arXiv subjects

Raul Tempone

Publications and source records attributed to Raul Tempone.

At least 37 records · Page 2Linked to original sources

arXiv2023

Uncertainty quantification in coastal aquifers using the multilevel Monte Carlo method

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.

math.NA↗

arXiv2022

Principal Component Density Estimation for Scenario Generation Using Normalizing Flows

Neural networks-based learning of the distribution of non-dispatchable renewable electricity generation from sources such as photovoltaics (PV) and wind as well as load demands has recently gained attention. Normalizing flow density models are particularly well suited for this task due to the training through direct log-likelihood maximization. However, research from the field of image generation has shown that standard normalizing flows can only learn smeared-out versions of manifold distributions. Previous works on normalizing flow-based scenario generation do not address this issue, and the smeared-out distributions result in the sampling of noisy time series. In this paper, we exploit the isometry of the principal component analysis (PCA), which sets up the normalizing flow in a lower-dimensional space while maintaining the direct and computationally efficient likelihood maximization. We train the resulting principal component flow (PCF) on data of PV and wind power generation as well as load demand in Germany in the years 2013 to 2015. The results of this investigation show that the PCF preserves critical features of the original distributions, such as the probability density and frequency behavior of the time series. The application of the PCF is, however, not limited to renewable power generation but rather extends to any data set, time series, or otherwise, which can be efficiently reduced using PCA.

cs.LG↗

arXiv2021

Generalized Parallel Tempering on Bayesian Inverse Problems

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 over more traditional sampling algorithms such as Random Walk Metropolis and (traditional) Parallel Tempering.

math.NA↗

arXiv2021

Efficient Importance Sampling for Large Sums of Independent and Identically Distributed Random Variables

We discuss estimating the probability that the sum of nonnegative independent and identically distributed random variables falls below a given threshold, i.e., $\mathbb{P}(\sum_{i=1}^{N}{X_i} \leq γ)$, via importance sampling (IS). We are particularly interested in the rare event regime when $N$ is large and/or $γ$ is small. The exponential twisting is a popular technique for similar problems that, in most cases, compares favorably to other estimators. However, it has some limitations: i) it assumes the knowledge of the moment generating function of $X_i$ and ii) sampling under the new IS PDF is not straightforward and might be expensive. The aim of this work is to propose an alternative IS PDF that approximately yields, for certain classes of distributions and in the rare event regime, at least the same performance as the exponential twisting technique and, at the same time, does not introduce serious limitations. The first class includes distributions whose probability density functions (PDFs) are asymptotically equivalent, as $x \rightarrow 0$, to $bx^{p}$, for $p>-1$ and $b>0$. For this class of distributions, the Gamma IS PDF with appropriately chosen parameters retrieves approximately, in the rare event regime corresponding to small values of $γ$ and/or large values of $N$, the same performance of the estimator based on the use of the exponential twisting technique. In the second class, we consider the Log-normal setting, whose PDF at zero vanishes faster than any polynomial, and we show numerically that a Gamma IS PDF with optimized parameters clearly outperforms the exponential twisting IS PDF. Numerical experiments validate the efficiency of the proposed estimator in delivering a highly accurate estimate in the regime of large $N$ and/or small $γ$.

stat.CO↗

arXiv2021

Analysis of a class of Multi-Level Markov Chain Monte Carlo algorithms based on Independent Metropolis-Hastings

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," \textit{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 \emph{pathological} posteriors, for which some of the previously proposed ML-MCMC algorithms fail.

math.NA↗

arXiv2020

Importance sampling for a robust and efficient multilevel Monte Carlo estimator for stochastic reaction networks

