Umberto Villa

A Mixed Formulation for the Brinkman Problem

The Brinkman model is a unified law governing the flow of a viscous fluid in an inhomogeneous medium, where fractures, bubbles, or channels alternate inside a porous matrix. In this work, we explore a novel mixed formulation of the Brinkman problem based on the Hodge decomposition of the vector Laplacian. Introducing the flows vorticity as an additional unknown, this formulation allows for a uniformly stable and conforming discretization by standard finite elements (Nedelec, Raviart--Thomas, piecewise discontinuous). A priori error estimates for the discretization error in the

Multilevel techniques lead to accurate numerical upscaling and scalable robust solvers for reservoir simulation

H({\rm curl}; \Omega)-H({\rm div}; \Omega)-L^2(\Omega)

Upscaling of Mixed Finite Element Discretization Problems by the Spectral AMGe Method

norm of the solution, which are optimal with respect to the approximation properties of finite element spaces, are obtained. The theoretical results are illustrated with numerical experiments. Finally, the proposed formulation allows for a scalable block diagonal preconditioner which takes advantage of the auxiliary space algebraic multigrid solvers for

Numerical Multilevel Upscaling for Incompressible Flow in Reservoir Simulation: An Element-Based Algebraic Multigrid (AMGe) Approach

H({\rm curl})

A Multilevel, Hierarchical Sampling Technique for Spatially Correlated Random Fields

and

Space-Time CFOSLS Methods with AMGe Upscaling

H...

Scalable hierarchical PDE sampler for generating spatially correlated random fields using nonmatching meshes: Scalable hierarchical PDE sampler using nonmatching meshes

reservoir simulation DTU Orbit (25/02/2019) Multilevel techniques lead to accurate numerical upscaling and scalable robust solvers for reservoir simulation This paper demonstrates an application of element-based Algebraic Multigrid (AMGe) technique developed at LLNL (19) to the numerical upscaling and preconditioning of subsurface porous media flow problems. The upscaling results presented here are further extension of our recent work in 3. The AMGe approach is well suited for the solution of large problems coming from finite element discretizations of systems of partial differential equations. The AMGe technique from 10,9 allows for the construction of operator-dependent coarse (upscaled) models and guarantees approximation properties of the coarse velocity spaces by introducing additional degrees of freedom associated with non-planar interfaces between agglomerates. This leads to coarse spaces which maintain the specific desirable properties of the original pair of Raviart-Thomas and piecewise discontinuous polynomial spaces. These coarse spaces can be used both as an upscaling tool and as a robust and scalable solver. The methods employed in the present paper have provable O(N) scaling and are particularly well suited for modern multicore architectures, because the construction of the coarse spaces by solving many small local problems offers a high level of concurrency in the computations. Numerical experiments demonstrate the accuracy of using AMGe as an upscaling tool and comparisons are made to more traditional flow-based upscaling techniques. The efficient solution of both the original and upscaled problem is also addressed, and a specialized AMGe preconditioner for saddle point problems is compared to state-of-the-art algebraic multigrid block preconditioners. In particular, we show that for the algebraically upscaled systems, our AMGe preconditioner outperforms traditional solvers. Lastly, parallel strong scaling of a distributed memory implementation of the reservoir simulator is demonstrated.

A Data Scalable Augmented Lagrangian KKT Preconditioner for Large-Scale Inverse Problems

We propose two multilevel spectral techniques for constructing coarse discretization spaces for saddle-point problems corresponding to PDEs involving a divergence constraint, with a focus on mixed finite element discretizations of scalar self-adjoint second order elliptic equations on general unstructured grids. We use element agglomeration algebraic multigrid (AMGe), which employs coarse elements that can have nonstandard shape since they are agglomerates of fine-grid elements. The coarse basis associated with each agglomerated coarse element is constructed by solving local eigenvalue problems and local mixed finite element problems. This construction leads to stable upscaled coarse spaces and guarantees the inf-sup compatibility of the upscaled discretization. Also, the approximation properties of these upscaled spaces improve by adding more local eigenfunctions to the coarse spaces. The higher accuracy comes at the cost of additional computational effort, as the sparsity of the resulting upscaled coars...

Nonlinear multigrid solvers exploiting AMGe coarse spaces with approximation properties

We study the application of a finite element numerical upscaling technique to the incompressible two-phase porous media total velocity formulation. Specifically, an element-agglomeration-based algebraic multigrid (AMGe) technique with improved approximation properties [I. Lashuk and P. Vassilevski, Numer. Linear Algebra Appl., 19 (2012), pp. 414--426] is used, for the first time, to generate upscaled and accurate coarse systems for the reservoir simulation equations. The upscaling technique is applied to both the mixed system for velocity and pressure and to the hyperbolic transport equations, providing fully upscaled systems. By introducing additional degrees of freedom associated with nonplanar interfaces between agglomerates, the coarse velocity space has guaranteed approximation properties. The employed AMGe technique provides coarse spaces with desirable local mass conservation and stability properties analogous to the original pair of Raviart--Thomas and piecewise discontinuous polynomial spaces, re...

Hessian-based adaptive sparse quadrature for infinite-dimensional Bayesian inverse problems☆

We propose an alternative method to generate samples of a spatially correlated random field with applications to large-scale problems for forward propagation of uncertainty. A classical approach for generating these samples is the Karhunen--Loeve (KL) decomposition. However, the KL expansion requires solving a dense eigenvalue problem and is therefore computationally infeasible for large-scale problems. Sampling methods based on stochastic partial differential equations provide a highly scalable way to sample Gaussian fields, but the resulting parametrization is mesh dependent. We propose a multilevel decomposition of the stochastic field to allow for scalable, hierarchical sampling based on solving a mixed finite element formulation of a stochastic reaction-diffusion equation with a random, white noise source function. Numerical experiments are presented to demonstrate the scalability of the sampling method as well as numerical results of multilevel Monte Carlo simulations for a subsurface porous media f...