Block Diagonal Preconditioner Research Articles

Efficient and reliable divergence-conforming methods for an elasticity-poroelasticity interface problem

We present a finite element discretization to model the interaction between a poroelastic structure and an elastic medium. The consolidation problem considers fully coupled deformations across an interface, ensuring continuity of displacement and total traction, as well as no-flux for the fluid phase. Our formulation of the poroelasticity equations incorporates displacement, fluid pressure, and total pressure, while the elasticity equations adopt a displacement-pressure formulation. Notably, the transmission conditions at the interface are enforced without the need for Lagrange multipliers. We demonstrate the stability and convergence of the divergence-conforming finite element method across various polynomial degrees. The a priori error bounds remain robust, even when considering large variations in intricate model parameters such as Lamé constants, permeability, and storativity coefficient. To enhance computational efficiency and reliability, we develop residual-based a posteriori error estimators that are independent of the aforementioned coefficients. Additionally, we devise parameter-robust and optimal block diagonal preconditioners. Through numerical examples, including adaptive scenarios, we illustrate the scheme's properties such as convergence and parameter robustness.

Robust block diagonal preconditioners for poroelastic problems with strongly heterogeneous material

AbstractThis paper focuses on the analysis and the solution of the saddle‐point problem arising from a three‐field formulation of Biot's model of poroelasticity, discretized in time by the implicit Euler method. A block diagonal‐preconditioner, based on the Schur complement, is analyzed on a functional level and compared with two other block‐diagonal preconditioners having a similar structure. The problem is discretized in space using mixed finite elements and solved with appropriate iterative solvers, incorporating the investigated preconditioners. The solvers are tested on numerical examples inspired by geotechnical practice, with particular attention devoted to the solvers' robustness concerning strong heterogeneity in permeability.

Rotated block diagonal preconditioners for Navier-Stokes control problems

Fluid flow control problems play a crucial role in various industrial applications. The optimal control of the Navier-Stokes equations poses a significant challenge. By employing Oseen's approximation to linearize the problem, we encounter a series of large sparse structured linear systems. These coefficient matrices exhibit a two-by-two block structure with square blocks. Leveraging this block structure, we exploit matrix splitting preconditioners. Theoretical analysis indicates that the real parts of all eigenvalues of preconditioned matrix are equal to 1/2. A large number of eigenvalues are clustered in (1±i)/2. The imaginary part of the eigenvalues is bounded explicitly. The eigenvalue distribution predicts the fast convergence of Krylov subspace methods. To avoid solving the saddle point subsystems in the preconditioning procedure, we propose a practical variant of the preconditioner. The theoretical analysis demonstrates that if the approximation is sufficiently close to Schur complement, the eigenvalues of the modified preconditioned matrix will lie within a circle centered at the original eigenvalues with a radius less than 1. This implies that the modified preconditioner exhibits excellent performance in terms of preconditioning. The numerical experiments demonstrate that the generalized minimal residual method, when combined with the proposed preconditioners, proves to be an efficient and effective solution for Oseen's control optimization with a variable viscosity coefficient. The number of iterations required by the preconditioned methods remains unaffected by the mesh size used in finite element discretization and only marginally depends on the regularization parameter.

On symmetric positive definite preconditioners for multiple saddle-point systems

Abstract We consider symmetric positive definite preconditioners for multiple saddle-point systems of block tridiagonal form, which can be applied within the Minres algorithm. We describe such a preconditioner for which the preconditioned matrix has only two distinct eigenvalues, $1$ and $-1$, when the preconditioner is applied exactly. We discuss the relative merits of such an approach compared to a more widely studied block diagonal preconditioner, specify the computational work associated with applying the new preconditioner inexactly, and survey a number of theoretical results for the block diagonal case. Numerical results validate our theoretical findings.