The multilevel Monte Carlo (MLMC) method for continuous-time Markov chains, first introduced by Anderson and Higham (SIAM Multiscal Model. Simul. 10(1), 2012), is a highly efficient simulation technique that can be used to estimate various statistical quantities for stochastic reaction networks (SRNs), in particular for stochastic biological systems. Unfortunately, the robustness and performance of the multilevel method can be affected by the high kurtosis, a phenomenon observed at the deep levels of MLMC, which leads to inaccurate estimates of the sample variance. In this work, we address cases where the high-kurtosis phenomenon is due to \textit{catastrophic coupling (characteristic of pure jump processes where coupled consecutive paths are identical in most of the simulations, while differences only appear in a tiny proportion) and introduce a pathwise-dependent importance sampling (IS) technique that improves the robustness and efficiency of the multilevel method. Our theoretical results, along with the conducted numerical experiments, demonstrate that our proposed method significantly reduces the kurtosis of the deep levels of MLMC, and also improves the strong convergence rate from $β=1$ for the standard case (without IS), to $β=1+δ$, where $0<δ<1$ is a user-selected parameter in our IS algorithm. Due to the complexity theorem of MLMC, and given a pre-selected tolerance, $\text{TOL}$, this results in an improvement of the complexity from $\mathcal{O}\left(\text{TOL}^{-2} \log(\text{TOL})^2\right)$ in the standard case to $\mathcal{O}\left(\text{TOL}^{-2}\right)$, which is the optimal complexity of the MLMC estimator. We achieve all these improvements with a negligible additional cost since our IS algorithm is only applied a few times across each simulated path.

math.NA↗

arXiv2020

A Wasserstein Coupled Particle Filter for Multilevel Estimation

In this paper, we consider the filtering problem for partially observed diffusions, which are regularly observed at discrete times. We are concerned with the case when one must resort to time-discretization of the diffusion process if the transition density is not available in an appropriate form. In such cases, one must resort to advanced numerical algorithms such as particle filters to consistently estimate the filter. It is also well known that the particle filter can be enhanced by considering hierarchies of discretizations and the multilevel Monte Carlo (MLMC) method, in the sense of reducing the computational effort to achieve a given mean square error (MSE). A variety of multilevel particle filters (MLPF) have been suggested in the literature, e.g., in Jasra et al., SIAM J, Numer. Anal., 55, 3068--3096. Here we introduce a new alternative that involves a resampling step based on the optimal Wasserstein coupling. We prove a central limit theorem (CLT) for the new method. On considering the asymptotic variance, we establish that in some scenarios, there is a reduction, relative to the approach in the aforementioned paper by Jasra et al., in computational effort to achieve a given MSE. These findings are confirmed in numerical examples. We also consider filtering diffusions with unstable dynamics; we empirically show that in such cases a change of measure technique seems to be required to maintain our findings.

math.NA↗

arXiv2020

Multilevel ensemble Kalman filtering for spatio-temporal processes

We design and analyse the performance of a multilevel ensemble Kalman filter method (MLEnKF) for filtering settings where the underlying state-space model is an infinite-dimensional spatio-temporal process. We consider underlying models that needs to be simulated by numerical methods, with discretization in both space and time. The multilevel Monte Carlo (MLMC) sampling strategy, achieving variance reduction through pairwise coupling of ensemble particles on neighboring resolutions, is used in the sample-moment step of MLEnKF to produce an efficient hierarchical filtering method for spatio-temporal models. Under sufficient regularity, MLEnKF is proven to be more efficient for weak approximations than EnKF, asymptotically in the large-ensemble and fine-numerical-resolution limit. Numerical examples support our theoretical findings.

math.NA↗

arXiv2020

Nesterov-aided Stochastic Gradient Methods using Laplace Approximation for Bayesian Design Optimization

Finding the best setup for experiments is the primary concern for Optimal Experimental Design (OED). Here, we focus on the Bayesian experimental design problem of finding the setup that maximizes the Shannon expected information gain. We use the stochastic gradient descent and its accelerated counterpart, which employs Nesterov's method, to solve the optimization problem in OED. We adapt a restart technique, originally proposed for the acceleration in deterministic optimization, to improve stochastic optimization methods. We combine these optimization methods with three estimators of the objective function: the double-loop Monte Carlo estimator (DLMC), the Monte Carlo estimator using the Laplace approximation for the posterior distribution (MCLA) and the double-loop Monte Carlo estimator with Laplace-based importance sampling (DLMCIS). Using stochastic gradient methods and Laplace-based estimators together allows us to use expensive and complex models, such as those that require solving partial differential equations (PDEs). From a theoretical viewpoint, we derive an explicit formula to compute the gradient estimator of the Monte Carlo methods, including MCLA and DLMCIS. From a computational standpoint, we study four examples: three based on analytical functions and one using the finite element method. The last example is an electrical impedance tomography experiment based on the complete electrode model. In these examples, the accelerated stochastic gradient descent method using MCLA converges to local maxima with up to five orders of magnitude fewer model evaluations than gradient descent with DLMC.

