arXiv subjects

Felipe Pereira

Publications and source records attributed to Felipe Pereira.

14 recordsLinked to original sources

arXiv2026

Comment on arXiv:2106.08363v3 [math.NA], E. Abreu, A. Espirito Santo, W. Lambert, and J. Perez, Convergence of a Lagrangian--Eulerian scheme by a weak asymptotic analysis for one-dimensional hyperbolic problems

This Comment concerns arXiv:2106.08363v3 [math.NA] by E. Abreu, A. Espirito Santo, W. Lambert and J. Perez, published in Numer. Methods Partial Differential Equations 39 (2023) 2400-2443. That article builds its scheme on space-time control volumes whose lateral boundaries, called "no-flow curves", solve dsigma/dt = H(u)/u and are presented as new. We show, equation by equation, that this object is the space-time integral curve of the locally conservative Eulerian-Lagrangian method of Douglas, Pereira and Yeh [Comput. Geosci. 4 (2000) 1-40], that the no-flow region is the tube those curves bound, and that its scalar forward-tracked form appears in Mancuso, Pereira and de Souza [TEMA 8 (2007) 269-276, 277-286]. At issue is not moving a control volume, which is generic, but which curve moves it: H'(u) and H(u)/u coincide identically only for a linear flux, and only the latter makes lateral mass flux vanish. The commented article itself calls the 2000 paper the first to introduce space-time local conservation, with an integral tube bounded by integral curves, while its abstract calls the same object introduced by the authors. A full-text corpus documents the changing terminology and attribution. Papers published in 2025 and 2026 use the same construction while citing later work but not DPY. Another 2026 paper gives mixed attribution: it calls the no-flow curve an "extension" of the DPY integral curve, although its own zero-flux definition and ratio ODE show identity, and it states that diffusion and dispersion do not modify the defining vector field. It extends the model, scheme and analysis, not the continuous curve. The later discrete contributions are not challenged.

math.NA↗

arXiv2026

AlgMortar: a fully algebraic multiscale mortar preconditioner

The solution of large-scale symmetric positive definite linear systems arising from discretizations of second-order elliptic equations is challenging, especially in applications with highly heterogeneous coefficients, such as flow in porous media, which can lead to severely ill-conditioned systems. In this context, multiscale methods have recently been used to accelerate Krylov subspace methods, owing to their favorable parallel scalability. In this work, we present AlgMortar, a fully algebraic realization of the Multiscale Mortar Mixed Finite Element Method (MMMFEM). The method uses only information extracted from the fine-grid system matrix, which facilitates its implementation in existing solvers. AlgMortar uses graph partitioning to define a domain decomposition directly from the matrix graph. On each subdomain, it builds local linear systems that mimic Dirichlet problems, and couples the resulting local solutions through an algebraic interface condition that recovers the weak flux-continuity mechanism of MMMFEM. We prove that the method is well posed when the fine-grid matrix is symmetric positive definite and has nonpositive off-diagonal entries, a structure commonly arising from discretizations of elliptic problems. Numerical experiments on fine-grid linear systems arising from finite-volume discretizations of Darcy flow problems show that, when used as a preconditioner for the conjugate gradient method, the proposed approach exhibits good scalability and is competitive with state-of-the-art algebraic multigrid methods for challenging heterogeneous, high-contrast test cases, including highly irregular corner-point grids.

math.NA↗

arXiv2026

Accelerating GMRES with Matrix-Free Multiscale Robin Preconditioners

We propose a matrix-free right-preconditioning strategy for the Generalized Minimal Residual (GMRES) method based on the Multiscale Robin Coupled Method with oversampling (MRCM-OS) for the numerical solution of elliptic problems arising in subsurface flow. The resulting preconditioner is constructed through local subdomain solves with oversampling and smoothing, and can be applied without explicit assembly of the global operator. After a careful presentation of the new procedure, it is used in extensive numerical experiments. Our results demonstrate that the proposed approach substantially reduces iteration counts across a range of challenging, high-contrast subsurface flow problems. In many cases, convergence is obtained in one or two GMRES iterations when oversampling and smoothing are employed. The results indicate that combining GMRES with multiscale Robin-based operators is a promising direction for the construction of rapidly convergent preconditioning strategies.

math.NA↗

arXiv2025

Variational Autoencoder for Generating Broader-Spectrum prior Proposals in Markov chain Monte Carlo Methods

