Matrix-free Strategies Research Articles

An efficient framework for matrix-free SpMV computation on GPU for elastoplastic problems

High computational cost in elastoplastic analysis is often handled by the use of high performance parallel computers. However, the presence of both elastic and plastic states leads to the branching issue, which prevents the realization of true parallel performance. The computational efficiency of an elastoplastic analysis using finite element method is largely determined by the performance of repeated solution of linearized system of equations. In this paper, we propose GPU-based matrix-free strategies to compute sparse matrix–vector multiplication (SpMV) in Conjugate Gradient (CG) iterative solver for the acceleration of solution of linear system of equations. Matrix-free solvers never assemble large sparse global tangent matrix and perform the computation directly with small dense elemental matrices, reducing the storage requirement and preventing the use of problematic sparse storage formats. A uniform treatment of elements in elastic and plastic regions is achieved by the proposed single kernel strategy, which prevents branching, avoids redundant computation and provides efficient memory access. In addition, we propose node-based and degrees-of-freedom (DOF)-based parallel strategies for effective implementation of matrix-free SpMV on a GPU. The proposed strategies use single elemental tangent matrix for all elements in elastic region and individual tangent matrices in plastic region. The computational experiments over three large-scale benchmark examples of elastoplasticity reveal that the performance of the node-based and DOF-based parallelization strategies depends on the amount of plasticity in the body. At low plasticity levels, node-based strategy performs best, achieving 3.2× speedup over an existing GPU-based matrix-free SpMV strategy in the literature. For moderate to high amount of plasticity, the DOF-based strategy outperforms every other strategy and obtains speedups of up to 3.5× over the existing SpMV strategy.

Strategies of Preconditioner Updates for Sequences of Linear Systems Associated with the Neutron Diffusion

The time-dependent neutron diffusion equation approximates the neutronic power evolution inside a nuclear reactor core. Applying a Galerkin finite element method for the spatial discretization of these equations leads to a stiff semi-discrete system of ordinary differential equations. For time discretization, an implicit scheme is used, which implies solving a large and sparse linear system of equations for each time step. The GMRES method is used to solve these systems because of its fast convergence when a suitable preconditioner is provided. This work explores several matrix-free strategies based on different updated preconditioners, which are constructed by low-rank updates of a given initial preconditioner. They are two tuned preconditioners based on the bad and good Broyden’s methods, initially developed for nonlinear equations and optimization problems, and spectral preconditioners. The efficiency of the resulting preconditioners under study is closely related to the selection of the subspace used to construct the update. Our numerical results show the effectiveness of these methodologies in terms of CPU time and storage for different nuclear benchmark transients, even if the initial preconditioner is not good enough.

Truncated trust region method for nonlinear inverse problems and application in full-waveform inversion

We present a general truncated trust region method to solve large-scale nonlinear inverse problems. The truncated trust region method can serve as an implicit regularization method, and it can take advantage of the second-order derivative information of the misfit functional. The convergence of the truncated trust region method is provided under some smoothness assumptions. To improve the computational efficiency and reduce the memory requirement, we develop a second-order adjoint-state method to efficiently estimate matrix–vector products, and solve the trust region subproblem using the truncated conjugate gradient method with the matrix-free strategy. The full-waveform inversion problem is used to test the numerical performance of the proposed method. Numerical results show that the truncated trust region method can perform better than conventional methods (e.g. nonlinear conjugate gradient method and L-BFGS) for highly nonlinear inverse problems, in terms of inverting resolution.

Chemical adsorption on 2D dielectric nanosheets for matrix free nanocomposites with ultrahigh electrical energy storage

Relaxor ferroelectric polymers display great potential in capacitor dielectric applications because of their excellent flexibility, light weight, and high dielectric constant. However, their electrical energy storage capacity is limited by their high conduction losses and low dielectric strength, which primarily originates from the impact-ionization-induced electronmultiplication, low mechanical modulus, and low thermal conductivity of the dielectric polymers. Here a matrix free strategy is developed to effectively suppress electron multiplication effects and to enhance mechanical modulus and thermal conductivity of a dielectric polymer, which involves the chemical adsorption of an electron barrier layer on boron nitride nanosheet surfaces by chemically adsorbing an amino-containing polymer. A dramatic decrease of leakage current (from 2.4 × 10−6 to 1.1 × 10−7 A cm−2 at 100 MV m−1) and a substantial increase of breakdown strength (from 340 to 742 MV m−1) were achieved in the nanocompostes, which result in a remarkable increase of discharge energy density (from 5.2 to 31.8 J cm−3). Moreover, the dielectric strength of the nanocomposites suffering an electrical breakdown could be restored to 88% of the original value. This study demonstrates a rational design for fabricating dielectric polymer nanocomposites with greatly enhanced electric energy storage capacity.