math.NA↗

arXiv2020

Multilevel Double Loop Monte Carlo and Stochastic Collocation Methods with Importance Sampling for Bayesian Optimal Experimental Design

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.

math.NA↗

arXiv2020

Hierarchical adaptive sparse grids and quasi Monte Carlo for option pricing under the rough Bergomi model

The rough Bergomi (rBergomi) model, introduced recently in [5], is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet remarkably fits with empirical implied volatility surfaces. In the absence of analytical European option pricing methods for the model, and due to the non-Markovian nature of the fractional driver, the prevalent option is to use the Monte Carlo (MC) simulation for pricing. Despite recent advances in the MC method in this context, pricing under the rBergomi model is still a time-consuming task. To overcome this issue, we have designed a novel, hierarchical approach, based on i) adaptive sparse grids quadrature (ASGQ), and ii) quasi-Monte Carlo (QMC). Both techniques are coupled with a Brownian bridge construction and a Richardson extrapolation on the weak error. By uncovering the available regularity, our hierarchical methods demonstrate substantial computational gains with respect to the standard MC method, when reaching a sufficiently small relative error tolerance in the price estimates across different parameter constellations, even for very small values of the Hurst parameter. Our work opens a new research direction in this field, i.e., to investigate the performance of methods other than Monte Carlo for pricing and calibrating under the rBergomi model.

q-fin.CP↗

arXiv2019

Multilevel Monte Carlo Acceleration of Seismic Wave Propagation under Uncertainty

We interpret uncertainty in a model for seismic wave propagation by treating the model parameters as random variables, and apply the Multilevel Monte Carlo (MLMC) method to reduce the cost of approximating expected values of selected, physically relevant, quantities of interest (QoI) with respect to the random variables. Targeting source inversion problems, where the source of an earthquake is inferred from ground motion recordings on the Earth's surface, we consider two QoI that measure the discrepancies between computed seismic signals and given reference signals: one QoI, $\hbox{QoI}_E$, is defined in terms of the $L^2$-misfit, which is directly related to maximum likelihood estimates of the source parameters; the other, $\hbox{QoI}_W$, is based on the quadratic Wasserstein distance between probability distributions, and represents one possible choice in a class of such misfit functions that have become increasingly popular to solve seismic inversion in recent years. We simulate seismic wave propagation, including seismic attenuation, using a publicly available code in widespread use, based on the spectral element method. Using random coefficients and deterministic initial and boundary data, we present benchmark numerical experiments with synthetic data in a two-dimensional physical domain and a one-dimensional velocity model where the assumed parameter uncertainty is motivated by realistic Earth models. Here, the computational cost of the standard Monte Carlo method was reduced by up to 97% for $\hbox{QoI}_E$, and up to 78% for $\hbox{QoI}_W$, using a relevant range of tolerances. Shifting to three-dimensional domains is straight-forward and will further increase the relative computational work reduction.

math.NA↗

arXiv2019

A Universal Splitting Estimator for the Performance Evaluation of Wireless Communications Systems

We propose a unified rare-event estimator for the performance evaluation of wireless communication systems. The estimator is derived from the well-known multilevel splitting algorithm. In its original form, the splitting algorithm cannot be applied to the simulation and estimation of time-independent problems, because splitting requires an underlying continuous-time Markov process whose trajectories can be split. We tackle this problem by embedding the static problem of interest within a continuous-time Markov process, so that the target time-independent distribution becomes the distribution of the Markov process at a given time instant. The main feature of the proposed multilevel splitting algorithm is its large scope of applicability. For illustration, we show how the same algorithm can be applied to the problem of estimating the cumulative distribution function (CDF) of sums of random variables (RVs), the CDF of partial sums of ordered RVs, the CDF of ratios of RVs, and the CDF of weighted sums of Poisson RVs. We investigate the computational efficiency of the proposed estimator via a number of simulation studies and find that it compares favorably with existing estimators.

cs.IT↗

arXiv2019

Solution of the 3D density-driven groundwater flow problem with uncertain porosity and permeability