This study uses a Variational Autoencoder method to enhance the efficiency and applicability of Markov Chain Monte Carlo (McMC) methods by generating broader-spectrum prior proposals. Traditional approaches, such as the Karhunen-Lo\`eve Expansion (KLE), require previous knowledge of the covariance function, often unavailable in practical applications. The VAE framework enables a data-driven approach to flexibly capture a broader range of correlation structures in Bayesian inverse problems, particularly subsurface flow modeling. The methodology is tested on a synthetic groundwater flow inversion problem, where pressure data is used to estimate permeability fields. Numerical experiments demonstrate that the VAE-based parameterization achieves comparable accuracy to KLE when the correlation length is known and outperforms KLE when the assumed correlation length deviates from the true value. Moreover, the VAE approach significantly reduces stochastic dimensionality, improving computational efficiency. The results suggest that leveraging deep generative models in McMC methods can lead to more adaptable and efficient Bayesian inference in high-dimensional problems.

cs.LG↗

arXiv2025

A Framework to Analyze Multiscale Sampling MCMC Methods

We consider the theoretical analysis of Multiscale Sampling Methods, which are a new class of gradient-free Markov chain Monte Carlo (MCMC) methods for high dimensional inverse differential equation problems. A detailed presentation of those methods is given, including a review of each MCMC technique that they employ. Then, we propose a two-part framework to study and compare those methods. The first part identifies the new corresponding state space for the chain of random fields, and the second assesses convergence conditions on the instrumental and target distributions. Three Multiscale Sampling Methods are then analyzed using this new framework.

stat.ME↗

arXiv2025

Fast Converging Parallel Offline-Online Iterative Multiscale Mixed Methods

In this work, we build upon the recently introduced Multiscale Robin Coupled Method with Oversampling and Smoothing (MRCM-OS) to develop two highly efficient iterative multiscale methods. The MRCM-OS methodology demonstrated the ability to achieve flux error magnitudes on the order of $10^{-4}$ in a challenging industry benchmark, namely the SPE10 permeability field. The two newly proposed iterative procedures, through the construction of online informed spaces, significantly enhance the solution accuracy, reaching flux error magnitudes of order $10^{-10}$ for a reduced number of steps. The proposed methods are based on the construction of online informed spaces, which are iteratively refined to improve solution accuracy. Following an initial offline stage, where known boundary conditions are applied to construct multiscale basis functions, the informed spaces are updated through iterative procedures that utilize boundary conditions defined by the most recently computed solution variables. Two distinct approaches are introduced, each leveraging this framework to deliver efficient and accurate iterative solutions. A series of numerical simulations, conducted on the SPE10 benchmark, demonstrates the very rapid convergence of the iterative solutions. These results highlight the computational efficiency and competitiveness of the two proposed methods, which are thoroughly compared to each other and to an existing multiscale iterative method from the literature.

math.NA↗

arXiv2024

Estimating the Effective Sample Size for an inverse problem in subsurface flows

The Effective Sample Size (ESS) and Integrated Autocorrelation Time (IACT) are two popular criteria for comparing Markov Chain Monte Carlo (MCMC) algorithms and detecting their convergence. Our goal is to assess those two quantities in the context of an inverse problem in subsurface flows. We begin by presenting a review of some popular methods for their estimation, and then simulate their sample distributions on AR(1) sequences for which the exact values were known. We find that those ESS estimators may not be statistically consistent, because their variance grows linearly in the number of sample values of the MCMC. Next, we analyze the output of two distinct MCMC algorithms for the Bayesian approach to the simulation of an elliptic inverse problem. Here, the estimators cannot even agree about the order of magnitude of the ESS. Our conclusion is that the ESS has major limitations and should not be used on MCMC outputs of complex models.

stat.ME↗

arXiv2024

Multiscale Mixed Methods with Improved Accuracy: The Role of Oversampling and Smoothing