EMFEM: A parallel 3D modeling code for frequency-domain electromagnetic method using goal-oriented adaptive finite element method

We have developed an open-source parallel 3D forward modeling code called EMFEM for the frequency-domain electromagnetic method (FDEM). This code is based on the adaptive finite element method and uses the total-field approach to solve the time-harmonic Maxwell equations. The use of tetrahedral meshes enables modeling of problems with complex geometry and topography. We utilize an iterative solver and an optimal block-diagonal preconditioner and auxiliary Maxwell solver to accelerate the convergence rate. We also use goal-oriented error estimation to adaptively refine the mesh and improve the accuracy of the solution. In addition, we develop a parallel algorithm based on the mesh partitioning approach. The effectiveness of the adaptive mesh refinement is demonstrated through a 1D layered model for marine controlled-source electromagnetic (CSEM) problems. We further present two 3D models for CSEM and magnetotelluric (MT) problems to study the scalability of the parallel algorithm, and the accuracy of the solutions is cross-validated with other open-source codes. The code is implemented in C++ and follows the modular design principle, making it easy to extend and maintain. The code is capable of modeling large-scale 3D problems and utilizing high-performance computing platforms. Furthermore, as the code is freely available, it can serve as a cross-validation tool for other 3D modeling codes and can also be utilized to implement 3D inversion routines.

On Preconditioners of the FFT-JVIE for Inhomogeneous Dielectric Objects

An improved block-diagonal preconditioner (IBDP) is developed for the FFT accelerated the volume integral equation formulation based on equivalent volumetric currents (FFT-JVIE). A comparative study on the realistic models is carried out among the IBDP, BDP, the previously developed analytical preconditioner (AnP) and the circulant-based preconditioners (CBP).

3-D anisotropic modelling of geomagnetic depth sounding based on unstructured edge-based finite-element method

SUMMARY Geomagnetic depth sounding (GDS) is a geophysical electromagnetic (EM) method that studies the deep structure and composition of the Earth by using long-period EM signals from geomagnetic observatories and satellites. In this paper, a 3-D anisotropic GDS modelling algorithm is developed. The curl–curl equation is discretized using the edge-based finite-element method on unstructured tetrahedral grids. In order to solve the computationally demanding problem of EM modelling on a global scale, the complex linear system is first separated into the equivalent real linear systems and then the real system is iteratively solved by the flexible generalized minimum residual method with a block diagonal pre-conditioner. This will greatly reduce the condition number of the linear system and thus speed up the solution process. We verify the accuracy of the proposed algorithm by comparing our results with the existing methods. After that, we design a subduction zone model to simulate the EM responses under isotropic and anisotropic environments, respectively. The numerical results show the high efficiency of the proposed algorithm and the response differences between isotropic and anisotropic models. This research can provide theoretical and technical support for the high-accuracy and efficient inversion of GDS data for the geo-dynamic study.

An efficient hybrid direct-iterative solver for three-dimensional higher-order edge-based finite element simulation for magnetotelluric data in anisotropic media

This work investigates the use of the high-order finite element method (FEM) for 3D magnetotelluric (MT) modeling in anisotropic media. The high-order FE is more accurate but also more difficult than the linear elements. Numerically, the resulting linear system is larger and denser than the linear case with the same grid. In 3D cases, finite element discretization of Maxwell's equations often leads to large but sparse linear systems of equations, requiring effective methods to solve them. However, for the domain-frequency Maxwell's equations, the discrete rotor operator with a large kernel hampers the convergence of common iterative solvers. Robust and effective solvers are needed to solve 3D electromagnetic (EM) problems. In this study, we presented a robust direct-iterative solver based on a block-diagonal preconditioner. The proposed solver is efficient for high-order FE modeling with arbitrary anisotropy. We implemented the algorithm in a high-level Julia language, which is easy to read, maintain, and extend. Numerical examples show that the high-order elements produce a more accurate solution on a coarse grid than the linear case with the same grid. Compared with the direct solver, the hybrid direct-iterative solver reduces the computation time and memory usage. In addition, the solver only needs a few iterations to converge compared with the traditional BCISGTAB solver with an SSOR preconditioner. Our modeling results suggested the FGMRES solver with the block-diagonal preconditioner is quite effective for 3D MT high-order FE modeling.

