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We consider a class of density-driven flow problems. We are particularly interested in the problem of the salinization of coastal aquifers. We consider the Henry saltwater intrusion problem with uncertain porosity, permeability, and recharge parameters as a test case. The reason for the presence of uncertainties is the lack of knowledge, inaccurate measurements, and inability to measure parameters at each spatial or time location. This problem is nonlinear and time-dependent. The solution is the salt mass fraction, which is uncertain and changes in time. Uncertainties in porosity, permeability, recharge, and mass fraction are modeled using random fields. This work investigates the applicability of the well-known multilevel Monte Carlo (MLMC) method for such problems. The MLMC method can reduce the total computational and storage costs. Moreover, the MLMC method runs multiple scenarios on different spatial and time meshes and then estimates the mean value of the mass fraction. The parallelization is performed in both the physical space and stochastic space. To solve every deterministic scenario, we run the parallel multigrid solver ug4 in a black-box fashion. We use the solution obtained from the quasi-Monte Carlo method as a reference solution., Comment: 24 pages, 3 tables, 11 figures

This work employs the Birnbaum--Saunders distribution to model the fatigue life of metallic materials under cyclic loading and compares it with the normal distribution. Fatigue-limit models are fitted to three datasets of unnotched specimens of 75S-T6 aluminum alloys and carbon laminate with different loading types. A new equivalent stress definition that accounts for the effect of the experiment type is proposed. The results show that the Birnbaum--Saunders distribution consistently outperforms the normal distribution in fitting the fatigue data and provides more accurate predictions of fatigue life and survival probability.

Using quasi-Newton methods in stochastic optimization is not a trivial task. In deterministic optimization, these methods are often a common choice due to their excellent performance regardless of the problem's condition number. However, standard quasi-Newton methods fail to extract curvature information from noisy gradients in stochastic optimization. Moreover, pre-conditioning noisy gradient observations tend to amplify the noise by a factor given by the largest eigenvalue of the pre-conditioning matrix. We propose a Bayesian approach to obtain a Hessian matrix approximation for stochastic optimization that minimizes the secant equations residue while retaining the smallest eigenvalue above a specified limit. Thus, the proposed approach assists stochastic gradient descent to converge to local minima without augmenting gradient noise. The prior distribution is modeled as the exponential of the negative squared distance to the previous Hessian approximation, in the Frobenius sense, with logarithmic barriers imposing extreme eigenvalues constraints. The likelihood distribution is derived from the secant equations, i.e., the probability of obtaining the observed gradient evaluations for a given candidate Hessian approximation. We propose maximizing the log-posterior using the Newton-CG method. Numerical results on a stochastic quadratic function and a $\ell_2$ regularized logistic regression problem are presented. In all the cases tested, our approach improves the convergence of stochastic gradient descent, compensating for the overhead of solving the log-posterior maximization. In particular, pre-conditioning the stochastic gradient with the inverse of our Hessian approximation becomes more advantageous the larger the condition number of the problem is.

In the current work we present two generalizations of the Parallel Tempering algorithm, inspired by the so-called continuous-time Infinite Swapping algorithm. Such a method, found its origins in the molecular dynamics community, and can be understood as the limit case of the continuous-time Parallel Tempering algorithm, where the (random) time between swaps of states between two parallel chains goes to zero. Thus, swapping states between chains occurs continuously. In the current work, we extend this idea to the context of time-discrete Markov chains and present two Markov chain Monte Carlo algorithms that follow the same paradigm as the continuous-time infinite swapping procedure. We analyze the convergence properties of such discrete-time algorithms in terms of their spectral gap, and implement them to sample from different target distributions. Numerical results show that the proposed methods significantly improve sampling efficiency over more traditional sampling algorithms such as Random Walk Metropolis and (traditional) Parallel Tempering.

We here introduce Multi-Iteration Stochastic Optimizers, a novel class of first-order stochastic optimizers where the coefficient of variation of the mean gradient approximation, its relative statistical error, is estimated and controlled using successive control variates along the path of iterations. By exploiting the correlation between iterates, control variates may reduce the estimator's variance so that an accurate estimation of the mean gradient becomes computationally affordable. We name the estimator of the mean gradient Multi-Iteration stochastiC Estimator-MICE. In principle, MICE can be flexibly coupled with any first-order stochastic optimizer given its non-intrusive nature. Optimally, our generic algorithm decides whether to drop a particular iteration out of the control variates hierarchy, restart it, or clip it somewhere in the hierarchy, discarding previous iterations. We present a simplified study of the convergence and complexity of Multi-Iteration Stochastic Optimizers for strongly-convex and L-smooth functions. Motivated by this analysis, we provide a generic step-size choice for Stochastic Gradient Descent (SGD) with MICE, which, combined with the above characteristics, yields an efficient and robust stochastic optimizer. To assess the efficiency of MICE, we present several examples in which we use SGD-MICE and Adam-MICE. We include one example based on a stochastic adaptation of the Rosenbrock function and logistic regression training for various datasets. When compared to SGD, SAG, SAGA, SRVG, and SARAH methods, the Multi-Iteration Stochastic Optimizers reduced, without the need to tune parameters for each example, the gradient sampling cost in all cases tested, also lowering the total runtime in some cases.