As groundwater is an essential nutrition and irrigation resource, its pollution may lead to catastrophic consequences. Therefore, accurate modeling of the pollution of the soil and groundwater aquifer is highly important. As a model, we consider a density-driven groundwater flow problem with uncertain porosity and permeability. This problem may arise in geothermal reservoir simulation, natural saline-disposal basins, modeling of contaminant plumes, and subsurface flow. This strongly nonlinear time-dependent problem describes the convection of the two-phase flow. This liquid streams under the gravity force, building so-called "fingers". The accurate numerical solution requires fine spatial resolution with an unstructured mesh and, therefore, high computational resources. Consequently, we run the parallelized simulation toolbox \myug with the geometric multigrid solver on Shaheen II supercomputer. The parallelization is done in physical and stochastic spaces. Additionally, we demonstrate how the \myug toolbox can be run in a black-box fashion for testing different scenarios in the density-driven flow. As a benchmark, we solve the Elder-like problem in a 3D domain. For approximations in the stochastic space, we use the generalized polynomial chaos expansion. We compute the mean, variance, and exceedance probabilities of the mass fraction. As a reference solution, we use the solution, obtained from the quasi-Monte Carlo method.

math.NA↗

arXiv2019

Propagation of Uncertainties in Density-Driven Flow

Accurate modeling of contamination in subsurface flow and water aquifers is crucial for agriculture and environmental protection. Here, we demonstrate a parallel method to quantify the propagation of the uncertainty in the dispersal of pollution in subsurface flow. Specifically, we consider the density-driven flow and estimate how uncertainty from permeability and porosity propagates to the solution. We take an Elder-like problem as a numerical benchmark and we use random fields to model the limited knowledge on the porosity and permeability. We construct a low-cost generalized polynomial chaos expansion (gPC) surrogate model, where the gPC coefficients are computed by projection on sparse and full tensor grids. We parallelize both the numerical solver for the deterministic problem based on the multigrid method, and the quadrature over the parametric space

math.NA↗

arXiv2019

An Accurate Sample Rejection Estimator for the Estimation of Outage Probability of EGC Receivers

In this work, we evaluate the outage probability (OP) for L-branch equal gain combining (EGC) diversity receivers operating over fading channels, i.e. equivalently the cumulative distribution function (CDF) of the sum of the L channel envelopes. In general, closed form expressions of OP values are unobtainable. The use of Monte Carlo (MC) simulations is not considered a good alternative as it requires a large number of samples for small values of OP, making MC simulations very expensive. In this paper, we use the concept of importance sampling (IS), being known to yield accurate estimates using fewer simulation runs. Our proposed IS scheme is essentially based on sample rejection where the IS probability density function (PDF) is the truncation of the underlying PDF over the L dimensional sphere. It assumes the knowledge of the CDF of the sum of the L channel gains in a closed-form expression. Such an assumption is not restrictive since it holds for various challenging fading models. We apply our approach to the case of independent Rayleigh, correlated Rayleigh, and independent and identically distributed Rice fading models. Next, we extend our approach to the interesting scenario of generalised selection combining receivers combined with EGC under the independent Rayleigh fading environment. For each case, we prove the desired bounded relative error property. Finally, we validate these theoretical results through some selected experiments.

eess.SP↗

arXiv2019

Eficient Monte Carlo Simulation of the Left Tail of Positive Gaussian Quadratic Forms

Estimating the left tail of quadratic forms in Gaussian random vectors is of major practical importance in many applications. In this paper, we propose an efficient and robust importance sampling estimator that is endowed with the bounded relative error property. This property significantly reduces the number of simulation runs required by the proposed estimator compared to naive Monte Carlo. Thus, our importance sampling estimator is especially useful when the probability of interest is very small. Selected simulation results are presented to illustrate the efficiency of our estimator compared to naive Monte Carlo in both central and non-central cases, as well as both real and complex settings.

stat.ME↗

arXiv2018

Sparse approximation of multilinear problems with applications to kernel-based methods in UQ

We provide a framework for the sparse approximation of multilinear problems and show that several problems in uncertainty quantification fit within this framework. In these problems, the value of a multilinear map has to be approximated using approximations of different accuracy and computational work of the arguments of this map. We propose and analyze a generalized version of Smolyak's algorithm, which provides sparse approximation formulas with convergence rates that mitigate the curse of dimension that appears in multilinear approximation problems with a large number of arguments. We apply the general framework to response surface approximation and optimization under uncertainty for parametric partial differential equations using kernel-based approximation. The theoretical results are supplemented by numerical experiments.

math.NA↗