Linear Solvers

Introduction

Any physics solver relying on standard finite element and finite volume techniques requires the solution of algebraic linear systems, which are obtained upon linearization and discretization of the governing equations, of the form:

$\mathsf{A} \mathsf{x} = \mathsf{b}$

where $\mathsf{A}$ is the square sparse matrix, $\mathsf{x}$ the solution vector, and $\mathsf{b}$ the right-hand side. For example, in a classical linear elastostatics problem $\mathsf{A}$ is the stiffness matrix, and $\mathsf{x}$ and $\mathsf{b}$ are the displacement and nodal force vectors, respectively.

This solution stage represents the most computationally expensive portion of a typical simulation. Solution algorithms generally belong to two families of methods: direct methods and iterative methods. In GEOS both options are made available wrapping around well-established open-source linear algebra libraries, namely HYPRE, PETSc, SuperLU, and Trilinos.

Direct methods

The major advantages are their reliability, robustness, and ease of use. However, they have large memory requirements and exhibit poor scalability. Direct methods should be used in a prototyping stage, for example when developing a new formulation or algorithm, when the dimension of the problem, namely the size of matrix $\mathsf{A}$ , is small. Irrespective of the selected direct solver implementation, three stages can be idenitified:

Setup Stage: the matrix is first analyzed and then factorized
Solve Stage: the solution to the linear systems involving the factorized matrix is computed
Finalize Stage: the systems involving the factorized matrix have been solved and the direct solver lifetime ends

The default option in GEOS relies on SuperLU, a general purpose library for the direct solution of large, sparse, nonsymmetric systems of linear equations, that is called taking advantage of the interface provided in HYPRE.

Iterative methods

As the problem size (number of computational cells) increases, global iterative solution strategies are the method of choice—typically nonsymmetric Krylov solvers. Because of the possible poor conditioning of $\mathsf{A}$ , preconditioning is essential to solve such systems efficiently. ‘’Preconditioning is simply a means of transforming the original linear system into one which has the same solution, but which is likely to be easier to solve with an iterative solver’’ [Saad (2003)].

The design of a robust and efficient preconditioner is based on a trade-off between two competing objectives:

Robustness: reducing the number of iterations needed by the preconditioned solver to achieve convergence;
Efficiency: limiting the time required to construct and apply the preconditioner.

Assuming a preconditioning matrix $\mathsf{M}$ is available, three standard approaches are used to apply the preconditioner:

Left preconditioning: the preconditioned system is $\mathsf{M}^{-1} \mathsf{A} \mathsf{x} = \mathsf{M}^{-1} \mathsf{b}$
Right preconditioning: the preconditioned system is $\mathsf{A} \mathsf{M}^{-1} \mathsf{y} = \mathsf{b}$ , with $\mathsf{x} = \mathsf{M}^{-1} \mathsf{y}$
Split preconditioning: the preconditioned system is $\mathsf{M}^{-1}_L \mathsf{A} \mathsf{M}^{-1}_R \mathsf{y} = \mathsf{M}^{-1}_L \mathsf{b}$ , with $\mathsf{x} = \mathsf{M}^{-1}_R \mathsf{y}$

Summary

The following table summarizes the available input parameters for the linear solver.

