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We study an element agglomeration coarsening strategy that requires data redistribution at coarse levels when the number of coarse elements becomes smaller than the used computational units (cores). The overall procedure generates coarse elements (general unstructured unions of fine grid elements) within the framework of element-based algebraic multigrid methods (or AMGe) studied previously. The AMGe generated coarse spaces have the ability to exhibit approximation properties of the same order as the fine-level ones since by construction they contain the piecewise polynomials of the same order as the fine level ones. These approximation properties are key for the successful use of AMGe in multilevel solvers for nonlinear partial differential equations as well as for multilevel Monte Carlo (MLMC) simulations. The ability to coarsen without being constrained by the number of available cores, as described in the present paper, allows to improve the scalability of these solvers as well as in the overall MLMC method. The paper illustrates this latter fact with detailed scalability study of MLMC simulations applied to model Darcy equations with a stochastic log-normal permeability field.

A full approximation scheme (FAS) nonlinear multigrid solver for two-phase flow and transport problems driven by wells with multiple perforations is developed. It is an extension to our previous work on FAS solvers for diffusion and transport problems. The solver is applicable to discrete problems defined on unstructured grids as the coarsening algorithm is aggregation-based and algebraic. To construct coarse basis that can better capture the radial flow near wells, coarse grids in which perforated well cells are not near the coarse-element interface are desired. This is achieved by an aggregation algorithm proposed in this paper that makes use of the location of well cells in the cell-connectivity graph. Numerical examples in which the FAS solver is compared against Newton's method on benchmark problems are given. In particular, for a refined version of the SAIGUP model, the FAS solver is at least 35% faster than Newton's method for time steps with a CFL number greater than 10., Comment: arXiv admin note: text overlap with arXiv:2109.07546

This work describes the development of matrix-free GPU-accelerated solvers for high-order finite element problems in $H(\mathrm{div})$. The solvers are applicable to grad-div and Darcy problems in saddle-point formulation, and have applications in radiation diffusion and porous media flow problems, among others. Using the interpolation-histopolation basis (cf. SIAM J. Sci. Comput., 45 (2023), A675-A702, arXiv:2203.02465), efficient matrix-free preconditioners can be constructed for the $(1,1)$-block and Schur complement of the block system. With these approximations, block-preconditioned MINRES converges in a number of iterations that is independent of the mesh size and polynomial degree. The approximate Schur complement takes the form of an M-matrix graph Laplacian, and therefore can be well-preconditioned by highly scalable algebraic multigrid methods. High-performance GPU-accelerated algorithms for all components of the solution algorithm are developed, discussed, and benchmarked. Numerical results are presented on a number of challenging test cases, including the "crooked pipe" grad-div problem, the SPE10 reservoir modeling benchmark problem, and a nonlinear radiation diffusion test case., Comment: 21 pages, 10 figures

A common challenge in regression is that for many problems, the degrees of freedom required for a high-quality solution also allows for overfitting. Regularization is a class of strategies that seek to restrict the range of possible solutions so as to discourage overfitting while still enabling good solutions, and different regularization strategies impose different types of restrictions. In this paper, we present a multilevel regularization strategy that constructs and trains a hierarchy of neural networks, each of which has layers that are wider versions of the previous network's layers. We draw intuition and techniques from the field of Algebraic Multigrid (AMG), traditionally used for solving linear and nonlinear systems of equations, and specifically adapt the Full Approximation Scheme (FAS) for nonlinear systems of equations to the problem of deep learning. Training through V-cycles then encourage the neural networks to build a hierarchical understanding of the problem. We refer to this approach as \emph{multilevel-in-width} to distinguish from prior multilevel works which hierarchically alter the depth of neural networks. The resulting approach is a highly flexible framework that can be applied to a variety of layer types, which we demonstrate with both fully-connected and convolutional layers. We experimentally show with PDE regression problems that our multilevel training approach is an effective regularizer, improving the generalize performance of the neural networks studied., Comment: 24 pages, 9 figures, submitted to Numerical Linear Algebra with Applications

In this paper we propose a new approach to detect clusters in undirected graphs with attributed vertices. We incorporate structural and attribute similarities between the vertices in an augmented graph by creating additional vertices and edges as proposed in [1, 2]. The augmented graph is then embedded in a Euclidean space associated to its Laplacian and we cluster vertices via a modified K-means algorithm, using a new vector-valued distance in the embedding space. Main novelty of our method, which can be classified as an early fusion method, i.e., a method in which additional information on vertices are fused to the structure information before applying clustering, is the interpretation of attributes as new realizations of graph vertices, which can be dealt with as coordinate vectors in a related Euclidean space. This allows us to extend a scalable generalized spectral clustering procedure which substitutes graph Laplacian eigenvectors with some vectors, named algebraically smooth vectors, obtained by a linear-time complexity Algebraic MultiGrid (AMG) method. We discuss the performance of our proposed clustering method by comparison with recent literature approaches and public available results. Extensive experiments on different types of synthetic datasets and real-world attributed graphs show that our new algorithm, embedding attributes information in the clustering, outperforms structure-only-based methods, when the attributed network has an ambiguous structure. Furthermore, our new method largely outperforms the method which originally proposed the graph augmentation, showing that our embedding strategy and vector-valued distance are very effective in taking advantages from the augmented-graph representation., Comment: 32 pages, 12 figures, preprint