A Wave Equation-Based Hybridizable Discontinuous Galerkin–Robin Transmission Condition Algorithm for Electromagnetic Problems Analyzing

In this work, a wave-equation-based discontinuous Galerkin (DG) method hybridized with the Robin transmission condition (DG-RTC) is developed to solve the frequency-domain electromagnetic (EM) problems. The proposed DG method directly discretizes the vector electric field wave equation in each subdomain, and subsequently, a term named numerical flux is introduced at the subdomain interfaces to connect the solutions between neighboring subdomains. However, the numerical flux depends not only on the electric field <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {E}$ </tex-math></inline-formula> but also on the magnetic field <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {H}$ </tex-math></inline-formula> residing over the interface. Thereby, another equation is essential for solving <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {E}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {H}$ </tex-math></inline-formula> simultaneously. Realizing that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {H}$ </tex-math></inline-formula> only situates at the interface of adjacent subdomains, it is thus desired that the auxiliary equation also merely involves <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {H}$ </tex-math></inline-formula> at the interfaces. To reach this aim, at the subdomain interfaces, a Robin transmission condition (RTC) based on the tangential continuity of EM fields across the interface is introduced as the second equation. With this proposed DG-RTC algorithm, only <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {E}$ </tex-math></inline-formula> is a volume variable, while <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {H}$ </tex-math></inline-formula> is a two-dimensional (2-D) one, and the degrees of freedom (DoFs) are thus greatly reduced compared with the traditional DG method. On the other hand, in the frequency domain, the established subdomain matrix equations are implicitly coupled with each other, resulting in a global matrix system. Directly solving it is not yet cheap. To alleviate the computational cost, a finite-element tearing and interconnecting (FETI)-like approach resorts. With the block-diagonal preconditioner and a restriction operator, the global matrix is decomposed into a surface matrix equation and several local matrix equations pertinent to each subdomain. In this way, a direct solver can be applied to solve these small matrix equations efficiently. To validate the proposed DG method in solving various EM problems, several representative examples are presented.

A Cartesian FMM-accelerated Galerkin boundary integral Poisson-Boltzmann solver

The Poisson-Boltzmann model is an effective and popular approach for modeling solvated biomolecules in continuum solvent with dissolved electrolytes. In this paper, we report our recent work in developing a Galerkin boundary integral method for solving the linear Poisson-Boltzmann (PB) equation. The solver has combined advantages in accuracy, efficiency, and memory usage as it applies a well-posed boundary integral formulation to circumvent many numerical difficulties associated with the PB equation and uses an O(N) Cartesian Fast Multipole Method (FMM) to accelerate the GMRES iteration. In addition, special numerical treatments such as adaptive FMM order, block diagonal preconditioners, Galerkin discretization, and Duffy's transformation are combined to improve the performance of the solver, which is validated on benchmark Kirkwood's sphere and a series of testing proteins.

Eigenvalue bounds for double saddle-point systems

Abstract We derive bounds on the eigenvalues of a generic form of double saddle-point matrices. The bounds are expressed in terms of extremal eigenvalues and singular values of the associated block matrices. Inertia and algebraic multiplicity of eigenvalues are considered as well. The analysis includes bounds for preconditioned matrices based on block diagonal preconditioners using Schur complements, and it is shown that in this case the eigenvalues are clustered within a few intervals bounded away from zero. Analysis for approximations of Schur complements is included. Some numerical experiments validate our analytical findings.