XML Element: LinearSolverParameters

Name	Type	Default	Description
adaptiveExponent	real64	1	Exponent parameter for adaptive method
adaptiveGamma	real64	0.1	Gamma parameter for adaptive method
amgAggressiveCoarseningLevels	integer	0	AMG number of levels for aggressive coarsening
amgAggressiveCoarseningPaths	integer	1	AMG number of paths for aggressive coarsening
amgAggressiveInterpType	geos_LinearSolverParameters_AMG_AggInterpType	multipass	AMG aggressive interpolation algorithm. Available options are: `default\\|extendedIStage2\\|standardStage2\\|extendedStage2\\|multipass\\|modifiedExtended\\|modifiedExtendedI\\|modifiedExtendedE\\|modifiedMultipass`
amgCoarseSolver	geos_LinearSolverParameters_AMG_CoarseType	direct	AMG coarsest level solver/smoother type. Available options are: `default\\|jacobi\\|l1jacobi\\|fgs\\|sgs\\|l1sgs\\|chebyshev\\|direct\\|bgs\\|gsElimWPivoting\\|gsElimWInverse`
amgCoarseningType	geos_LinearSolverParameters_AMG_CoarseningType	HMIS	AMG coarsening algorithm. Available options are: `default\\|CLJP\\|RugeStueben\\|Falgout\\|PMIS\\|HMIS`
amgInterpolationMaxNonZeros	integer	4	AMG interpolation maximum number of nonzeros per row
amgInterpolationType	geos_LinearSolverParameters_AMG_InterpType	extendedI	AMG interpolation algorithm. Available options are: `default\\|modifiedClassical\\|direct\\|multipass\\|extendedI\\|standard\\|extended\\|directBAMG\\|modifiedExtended\\|modifiedExtendedI\\|modifiedExtendedE`
amgNullSpaceType	geos_LinearSolverParameters_AMG_NullSpaceType	constantModes	AMG near null space approximation. Available options are:`constantModes\\|rigidBodyModes`
amgNumFunctions	integer	1	AMG number of functions
amgNumSweeps	integer	1	AMG smoother sweeps
amgRelaxWeight	real64	1	AMG relaxation factor for the smoother
amgSeparateComponents	integer	0	AMG apply separate component filter for multi-variable problems
amgSmootherType	geos_LinearSolverParameters_AMG_SmootherType	l1sgs	AMG smoother type. Available options are: `default\\|jacobi\\|l1jacobi\\|fgs\\|bgs\\|sgs\\|l1sgs\\|chebyshev\\|ilu0\\|ilut\\|ic0\\|ict`
amgThreshold	real64	0	AMG strength-of-connection threshold
directCheckResidual	integer	0	Whether to check the linear system solution residual
directColPerm	geos_LinearSolverParameters_Direct_ColPerm	metis	How to permute the columns. Available options are: `none\\|MMD_AtplusA\\|MMD_AtA\\|colAMD\\|metis\\|parmetis`
directEquil	integer	1	Whether to scale the rows and columns of the matrix
directIterRef	integer	1	Whether to perform iterative refinement
directParallel	integer	1	Whether to use a parallel solver (instead of a serial one)
directReplTinyPivot	integer	1	Whether to replace tiny pivots by sqrt(epsilon)*norm(A)
directRowPerm	geos_LinearSolverParameters_Direct_RowPerm	mc64	How to permute the rows. Available options are: `none\\|mc64`
iluFill	integer	0	ILU(K) fill factor
iluThreshold	real64	0	ILU(T) threshold factor
krylovAdaptiveTol	integer	0	Use Eisenstat-Walker adaptive linear tolerance
krylovMaxIter	integer	200	Maximum iterations allowed for an iterative solver
krylovMaxRestart	integer	200	Maximum iterations before restart (GMRES only)
krylovStrongestTol	real64	1e-08	Strongest-allowed tolerance for adaptive method
krylovTol	real64	1e-06	Relative convergence tolerance of the iterative method If the method converges, the iterative solution $\mathsf{x}_k$ is such that the relative residual norm satisfies: $\left\lVert \mathsf{b} - \mathsf{A} \mathsf{x}_k \right\rVert_2$ < `krylovTol` * $\left\lVert\mathsf{b}\right\rVert_2$
krylovWeakestTol	real64	0.001	Weakest-allowed tolerance for adaptive method
logLevel	integer	0	Log level
preconditionerType	geos_LinearSolverParameters_PreconditionerType	iluk	Preconditioner type. Available options are: `none\\|jacobi\\|l1jacobi\\|fgs\\|sgs\\|l1sgs\\|chebyshev\\|iluk\\|ilut\\|icc\\|ict\\|amg\\|mgr\\|block\\|direct\\|bgs`
solverType	geos_LinearSolverParameters_SolverType	direct	Linear solver type. Available options are: `direct\\|cg\\|gmres\\|fgmres\\|bicgstab\\|preconditioner`
stopIfError	integer	1	Whether to stop the simulation if the linear solver reports an error

Preconditioner descriptions

This section provides a brief description of the available preconditioners.

None: no preconditioning is used, i.e., $\mathsf{M}^{-1} = \mathsf{I}$ .
Jacobi: diagonal scaling preconditioning, with $\mathsf{M}^{-1} = \mathsf{D}^{-1}$ , with $\mathsf{D}$ the matrix diagonal. Further details can be found in:
ILUK: incomplete LU factorization with fill level k of the original matrix: $\mathsf{M}^{-1} = \mathsf{U}^{-1} \mathsf{L}^{-1}$ . Further details can be found in:
ILUT: a dual threshold incomplete LU factorization: $\mathsf{M}^{-1} = \mathsf{U}^{-1} \mathsf{L}^{-1}$ . Further details can be found in:
- HYPRE documentation,
- not yet available through PETSc interface,
- Trilinos documentation.
ICC: incomplete Cholesky factorization of a symmetric positive definite matrix: $\mathsf{M}^{-1} = \mathsf{L}^{-T} \mathsf{L}^{-1}$ . Further details can be found in:
- not yet available through hypre interface,
- PETSc documentation,
- Trilinos documentation.
AMG: algebraic multigrid (can be classical or aggregation-based according to the specific package). Further details can be found in:
MGR: multigrid reduction. Available through hypre interface only. Further details can be found in MGR documentation, also see section below.
Block: custom preconditioner designed for a 2 x 2 block matrix.