Multiscale mixed methods based on non-overlapping domain decompositions can efficiently handle the solution of significant subsurface flow problems in very heterogeneous formations of interest to the industry, especially when implemented on multi-core supercomputers. Efficiency in obtaining numerical solutions is dictated by the choice of interface spaces that are selected: the smaller the dimension of these spaces, the better, in the sense that fewer multiscale basis functions need to be computed, and smaller interface linear systems need to be solved. Thus, in solving large computational problems, it is desirable to work with piecewise constant or linear polynomials for interface spaces. However, for these choices of interface spaces, it is well known that the flux accuracy is of the order of 10-1. This study is dedicated to advancing an efficient and accurate multiscale mixed method aimed at addressing industry-relevant problems. A distinctive feature of our approach involves subdomains with overlapping regions, a departure from conventional methods. We take advantage of the overlapping decomposition to introduce a computationally highly efficient smoothing step designed to rectify small-scale errors inherent in the multiscale solution. The effectiveness of the proposed solver, which maintains a computational cost very close to its predecessors, is demonstrated through a series of numerical studies. Notably, for scenarios involving modestly sized overlapping regions and employing just a few smoothing steps, a substantial enhancement of two orders of magnitude in flux accuracy is achieved with the new approach.

math.NA↗

arXiv2023

Multiscale Sampling for the Inverse Modeling of Partial Differential Equations

We are concerned with a novel Bayesian statistical framework for the characterization of natural subsurface formations, a very challenging task. Because of the large dimension of the stochastic space of the prior distribution in the framework, typically a dimensional reduction method, such as a Karhunen-Leove expansion (KLE), needs to be applied to the prior distribution to make the characterization computationally tractable. Due to the large variability of properties of subsurface formations (such as permeability and porosity) it may be of value to localize the sampling strategy so that it can better adapt to large local variability of rock properties. In this paper, we introduce the concept of multiscale sampling to localize the search in the stochastic space. We combine the simplicity of a preconditioned Markov Chain Monte Carlo method with a new algorithm to decompose the stochastic space into orthogonal subspaces, through a one-to-one mapping of the subspaces to subdomains of a non-overlapping domain decomposition of the region of interest. The localization of the search is performed by a multiscale blocking strategy within Gibbs sampling: we apply a KL expansion locally, at the subdomain level. Within each subdomain, blocking is applied again, for the sampling of the KLE random coefficients. The effectiveness of the proposed framework is tested in the solution of inverse problems related to elliptic partial differential equations arising in porous media flows. We use multi-chain studies in a multi-GPU cluster to show that the new algorithm clearly improves the convergence rate of the preconditioned MCMC method. Moreover, we illustrate the importance of a few conditioning points to further improve the convergence of the proposed method.

math.NA↗

arXiv2021

Conditioning by Projection for the Sampling from Prior Gaussian Distributions

In this work we are interested in the (ill-posed) inverse problem for absolute permeability characterization that arises in predictive modeling of porous media flows. We consider a Bayesian statistical framework with a preconditioned Markov Chain Monte Carlo (MCMC) algorithm for the solution of the inverse problem. Reduction of uncertainty can be accomplished by incorporating measurements at sparse locations (static data) in the prior distribution. We present a new method to condition Gaussian fields (the log of permeability fields) to available sparse measurements. A truncated Karhunen-Loève expansion (KLE) is used for dimension reduction. In the proposed method the imposition of static data is made through the projection of a sample (expressed as a vector of independent, identically distributed normal random variables) onto the nullspace of a data matrix, that is defined in terms of the KLE. The numerical implementation of the proposed method is straightforward. Through numerical experiments for a model of second-order elliptic equation, we show that the proposed method in multi-chain studies converges much faster than the MCMC method without conditioning. These studies indicate the importance of conditioning in accelerating the MCMC convergence.

math.NA↗

arXiv2021

A multiscale Robin-coupled implicit method for two-phase flows in high-contrast formations

In the presence of strong heterogeneities, it is well known that the use of explicit schemes for the transport of species in a porous medium suffers from severe restrictions on the time step. This has led to the development of implicit schemes that are increasingly favoured by practitioners for their computational efficiency. The transport equation requires knowledge of the velocity field, which results from an elliptic problem (Darcy problem) that is the most expensive part of the computation. When considering large reservoirs, a cost-effective way of approximating the Darcy problems is using multiscale domain decomposition (MDD) methods. They allow for the pressure and velocity fields to be computed on coarse meshes (large scale), while detailed basis functions are defined locally, usually in parallel, in a much finer grid (small scale). In this work we adopt the Multiscale Robin Coupled Method (MRCM, [Guiraldello, et al., J. Comput. Phys., 355 (2018) pp. 1-21], [Rocha, et al., J. Comput. Phys., (2020) 109316]), which is a generalization of previous MDD methods that allows for great flexibility in the choice of interface spaces. In this article we investigate the combination of the MRCM with implicit transport schemes. A sequentially implicit strategy is proposed, with different trust-region algorithms ensuring the convergence of the transport solver. The method is assessed on several very stringent 2D two-phase problems, demonstrating its stability even for large time steps. It is also shown that the best accuracy is achieved by considering recently introduced non-polynomial interface spaces, since polynomial spaces are not optimal for high-contrast channelized permeability fields.