35 Young Scholar Award Talk: Computing Strategies for National Beef Cattle Evaluations

Abstract The constantly growing size of the national beef cattle evaluations datasets creates computational challenges to estimate breeding values and their accuracies under the single-step methodology timely and in a robust way. This study aims to present strategies for making weekly national single-step beef cattle evaluations computationally feasible. The process for estimating breeding values in single-step can be divided into two parts. In the first part, hereafter called genomic setup, the genotype file is read, and the inverse of the genomic relationship matrix is calculated using the Algorithm for Proven and Young (APY). The genotype file is read in a stream-unformatted format, and the markers are stored bit-wise. A residual polygenic effect can be fitted with an algorithm whose computing cost is constant with the number of genotyped animals. The calculation of is done using parallel computing and optimized matrix algebra procedures. The second part estimates breeding values by iteration on data with a block diagonal preconditioner. Finally, the accuracy of the estimated breeding values is calculated by combining the pedigree and genomic information in terms of effective daughter contributions. For an Angus dataset with 10 million animals in the pedigree and 500,000 genotypes, the genomic setup took 4 hours. With the same dataset, solutions for a three-trait model with maternal effects including birth weight, weaning weight, and post-weaning gain were obtained after 12 hours. The approximation of the accuracies of breeding values took less than one hour. Our results show that weekly national beef cattle evaluations are computationally feasible.

An all‐at‐once algebraic multigrid method for finite element discretizations of Stokes problem

AbstractWe consider numerical solution of finite element discretizations of the Stokes problem. We focus on the transform‐then‐solve approach, which amounts to first apply a specific algebraic transformation to the linear system of equations arising from the discretization, and then solve the transformed system with an algebraic multigrid method. The approach has recently been applied to finite difference discretizations of the Stokes problem with constant viscosity, and has recommended itself as a robust and competitive solution method. In this work, we examine the extension of the approach to standard finite element discretizations of the Stokes problem, including problems with variable viscosity. The extension relies, on one hand, on the use of the successive over‐relaxation method as a multigrid smoother for some finite element schemes. On the other hand, we present strategies that allow us to limit the complexity increase induced by the transformation. Numerical experiments show that for stationary problems our method is competitive compared to a reference solver based on a block diagonal preconditioner and MINRES, and suggest that the transform‐then‐solve approach is also more robust. In particular, for problems with variable viscosity, the transform‐then‐solve approach demonstrates significant speed‐up with respect to the block diagonal preconditioner. The method is also particularly robust for time‐dependent problems whatever the time step size.

Solution of Volume–Surface Integral Equation Accelerated by MLFMA and Skeletonization

A fast algorithm for solving large multi-scale scattering problems by the volume–surface integral equation (VSIE) is proposed. The sub-matrices of the near-field interactions (NFIs) in terms of the multi-level fast multipole algorithm (MLFMA) are compressed by the skeletonization after the theoretical and numerical investigations on their low-rankness are carried out. A block-diagonal preconditioner (BDP) based on the recursive skeletonization (RS) is developed to improve the conditioning of the impedance matrix of the VSIE. The rules to select hyperparameters in the proposed method are given. Numerical experiments conducted on scattering problems show the performance of the proposed algorithm.

Uniform block-diagonal preconditioners for divergence-conforming HDG Methods for the generalized Stokes equations and the linear elasticity equations

Abstract We propose a uniform block-diagonal preconditioner for condensed $H$(div)-conforming hybridizable discontinuous Galerkin schemes for parameter-dependent saddle point problems, including the generalized Stokes equations and the linear elasticity equations. An optimal preconditioner is obtained for the stiffness matrix on the global velocity/displacement space via the auxiliary space preconditioning technique (Xu (1994) The Auxiliary Space Method and Optimal Multigrid Preconditioning Techniques for Unstructured Grids, vol. 56. International GAMM-Workshop on Multi-level Methods (Meisdorf), pp. 215–235). A spectrally equivalent approximation to the Schur complement on the element-wise constant pressure space is also constructed, and an explicit computable exact inverse is obtained via the Woodbury matrix identity. Finally, the numerical results verify the robustness of our proposed preconditioner with respect to model parameters and mesh size.

