We present an adaptive multigrid solver for application to the non-Hermitian Wilson-Dirac system of QCD. The key components leading success our proposed algorithm are use projection onto coarse grids that preserves near null space matrix together with a simplified form correction based on so-called γ5-Hermitian symmetry Dirac operator. demonstrate nearly eliminates critical slowing down in chiral limit and it has weak dependence lattice volume.

We develop an algebraic multigrid (AMG) setup scheme based on the bootstrap framework for multiscale scientific computation. Our approach uses a weighted least squares definition of interpolation, set test vectors that are computed by cycle and then improved eigensolver local residual-based adaptive relaxation process. To emphasize robustness, efficiency, flexibility individual components proposed approach, we include extensive numerical results method applied to scalar elliptic partial...

We present a new multigrid solver that is suitable for the Dirac operator in presence of disordered gauge fields. The key behind success algorithm an adaptive projection onto coarse grids preserves near null space. resulting has weak dependence on coupling and exhibits very little critical slowing down chiral limit. Results are presented Wilson-Dirac 2D U(1) Schwinger model.

We introduce a coarsening algorithm for algebraic multigrid (AMG) based on the concept of compatible relaxation (CR). The is significantly different from standard methods, most notably because it does not rely any notion strength connection. study its behavior number model problems and evaluate performance an AMG that incorporates approach. Finally, we variant CR provides sharper metric coarse-grid quality demonstrate potential with two simple examples.

We present an approach to constructing a practical coarsening algorithm and interpolation operator for the algebraic multigrid (AMG) method, tailored towards systems of partial differential equations (PDEs) with large near-kernels, such as H(curl) H(div). Our method builds on compatible relaxation (CR) ideal model within generalized AMG (GAMG) framework but introduces several modifications define PDE systems. construct through process that first coarsens nodal dual problem then coarse fine...

In this paper, we consider a classical algebraic multigrid (AMG) form of optimal interpolation that directly minimizes the two-grid convergence rate and compare it with so-called ideal weak approximation property coarse space. We study compatible relaxation type estimates for quality grid derive new sharp measure using provides guaranteed lower bound on resulting method given grid. addition, design generalized bootstrap AMG setup algorithm computes sparse to matrix. demonstrate numerically...

This paper presents estimates of the convergence rate and complexity an algebraic multilevel preconditioner based on piecewise constant coarse vector spaces applied to graph Laplacian. A bound is derived energy norm projection operator onto any space, which results in estimate two-level where level obtained by matching. The method then used establish iteration that uses scheme recursively. On structured grids, proved have $\approx (1-1/\log n)$ $O(n \log for each cycle, $n$ denotes number...

Abstract This paper analyzes a multigrid (MG) V ‐cycle scheme for solving the discretized 2D Poisson equation with corner singularities. Using weighted Sobolev spaces K m (Ω) and space decomposition based on elliptic projections, we prove that MG standard smoothers (Richardson, Jacobi, Gauss–Seidel, etc.) piecewise linear interpolation converges uniformly systems obtained by finite element discretization of graded meshes. In addition, provide numerical experiments to demonstrate optimality...

SUMMARY This paper presents a strength of connection measure for algebraic multilevel algorithms class linear systems corresponding to the graph Laplacian on general graph. The coarsening in algorithm is based partitioning subgraphs (using matching) underlying Our idea define local quality matching that follows from commutative diagram we introduce, whose maximum gives an upper bound stability (energy seminorm) orthogonal projection coarse space. As application, focus utilizing this as tool...

This work concerns the development of an algebraic multilevel method for computing state vectors Markov chains. We present efficient bootstrap multigrid (AMG) this task. In our proposed approach, we employ a eigensolver, with interpolation built using ideas based on compatible relaxation, distances, and least squares fitting test vectors. Our adaptive variational strategy computation vector given chain is then combination eigensolver associated additive preconditioned correction process....

Abstract This paper provides an overview of the main ideas driving bootstrap algebraic multigrid methodology, including compatible relaxation and distances for defining effective coarsening strategies, least squares method computing accurate prolongation operators cycles test vectors that are used in process. We review some recent research development, analysis application point to open problems these areas. Results from our previous as well new results model diffusion with highly...

Abstract In this paper, we introduce a diffuse interface model for describing the dynamics of mixtures involving multiple (two or more) phases. The coupled hydrodynamical system is derived through an energetic variational approach. total energy includes kinetic and mixing (interfacial) energies. least action principle (or virtual work) applied to derive conservative part dynamics, with focus on reversible stress tensor arising from dissipative then introduced dissipation function in law,...

This paper focuses on developing a reduction-based algebraic multigrid (AMG) method that is suitable for solving general (non)symmetric linear systems and naturally robust from pure advection to diffusion.Initial motivation comes new approach, ℓAIR (local approximate ideal restriction), was developed advection-dominated problems.Though this solver very effective in the dominated regime, its performance degrades cases where diffusion becomes dominant.This consistent with fact general, AMG...

We focus on the study of multigrid methods with aggressive coarsening and polynomial smoothers for solution linear systems corresponding to finite difference/element discretizations Laplace equation. Using local Fourier analysis we determine automatically optimal values parameters involved in defining achieve fast convergence cycles coarsening. also present numerical tests supporting theoretical results heuristic ideas. The introduce are highly parallelizable efficient algorithms structured...

In this paper we motivate, discuss the implementation and present resulting numerics for a new definition of strength connection which is based on notion algebraic distance. This distance measure, combined with compatible relaxation, used to choose suitable coarse grids accurate interpolation operators multigrid algorithms. The main tool proposed measure least squares functional defined using set relaxed test vectors. motivating application anisotropic diffusion problem, in particular...

We develop an algebraic multigrid method for solving the non-Hermitian Wilson discretization of two-dimensional Dirac equation. The proposed approach uses a bootstrap setup algorithm based on eigensolver. It computes test vectors which define least squares interpolation operators by working mainly coarse grids, leading to efficient and integrated self-learning process defining interpolation. is motivated $\gamma_5$-symmetry equation, carries over discretization. This discrete used reduce...