algebraic multigrid (rBAMG), that involves adding scaled residuals of the test vectors to the least‐squares equations. The basic rBAMG scheme was introduced in an earlier paper [1] and analyzed on a simple model problem. The purpose of the current paper is to further develop this algorithm by incorporating several new critical components and to systematically study its performance on an interesting model problem from quantum chromodynamics. Whereas the earlier paper introduced a new least‐squares principle involving the residuals of the test vectors, a simple extrapolation scheme is developed here to accurately estimate the convergence factors of the evolving algebraic multigrid solver. Such a capability is essential to the effective development of a fast solver, and the approach introduced here is shown numerically to be much more effective than the conventional approach of just observing successive error reduction factors. Another component of the setup process developed here is an adaptive cycling process. This component assesses the effectiveness of the V‐cycle constructed in the initial rBAMG phase by applying it to the homogeneous equation. When poor convergence is observed, the set of test vectors is enhanced with the resulting error, enabling the subsequent least‐squares fit of interpolation to produce an improved V‐cycle. A related component is the scaling and recombination Ritz process that targets the so‐called weak approximation property in an attempt to reveal the important elements of these evolving error and test vector spaces. The aim of the numerical study documented here is to provide insight into the various design choices that arise in the development of an rBAMG algorithm. With this in mind, the results for quantum chromodynamics focus on the behavior of rBAMG in terms of the number of initial test vectors used, the number of relaxation sweeps applied to them, and the size of the target matrices. Copyright © 2012 John Wiley & Sons, Ltd.
Abstract. A significant amount of the computational time in large Monte Carlo simulations of lattice field theory is spent inverting the discrete Dirac operator. Unfortunately, traditional covariant finite difference discretizations of the Dirac operator present serious challenges for standard iterative methods. For interesting physical parameters, the discretized operator is large and ill-conditioned, and has random coefficients. More recently, adaptive algebraic multigrid (AMG) methods have been shown to be effective preconditioners for Wilson's discretization [1] [2] of the Dirac equation. This paper presents an alternate discretization of the 2D Dirac operator of Quantum Electrodynamics (QED) based on least-squares finite elements. The discretization is systematically developed and physical properties of the resulting matrix system are discussed. Finally, numerical experiments are presented that demonstrate the effectiveness of adaptive smoothed aggregation (αSA ) multigrid as a preconditioner for the discrete field equations.
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