A Mass Conserving Mixed $hp$-FEM Scheme for Stokes Flow. Part III: Implementation and Preconditioning

This is the third part in a series on a mass conserving, high order, mixed finite element method for 2D Stokes flow. In this part, we study a block-diagonal preconditioner for the indefinite Schur complement system arising from the discretization of the Stokes equations using these elements. The underlying finite element method is uniformly stable in both the mesh size $h$ and the polynomial order $p$, and we prove bounds on the eigenvalues of the preconditioned system which are independent of $h$ and grow modestly in $p$. The analysis relates the Schur complement system to an appropriate variational setting with subspaces for which exact sequence properties and inf-sup stability hold. Several numerical examples demonstrate agreement with the theoretical results.

A MultiHarmonic Finite Element Method for Scattering Problems with Small-Amplitude Boundary Deformations

A finite element method in the frequency domain is proposed for solving scattering problems with moving or, more generally, deforming boundaries. First, the original problem is rewritten as an equivalent weak formulation set in a fixed domain. Next, this formulation is approximated as a simpler weak form based on asymptotic expansions when the amplitude of the movements or the deformations is small. Fourier series expansions of some geometrical quantities and of the solution are next introduced to obtain a coupled multi-harmonic frequency domain formulation. Standard finite element methods can then be applied to solve the resulting problem and a block diagonal preconditioner is proposed to accelerate the Krylov subspace solution of the linear system for high frequency problems. The efficiency of the resulting method is demonstrated on a radar sensing application for the automotive industry.

Rational Approximations in Robust Preconditioning of Multiphysics Problems

Multiphysics or multiscale problems naturally involve coupling at interfaces which are manifolds of lower dimensions. The block-diagonal preconditioning of the related saddle-point systems is among the most efficient approaches for numerically solving large-scale problems in this class. At the operator level, the interface blocks of the preconditioners are fractional Laplacians. At the discrete level, we propose to replace the inverse of the fractional Laplacian with its best uniform rational approximation (BURA). The goal of the paper is to develop a unified framework for analysis of the new class of preconditioned iterative methods. As a final result, we prove that the proposed preconditioners have optimal computational complexity O(N), where N is the number of unknowns (degrees of freedom) of the coupled discrete problem. The main theoretical contribution is the condition number estimates of the BURA-based preconditioners. It is important to note that the obtained estimates are completely analogous for both positive and negative fractional powers. At the end, the analysis of the behavior of the relative condition numbers is aimed at characterizing the practical requirements for minimal BURA orders for the considered Darcy–Stokes and 3D–1D examples of coupled problems.

A Cartesian Fmm-Accelerated Galerkin Boundary Integral Poisson-Boltzmann Solver

The Poisson-Boltzmann model is an effective and popular approach for modeling solvated biomolecules in continuum solvent with dissolved electrolytes. In this paper, we report our recent work in developing a Galerkin boundary integral method for solving the Poisson-Boltzmann (PB) equation. The solver has combined advantages in accuracy, efficiency, and memory usage as it applies a well-posed boundary integral formulation to circumvent many numerical difficulties associated with the PB equation and uses an O(N) Cartesian Fast Multipole Method (FMM) to accelerate the GMRES iteration. In addition, special numerical treatments such as adaptive FMM order, block diagonal preconditioners, Galerkin discretization, and Duffy's transformation are combined to improve the performance of the solver, which is validated on benchmark Kirkwood's sphere and a series of testing proteins.

Efficient block preconditioners for integral constrained elliptic optimal control problems with finite element approximations

In this paper, focusing on a distributed optimal control problem for the elliptic equations with integral control constraint, we propose efficient block diagonal preconditioners to solve the corresponding linearized algebraic system with finite element methods. We derive the first order necessary and sufficient optimality conditions for an integral constraint on the control, and divide the discretized optimal conditions into three matrix forms. Then block-diagonal preconditioners for the corresponding linear algebraic systems are constructed. With respect to both the mesh size and the regularization parameter, the robustness of our preconditioners is proved in detail. In fact, the condition numbers of the preconditioned matrices are a constant for different parameters. Meanwhile, based on the equivalent matrix forms, an algorithm is proposed for this kind of constrained optimal control problems. Numerical experiments are given to depict the efficiency of our proposed preconditioners.