An optimal experimental set-up maximizes the value of data for statistical inferences and predictions. The efficiency of strategies for finding optimal experimental set-ups is particularly important for experiments that are time-consuming or expensive to perform. For instance, in the situation when the experiments are modeled by Partial Differential Equations (PDEs), multilevel methods have been proven to dramatically reduce the computational complexity of their single-level counterparts when estimating expected values. For a setting where PDEs can model experiments, we propose two multilevel methods for estimating a popular design criterion known as the expected information gain in simulation-based Bayesian optimal experimental design. The expected information gain criterion is of a nested expectation form, and only a handful of multilevel methods have been proposed for problems of such form. We propose a Multilevel Double Loop Monte Carlo (MLDLMC), which is a multilevel strategy with Double Loop Monte Carlo (DLMC), and a Multilevel Double Loop Stochastic Collocation (MLDLSC), which performs a high-dimensional integration by deterministic quadrature on sparse grids. For both methods, the Laplace approximation is used for importance sampling that significantly reduces the computational work of estimating inner expectations. The optimal values of the method parameters are determined by minimizing the average computational work, subject to satisfying the desired error tolerance. The computational efficiencies of the methods are demonstrated by estimating the expected information gain for Bayesian inference of the fiber orientation in composite laminate materials from an electrical impedance tomography experiment. MLDLSC performs better than MLDLMC when the regularity of the quantity of interest, with respect to the additive noise and the unknown parameters, can be exploited.

This work embeds a multilevel Monte Carlo (MLMC) sampling strategy into the Monte Carlo step of the ensemble Kalman filter (EnKF), thereby yielding a multilevel ensemble Kalman filter (MLEnKF) which has provably superior asymptotic cost to a given accuracy level. The development of MLEnKF for finite-dimensional state-spaces in the work [20] is here extended to models with infinite-dimensional state- spaces in the form of spatial fields. A concrete example is given to illustrate the results., Comment: arXiv admin note: substantial text overlap with arXiv:1502.06069

In this work we consider quasi-optimal versions of the Stochastic Galerkin method for solving linear elliptic PDEs with stochastic coefficients. In particular, we consider the case of a finite number $N$ of random inputs and an analytic dependence of the solution of the PDE with respect to the parameters in a polydisc of the complex plane $C^N$. We show that a quasi-optimal approximation is given by a Galerkin projection on a weighted (anisotropic) total degree space and prove a (sub)exponential convergence rate. As a specific application we consider a thermal conduction problem with non-overlapping inclusions of random conductivity. Numerical results show the sharpness of our estimates.

In the current work we present two generalizations of the Parallel Tempering algorithm, inspired by the so-called continuous-time Infinite Swapping algorithm. Such a method, found its origins in the molecular dynamics community, and can be understood as the limit case of the continuous-time Parallel Tempering algorithm, where the (random) time between swaps of states between two parallel chains goes to zero. Thus, swapping states between chains occurs continuously. In the current work, we extend this idea to the context of time-discrete Markov chains and present two Markov chain Monte Carlo algorithms that follow the same paradigm as the continuous-time infinite swapping procedure. We analyze the convergence properties of such discrete-time algorithms in terms of their spectral gap, and implement them to sample from different target distributions. Numerical results show that the proposed methods significantly improve sampling efficiency over more traditional sampling algorithms such as Random Walk Metropolis and (traditional) Parallel Tempering.

In this work, we present, analyze, and implement a class of Multi-Level Markov chain Monte Carlo (ML-MCMC) algorithms based on independent Metropolis-Hastings proposals for Bayesian inverse problems. In this context, the likelihood function involves solving a complex differential model, which is then approximated on a sequence of increasingly accurate discretizations. The key point of this algorithm is to construct highly coupled Markov chains together with the standard Multi-level Monte Carlo argument to obtain a better cost-tolerance complexity than a single-level MCMC algorithm. Our method extends the ideas of Dodwell, et al. "A hierarchical multilevel Markov chain Monte Carlo algorithm with applications to uncertainty quantification in subsurface flow," SIAM/ASA Journal on Uncertainty Quantification 3.1 (2015): 1075-1108, to a wider range of proposal distributions. We present a thorough convergence analysis of the ML-MCMC method proposed, and show, in particular, that (i) under some mild conditions on the (independent) proposals and the family of posteriors, there exists a unique invariant probability measure for the coupled chains generated by our method, and (ii) that such coupled chains are uniformly ergodic. We also generalize the cost-tolerance theorem of Dodwell et al., to our wider class of ML-MCMC algorithms. Finally, we propose a self-tuning continuation-type ML-MCMC algorithm (C-ML-MCMC). The presented method is tested on an array of academic examples, where some of our theoretical results are numerically verified. These numerical experiments evidence how our extended ML-MCMC method is robust when targeting some pathological posteriors, for which some of the previously proposed ML-MCMC algorithms fail.