A nonlinear multigrid solver for two-phase flow and transport in a mixed fractional-flow velocity-pressure-saturation formulation is proposed. The solver, which is under the framework of the full approximation scheme (FAS), extends our previous work on nonlinear multigrid for heterogeneous diffusion problems. The coarse spaces in the multigrid hierarchy are constructed by first aggregating degrees of freedom, and then solving some local flow problems. The mixed formulation and the choice of coarse spaces allow us to assemble the coarse problems without visiting finer levels during the solving phase, which is crucial for the scalability of multigrid methods. Specifically, a natural generalization of the upwind flux can be evaluated directly on coarse levels using the precomputed coarse flux basis vectors. The resulting solver is applicable to problems discretized on general unstructured grids. The performance of the proposed nonlinear multigrid solver in comparison with the standard single level Newton's method is demonstrated through challenging numerical examples. It is observed that the proposed solver is robust for highly nonlinear problems and clearly outperforms Newton's method in the case of high Courant-Friedrichs-Lewy (CFL) numbers.

This paper presents the use of element-based algebraic multigrid (AMGe) hierarchies, implemented in the ParELAG (Parallel Element Agglomeration Algebraic Multigrid Upscaling and Solvers) library, to produce multilevel preconditioners and solvers for H(curl) and H(div) formulations. ParELAG constructs hierarchies of compatible nested spaces, forming an exact de Rham sequence on each level. This allows the application of hybrid smoothers on all levels and AMS (Auxiliary-space Maxwell Solver) or ADS (Auxiliary-space Divergence Solver) on the coarsest levels, obtaining complete multigrid cycles. Numerical results are presented, showing the parallel performance of the proposed methods. As a part of the exposition, this paper demonstrates some of the capabilities of ParELAG and outlines some of the components and procedures within the library.

The paper introduces a fully discrete quasi-Lagrangian finite element method for a monolithic formulation of a fluid-porous structure interaction problem. The method is second order in time and allows a standard $P_2-P_1$ (Taylor--Hood) finite element spaces for fluid problems in both fluid and porous domains. The performance of the method is illustrated on a series of numerical experiments.

The paper discusses a stabilization of a finite element method for the equations of fluid motion in a time-dependent domain. After experimental convergence analysis, the method is applied to simulate a blood flow in the right ventricle of a post-surgery patient with the transposition of the great arteries disorder. The flow domain is reconstructed from a sequence of 4D CT images. The corresponding segmentation and triangulation algorithms are also addressed in brief., Comment: to be published in Russian Journal of Numerical Analysis and Mathematical Modelling

In this work we develop a new hierarchical multilevel approach to generate Gaussian random field realizations in an algorithmically scalable manner that is well-suited to incorporate into multilevel Markov chain Monte Carlo (MCMC) algorithms. This approach builds off of other partial differential equation (PDE) approaches for generating Gaussian random field realizations; in particular, a single field realization may be formed by solving a reaction-diffusion PDE with a spatial white noise source function as the righthand side. While these approaches have been explored to accelerate forward uncertainty quantification tasks, e.g. multilevel Monte Carlo, the previous constructions are not directly applicable to multilevel MCMC frameworks which build fine scale random fields in a hierarchical fashion from coarse scale random fields. Our new hierarchical multilevel method relies on a hierarchical decomposition of the white noise source function in $L^2$ which allows us to form Gaussian random field realizations across multiple levels of discretization in a way that fits into multilevel MCMC algorithmic frameworks. After presenting our main theoretical results and numerical scaling results to showcase the utility of this new hierarchical PDE method for generating Gaussian random field realizations, this method is tested on a four-level MCMC algorithm to explore its feasibility.

This work develops a nonlinear multigrid method for diffusion problems discretized by cell-centered finite volume methods on general unstructured grids. The multigrid hierarchy is constructed algebraically using aggregation of degrees of freedom and spectral decomposition of reference linear operators associated with the aggregates. For rapid convergence, it is important that the resulting coarse spaces have good approximation properties. In our approach, the approximation quality can be directly improved by including more spectral degrees of freedom in the coarsening process. Further, by exploiting local coarsening and a piecewise-constant approximation when evaluating the nonlinear component, the coarse level problems are assembled and solved without ever re-visiting the fine level, an essential element for multigrid algorithms to achieve optimal scalability. Numerical examples comparing relative performance of the proposed nonlinear multigrid solvers with standard single-level approaches -- Picard's and Newton's methods -- are presented. Results show that the proposed solver consistently outperforms the single-level methods, both in efficiency and robustness.