GPU-based matrix-free finite element solver exploiting symmetry of elemental matrices

Matrix-free solvers for finite element method (FEM) avoid assembly of elemental matrices and replace sparse matrix-vector multiplication required in iterative solution method by an element level dense matrix-vector product. In this paper, a novel matrix-free strategy for FEM is proposed which computes element level matrix-vector product by using only the symmetric part of the elemental matrices. The proposed strategy is developed to take advantage of the massive parallelism of Graphics Processing Unit (GPU). A unique data structure is also introduced which ensures localized and coalesced memory access suitable for a GPU while storing only the symmetric part of the elemental matrices. In addition, the proposed strategy emphasizes the efficient use of register cache, uniform workload distribution, reducing thread synchronization, and maintaining sufficient granularity to make the best use of GPU resources. The performance of the proposed strategy is evaluated by solving elasticity and heat conduction problems using 4-noded quadrilateral element with two degrees of freedom (DOFs) and one DOF per node, respectively. The performance is compared with the matrix-free solver strategies on GPU from the literature. It is found that a maximum speedup of 4.9 $$\times $$ is obtained for the elasticity problem and a maximum of 3.2 $$\times $$ speedup for the heat conduction problem. Further, the proposed strategy takes the least amount of GPU memory as compared to the existing strategies.

Matrix-free weighted quadrature for a computationally efficient isogeometric [formula omitted]-method

The k-method is the isogeometric method based on splines (or NURBS, etc.) with maximum regularity. When implemented following the paradigms of classical finite element methods, the computational resources required by the k-method are prohibitive even for moderate degree. In order to address this issue, we propose a matrix-free strategy combined with weighted quadrature, which is an ad-hoc strategy to compute the integrals of the Galerkin system. Matrix-free weighted quadrature (MF-WQ) speeds up matrix operations, and, perhaps even more important, greatly reduces memory consumption. Our strategy also requires an efficient preconditioner for the linear system iterative solver. In this work we deal with an elliptic model problem, and adopt a preconditioner based on the Fast Diagonalization method, an old idea to solve Sylvester-like equations. Our numerical tests show that the isogeometric solver based on MF-WQ is faster than standard approaches (where the main cost is the matrix formation by standard Gaussian quadrature) even for low degree. But the main achievement is that, with MF-WQ, the k-method gets orders of magnitude faster by increasing the degree, given a target accuracy. Therefore, we are able to show the superiority, in terms of computational efficiency, of the high-degree k-method with respect to low-degree isogeometric discretizations. What we present here is applicable to more complex and realistic differential problems, but its effectiveness will depend on the preconditioner stage, which is as always problem-dependent. This situation is typical of modern high-order methods: the overall performance is mainly related to the quality of the preconditioner.

Thermal diffusion desorption for the comprehensive analysis of organic compounds.

Comprehensive analysis of organic compounds is crucial yet challenging considering that information on elements, fragments, and molecules is unavailable simultaneously by current analytical techniques. Additionally, many compounds are insoluble or only dissolve in toxic solvents. A solvent- and matrix-free strategy has been developed which allows the organic compound analyzed in its original form. It utilizes thermal diffusion desorption with the solid analyte irradiated with high energy laser. It is capable of providing explicit elemental, fragmental, and molecular information simultaneously for a variety of organic compounds. Thermal diffusion desorption has many advantages compared to the electrospray and MALDI techniques. The protons that form the protonated molecular ions originate from the analyte itself. All the elements and fragments are also derived from the analyte itself, which provides abundant information and expedites the identification of organic compounds.

Hybrid finite element/volume method for 3D incompressible flows with heat transfer

An implicit hybrid finite element (FE)/volume solver has been extended to incompressible flows coupled with the energy equation. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centred finite volume (FV) method. The pressure Poisson equation is solved by the node-based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. Our current staggered-mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centres and the auxiliary variable at vertices. The Generalized Minimal Residual (GMRES) matrix-free strategy is adapted to solve the governing equations in both FE and FV methods. The presented 2D and 3D numerical examples show the robustness and accuracy of the numerical method.

