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Register a new userA Proposal of Multigrid Methods for Hermitian Positive Definite Linear Systems enjoying an order relation
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Given a multigrid procedure for linear systems with coefficient matrices $A_n$, we discuss the optimality of a related multigrid procedure with the same smoother and the same projector, when applied to properly related algebraic problems with coefficient matrices $B_n$: we assume that both $A_n$ and $B_n$ are positive definite with $A_nle vartheta B_n$, for some positive $vartheta$ independent of $n$. In this context we prove the Two-Grid method optimality. We apply this elementary strategy for designing a multigrid solution for modifications of multilevel structured (Toeplitz, circulants, Hartley, sine ($tau$ class) and cosine algebras) linear systems, in which the coefficient matrix is banded in a multilevel sense and Hermitian positive definite. In such a way, several linear systems arising from the approximation of integro-differential equations with various boundary conditions can be efficiently solved in linear time (with respect to the size of the algebraic problem). Some numerical experiments are presented and discussed, both with respect to Two-Grid and multigrid procedures.
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We present a class of reduced basis (RB) methods for the iterative solution of parametrized symmetric positive-definite (SPD) linear systems. The essential ingredients are a Galerkin projection of the underlying parametrized system onto a reduced basis space to obtain a reduced system; an adaptive greedy algorithm to efficiently determine sampling parameters and associated basis vectors; an offline-online computational procedure and a multi-fidelity approach to decouple the construction and application phases of the reduced basis method; and solution procedures to employ the reduced basis approximation as a {em stand-alone iterative solver} or as a {em preconditioner} in the conjugate gradient method. We present numerical examples to demonstrate the performance of the proposed methods in comparison with multigrid methods. Numerical results show that, when applied to solve linear systems resulting from discretizing the Poissons equations, the speed of convergence of our methods matches or surpasses that of the multigrid-preconditioned conjugate gradient method, while their computational cost per iteration is significantly smaller providing a feasible alternative when the multigrid approach is out of reach due to timing or memory constraints for large systems. Moreover, numerical results verify that this new class of reduced basis methods, when applied as a stand-alone solver or as a preconditioner, is capable of achieving the accuracy at the level of the {em truth approximation} which is far beyond the RB level.
We propose an adaptive multigrid preconditioning technology for solving linear systems arising from Discontinuous Petrov-Galerkin (DPG) discretizations. Unlike standard multigrid techniques, this preconditioner involves only trace spaces defined on the mesh skeleton, and it is suitable for adaptive hp-meshes. The key point of the construction is the integration of the iterative solver with a fully automatic and reliable mesh refinement process provided by the DPG technology. The efficacy of the solution technique is showcased with numerous examples of linear acoustics and electromagnetic simulations, including simulations in the high-frequency regime, problems which otherwise would be intractable. Finally, we analyze the one-level preconditioner (smoother) for uniform meshes and we demonstrate that theoretical estimates of the condition number of the preconditioned linear system can be derived based on well established theory for self-adjoint positive definite operators.
We consider a standard elliptic partial differential equation and propose a geometric multigrid algorithm based on Dirichlet-to-Neumann (DtN) maps for hybridized high-order finite element methods. The proposed unified approach is applicable to any locally conservative hybridized finite element method including multinumerics with different hybridized methods in different parts of the domain. For these methods, the linear system involves only the unknowns residing on the mesh skeleton, and constructing intergrid transfer operators is therefore not trivial. The key to our geometric multigrid algorithm is the physics-based energy-preserving intergrid transfer operators which depend only on the fine scale DtN maps. Thanks to these operators, we completely avoid upscaling of parameters and no information regarding subgrid physics is explicitly required on coarse meshes. Moreover, our algorithm is agglomeration-based and can straightforwardly handle unstructured meshes. We perform extensive numerical studies with hybridized mixed methods, hybridized discontinuous Galerkin method, weak Galerkin method, and a hybridized version of interior penalty discontinuous Galerkin methods on a range of elliptic problems including subsurface flow through highly heterogeneous porous media. We compare the performance of different smoothers and analyze the effect of stabilization parameters on the scalability of the multigrid algorithm.
In an iterative approach for solving linear systems with ill-conditioned, symmetric positive definite (SPD) kernel matrices, both fast matrix-vector products and fast preconditioning operations are required. Fast (linear-scaling) matrix-vector products are available by expressing the kernel matrix in an $mathcal{H}^2$ representation or an equivalent fast multipole method representation. Preconditioning such matrices, however, requires a structured matrix approximation that is more regular than the $mathcal{H}^2$ representation, such as the hierarchically semiseparable (HSS) matrix representation, which provides fast solve operations. Previously, an algorithm was presented to construct an HSS approximation to an SPD kernel matrix that is guaranteed to be SPD. However, this algorithm has quadratic cost and was only designed for recursive binary partitionings of the points defining the kernel matrix. This paper presents a general algorithm for constructing an SPD HSS approximation. Importantly, the algorithm uses the $mathcal{H}^2$ representation of the SPD matrix to reduce its computational complexity from quadratic to quasilinear. Numerical experiments illustrate how this SPD HSS approximation performs as a preconditioner for solving linear systems arising from a range of kernel functions.
Spatial symmetries and invariances play an important role in the description of materials. When modelling material properties, it is important to be able to respect such invariances. Here we discuss how to model and generate random ensembles of tensors where one wants to be able to prescribe certain classes of spatial symmetries and invariances for the whole ensemble, while at the same time demanding that the mean or expected value of the ensemble be subject to a possibly higher spatial invariance class. Our special interest is in the class of physically symmetric and positive definite tensors, as they appear often in the description of materials. As the set of positive definite tensors is not a linear space, but rather an open convex cone in the linear vector space of physically symmetric tensors, it may be advantageous to widen the notion of mean to the so-called Frechet mean, which is based on distance measures between positive definite tensors other than the usual Euclidean one. For the sake of simplicity, as well as to expose the main idea as clearly as possible, we limit ourselves here to second order tensors. It is shown how the random ensemble can be modelled and generated, with fine control of the spatial symmetry or invariance of the whole ensemble, as well as its Frechet mean, independently in its scaling and directional aspects. As an example, a 2D and a 3D model of steady-state heat conduction in a human proximal femur, a bone with high material anisotropy, is explored. It is modelled with a random thermal conductivity tensor, and the numerical results show the distinct impact of incorporating into the constitutive model different material uncertainties$-$scaling, orientation, and prescribed material symmetry$-$on the desired quantities of interest, such as temperature distribution and heat flux.
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