math.NA↗

arXiv2021

The Multiscale Perturbation Method for Two-Phase Reservoir Flow Problems

In this work we formulate and test a new procedure, the Multiscale Perturbation Method for Two-Phase Flows (MPM-2P), for the fast, accurate and naturally parallelizable numerical solution of two-phase, incompressible, immiscible displacement in porous media approximated by an operator splitting method. The proposed procedure is based on domain decomposition and combines the Multiscale Perturbation Method (MPM) with the Multiscale Robin Coupled Method (MRCM). When an update of the velocity field is called by the operator splitting algorithm, the MPM-2P may provide, depending on the magnitude of a dimensionless algorithmic parameter, an accurate and computationally inexpensive approximation for the velocity field by reusing previously computed multiscale basis functions. Thus, a full update of all multiscale basis functions required by the MRCM for the construction of a new velocity field is avoided. There are two main steps in the formulation of the MPM-2P. Initially, for each subdomain one local boundary value problem with trivial Robin boundary conditions is solved (instead of a full set of multiscale basis functions, that would be required by the MRCM). Then, the solution of an inexpensive interface problem provides the velocity field on the skeleton of the decomposition of the domain. The resulting approximation for the velocity field is obtained by downscaling. We consider challenging two-phase flow problems, with high-contrast permeability fields and water-oil finger growth in homogeneous media. Our numerical experiments show that the use of the MPM-2P gives exceptional speed-up - almost 90% of reduction in computational cost - of two-phase flow simulations. Hundreds of MRCM solutions can be replaced by inexpensive MPM-2P solutions, and water breakthrough can be simulated with very few updates of the MRCM set of multiscale basis functions.

math.NA↗

arXiv2021

Towards HPC simulations of Billion-cell Reservoirs by Multiscale Mixed Methods

A three dimensional parallel implementation of Multiscale Mixed Methods based on non-overlapping domain decomposition techniques is proposed for multi-core computers and its computational performance is assessed by means of numerical experimentation. As a prototypical method, from which many others can be derived, the Multiscale Robin Coupled Method is chosen and its implementation explained in detail. Numerical results for problems ranging from millions up to more than 2 billion computational cells in highly heterogeneous anisotropic rock formations based on the SPE10 benchmark are shown. The proposed implementation relies on direct solvers for both local problems and the interface coupling system. We find good weak and strong scalalability as compared against a state-of-the-art global fine grid solver based on Algebric Multigrid preconditioning in single and two-phase flow problems.

math.NA↗

arXiv2021

Interface spaces based on physics for multiscale mixed methods applied to flows in fractured-like porous media

It is well known that domain-decomposition-based multiscale mixed methods rely on interface spaces, defined on the skeleton of the decomposition, to connect the solution among the non-overlapping subdomains. Usual spaces, such as polynomial-based ones, cannot properly represent high-contrast channelized features such as fractures (high permeability) and barriers (low permeability) for flows in heterogeneous porous media. We propose here new interface spaces, which are based on physics, to deal with permeability fields in the simultaneous presence of fractures and barriers, accommodated respectively, by the pressure and flux spaces. Existing multiscale methods based on mixed formulations can take advantage of the proposed interface spaces, however, in order to present and test our results, we use the newly developed Multiscale Robin Coupled Method (MRCM) [Guiraldello, et al., J. Comput. Phys., 355 (2018) pp. 1-21], which generalizes most well-known multiscale mixed methods, and allows for the independent choice of the pressure and flux interface spaces. An adaptive version of the MRCM [Rocha, et al., J. Comput. Phys., 409 (2020), 109316] is considered that automatically selects the physics-based pressure space for fractured structures and the physics-based flux space for regions with barriers, resulting in a procedure with unprecedented accuracy. The features of the proposed approach are investigated through several numerical simulations of single-phase and two-phase flows, in different heterogeneous porous media. The adaptive MRCM combined with the interface spaces based on physics provides promising results for challenging problems with the simultaneous presence of fractures and barriers.

math.NA↗