We extend previously developed two-level coarsening procedures for graph Laplacian problems written in a mixed saddle point form to the fully recursive multilevel case. The resulting hierarchy of discretizations gives rise to a hierarchy of upscaled models, in the sense that they provide approximation in the natural norms (in the mixed setting). This property enables us to utilize them in three applications: (i) as an accurate reduced model, (ii) as a tool in multilevel Monte Carlo simulations (in application to finite volume discretizations), and (iii) for providing a sequence of nonlinear operators in FAS (full approximation scheme) for solving nonlinear pressure equations discretized by the conservative two-point flux approximation. We illustrate the potential of the proposed multilevel technique in all three applications on a number of popular benchmark problems used in reservoir simulation.

This paper presents and studies an approach for constructing auxiliary space preconditioners for finite element problems using a constrained nonconforming reformulation, that is based on a proposed modified version of the mortar method. The well-known mortar finite element discretization method is modified to admit a local structure, providing an element-by-element or subdomain-by-subdomain assembly property. This is achieved via the introduction of additional trace finite element spaces and degrees of freedom (unknowns) associated with the interfaces between adjacent elements or subdomains. The resulting nonconforming formulation and a reduced via static condensation Schur complement form on the interfaces are used in the construction of auxiliary space preconditioners for a given conforming finite element discretization problem. The properties of these preconditioners are studied and their performance is illustrated on model second order scalar elliptic problems utilizing high order elements., Comment: arXiv admin note: text overlap with arXiv:1910.06986

We investigate the potential of applying (D)NN ((deep) neural networks) for approximating nonlinear mappings arising in the finite element discretization of nonlinear PDEs (partial differential equations). As an application, we apply the trained DNN to replace the coarse nonlinear operator thus avoiding the need to visit the fine level discretization in order to evaluate the actions of the true coarse nonlinear operator. The feasibility of the studied approach is demonstrated in a two-level FAS (full approximation scheme) used to solve a nonlinear diffusion-reaction PDE.

We modify the well-known interior penalty finite element discretization method so that it allows for element-by-element assembly. This is possible due to the introduction of additional unknowns associated with the interfaces between neighboring elements. The resulting bilinear form, and a Schur complement (reduced) version of it, are utilized in a number of auxiliary space preconditioners for the original conforming finite element discretization problem. These preconditioners are analyzed on the fine scale and their performance is illustrated on model second order scalar elliptic problems discretized with high order elements.

This paper proposes improving the solve time of a bootstrap AMG designed previously by the authors. This is achieved by incorporating the information, set of algebraically smooth vectors, generated by the bootstrap algorithm, in a single hierarchy by using sufficiently large aggregates, and these aggregates are compositions of aggregates already built throughout the bootstrap algorithm. The modified AMG method has good convergence properties and shows significant reduction in both, memory and solve time. These savings with respect to the original bootstrap AMG are illustrated on some difficult (for standard AMG) linear systems arising from discretization of scalar and vector function elliptic partial differential equations (PDEs) in both 2d and 3d., Comment: 20 pages, 8 figures, accepted for journal publication

Algebraic multigrid (AMG) coarse spaces are commonly constructed so that they exhibit the so-called weak approximation (WAP) property which is necessary and sufficient condition for uniform two-grid convergence. This paper studies a modification of such coarse spaces so that the modified ones provide approximation in energy norm. Our modification is based on the projection in energy norm onto an orthogonal complement of original coarse space. This generally leads to dense modified coarse space matrices which is hence computationally infeasible. To remedy this, based on the fact that the projection involves inverse of a well-conditioned matrix, we use polynomials to approximate the projection and, therefore, obtain a practical, sparse modified coarse matrix and prove that the modified coarse space maintains computationally feasible approximation in energy norm. We present some numerical results for both, PDE discretization matrices as well as graph Laplacian ones, which are in accordance with our theoretical results.

We propose an unified algebraic approach for static condensation and hybridization, two popular techniques in finite element discretizations. The algebraic approach is supported by the construction of scalable solvers for problems involving H(div)-spaces discretized by conforming (Raviart-Thomas) elements of arbitrary order. We illustrate through numerical experiments the relative performance of the two (in some sense dual) techniques in comparison with a state-of-the-art parallel solver, ADS, available in the software hypre and MFEM. The superior performance of the hybridization technique is clearly demonstrated with increased benefit for higher order elements., Comment: 18 pages, 9 figures