HYPRE MGR Preconditioner

MGR stands for multigrid reduction, a multigrid method that uses the interpolation, restriction operators, and the Galerkin triple product, to reduce a linear system to a smaller one, similar to a Schur complement approach. As such, it is designed to target block linear systems resulting from discretizations of multiphysics problems. GEOS uses MGR through an implementation in HYPRE. More information regarding MGR can be found here. Currently, MGR strategies are implemented for hydraulic fracturing, singlephase and multiphase poromechanics, singlephase poromechanics with fractures, compositional flow with and without wells. More multiphysics solvers with MGR will be enabled in the future.

To use MGR for a specific block system, several components need to be specified.

The number of reduction levels and the coarse points (corresponding to fields) for each level. For example, for single-phase hydraulic fracturing, there are two fields, i.e. displacement and fluid pressure, a two-level MGR strategy can be used with the fluid pressure being the coarse degrees of freedom.
Interpolation/restriction operators and the coarse-grid computation strategy. A simple but effective strategy is to use Jacobi diagonalization for interpolation and injection for restriction. For most cases, a Galerkin coarse grid strategy can be used, but for special cases such as poroelastic, a non-Galerkin approach is preferable.
Global smoother. Depending on the problem, a global relaxation step could be beneficial. Some options include ILU(k), (block) Jacobi, (block) Gauss-Seidel.
Solvers for F-relaxation and coarse-grid correction. These solvers should be chosen carefully for MGR to be effective. The choice of these solvers should correspond to the properties of the blocks specified by the C- and F-points. For example, if the $\mathsf{A}_{FF}$ block is hyperbolic, a Jacobi smoother is sufficient while for an elliptic operator an AMG V-cycle might be required. For the single-phase hydraulic fracturing case, an AMG V-cycle is needed for both F-relaxation and coarse-grid correction.

Note that these are only general guidelines for designing a working MGR recipe. For complicated multiphysics problems, experimentation with different numbers of levels, choices of C- and F-points, and smoothers/solvers, etc., is typically needed to find the best strategy. Currently, these options are only available to developers. We are working on exposing these functionalities to the users in future releases.

Block preconditioner

This framework allows the user to design a block preconditioner for a 2 x 2 block matrix. The key component is the Schur complement $\mathsf{S} = \mathsf{A}_{11} - \mathsf{A}_{10} \mathsf{\widetilde{A}}_{00}^{-1} \mathsf{A}_{01}$ computation, that requires an approximation of the leading block. Currently, available options for $\mathsf{\widetilde{A}}_{00}^{-1}$ are:

diagonal with diagonal values (essentially, a Jacobi preconditioner);
diagonal with row sums as values (e.g., used for CPR-like preconditioners).

Once the Schur complement is computed, to properly define the block preconditioner we need:

the preconditioner for $\mathsf{A}_{00}$ (any of the above listed single-matrix preconditioner);
the preconditioner for $\mathsf{S}$ (any of the above listed single-matrix preconditioner);
the application strategy. This can be:
- diagonal: none of the coupling terms is used;
- upper triangular: only the upper triangular coupling term is used;
- lower-upper triangular: both coupling terms are used.

Moreover, a block scaling is available. Feasible options are:

none: keep the original scaling;
Frobenius norm: equilibrate Frobenius norm of the diagonal blocks;
user provided.

Adaptive tolerance

This feature is available for iterative solvers and can be enabled using krylovAdaptiveTol flag in LinearSolverParameters. It follows the Eisenstat-Walker inexact Newton approach described in [Eisenstat and Walker 1996]. The key idea is to relax the linear solver tolerance at the beginning of the nonlinear iterations loop and tighten it when getting closer to the final solution. The initial tolerance is defined by krylovWeakestTol and starting from second nonlinear iteration the tolerance is chosen using the following steps:

compute the current to previous nonlinear norm ratio: $\mathsf{nr} = \mathsf{min}( \mathsf{norm}^{curr} / \mathsf{norm}^{prev}, 1.0 )$
estimate the new linear solver tolerance: $\mathsf{tol}_{new} = \mathsf{\gamma} \cdot \mathsf{nr}^{ax}$
compute a safeguard to avoid too sharp tolerance reduction: $\mathsf{tol}_{alt} = \mathsf{tol}_{old}^{2}$ (the bound is the quadratic reduction with respect to the previous tolerance value)
apply safeguards and compute the final tolerance: $\mathsf{tol} = \mathsf{max}( \mathsf{tol}_{new}, \mathsf{tol}_{alt} )$ , $\mathsf{tol} = \mathsf{min}( \mathsf{tol}_{max}, \mathsf{max}( \mathsf{tol}_{min}, \mathsf{tol} ) )$

Here $\mathsf{\gamma}$ is the forcing term, $ax$ is the adaptivity exponent, $\mathsf{tol}_{min}$ and $\mathsf{tol}_{max}$ are prescribed tolerance bounds (defined by krylovStrongestTol and krylovWeakestTol, respectively).