An Efficient Numerical Method for General $L_{p}$ Regularization in Fluorescence Molecular Tomography

Reconstruction algorithms for fluorescence tomography have to address two crucial issues: 1) the ill-posedness of the reconstruction problem, 2) the large scale of numerical problems arising from imaging of 3-D samples. Our contribution is the design and implementation of a reconstruction algorithm that incorporates general Lp regularization (p ¿ 1). The originality of this work lies in the application of general Lp constraints to fluorescence tomography, combined with an efficient matrix-free strategy that enables the algorithm to deal with large reconstruction problems at reduced memory and computational costs. In the experimental part, we specialize the application of the algorithm to the case of sparsity promoting constraints (L (1)). We validate the adequacy of L (1) regularization for the investigation of phenomena that are well described by a sparse model, using data acquired during phantom experiments.

A Newton method for the resolution of steady stochastic Navier–Stokes equations

We present a Newton method to compute the stochastic solution of the steady incompressible Navier–Stokes equations with random data (boundary conditions, forcing term, fluid properties). The method assumes a spectral discretization at the stochastic level involving a orthogonal basis of random functionals (such as Polynomial Chaos or stochastic multi-wavelets bases). The Newton method uses the unsteady equations to derive a linear equation for the stochastic Newton increments. This linear equation is subsequently solved following a matrix-free strategy, where the iterations consist in performing integrations of the linearized unsteady Navier–Stokes equations, with an appropriate time scheme to allow for a decoupled integration of the stochastic modes. Various examples are provided to demonstrate the efficiency of the method in determining stochastic steady solution, even for regimes where it is likely unstable.

Technical Note: Computing Strategies in Genome-Wide Selection

Genome-wide genetic evaluation might involve the computation of BLUP-like estimations, potentially including thousands of covariates (i.e., single-nucleotide polymorphism markers) for each record. This implies dense Henderson's mixed-model equations and considerable computing resources in time and storage, even for a few thousand records. Possible computing options include the type of storage and the solving algorithm. This work evaluated several computing options, including half-stored Cholesky decomposition, Gauss-Seidel, and 3 matrix-free strategies: Gauss-Seidel, Gauss-Seidel with residuals update, and preconditioned conjugate gradients. Matrix-free Gauss-Seidel with residuals update adjusts the residuals after computing the solution for each effect. This avoids adjusting the left-hand side of the equations by all other effects at every step of the algorithm and saves considerable computing time. Any Gauss-Seidel algorithm can easily be extended for variance component estimation by Markov chain-Monte Carlo. Let m and n be the number of records and markers, respectively. Computing time for Cholesky decomposition is proportional to n3. Computing times per round are proportional to mn2 in matrix-free Gauss-Seidel, to n2 for half-stored Gauss-Seidel, and to n and m for the rest of the algorithms. Algorithms were tested on a real mouse data set, which included 1,928 records and 10,946 single-nucleotide polymorphism markers. Computing times were in the order of a few minutes for Gauss-Seidel with residuals update and preconditioned conjugate gradients, more than 1h for half-stored Gauss-Seidel, 2h for Cholesky decomposition, and 4 d for matrix-free Gauss-Seidel. Preconditioned conjugate gradients was the fastest. Gauss-Seidel with residuals update would be the method of choice for variance component estimation as well as solving.

Desorption-ionization mass spectrometry on porous silicon.

Desorption mass spectrometry has undergone significant improvements since the original experiments were performed more than 90 years ago. The most dramatic change occurred in the early 1980s with the introduction of an organic matrix to transfer energy to the analyte. This reduces ion fragmentation but also introduces background ions from the matrix. Here we describe a matrix-free strategy for biomolecular mass spectrometry based on pulsed-laser desorption-ionization from a porous silicon surface. Our method uses porous silicon to trap analytes deposited on the surface, and laser irradiation to vaporize and ionize them. We show that the method works at femtomole and attomole levels of analyte, and induces little or no fragmentation, in contrast to what is typically observed with other such approaches. The ability to perform these measurements without a matrix also makes it more amenable to small-molecule analysis. Chemical and structural modification of the porous silicon has enabled optimization of the ionization characteristics of the surface. Our technique offers good sensitivity as well as compatibility with silicon-based microfluidics and microchip technologies.