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Summary: This paper addresses the overwhelming computational resources needed with standard numerical approaches to simulate architected materials. Those multiscale heterogeneous lattice structures gain intensive interest in conjunction with the improvement of additive manufacturing as they offer, among many others, excellent stiffness‐to‐weight ratios. We develop here a dedicated HPC solver that benefits from the specific nature of the underlying problem in order to drastically reduce the computational costs (memory and time) for the full fine‐scale analysis of lattice structures. Our purpose is to take advantage of the natural domain decomposition into cells and, even more importantly, of the geometrical and mechanical similarities among cells. Our solver consists in a so‐called inexact FETI‐DP method where the local, cell‐wise operators and solutions are approximated with reduced order modeling techniques. Instead of considering independently every cell, we end up with only few principal local problems to solve and make use of the corresponding principal cell‐wise operators to approximate all the others. It results in a scalable algorithm that saves numerous local factorizations. Our solver is applied for the isogeometric analysis of lattices built by spline composition, which offers the opportunity to compute the reduced basis with macro‐scale data, thereby making our method also multiscale and matrix‐free. The solver is tested against various 2D and 3D analyses. It shows major gains compared to black‐box solvers; in particular, problems of several millions of degrees of freedom can be solved with a simple computer within few minutes. [ABSTRACT FROM AUTHOR]

In this paper we consider a multigrid approach for solving elliptic equations over non‐matching grids with domain decomposition methods. The domain is partitioned into subdomains with different mesh levels that do not match at the interface. The proposed algorithm searches for the global solution over different levels by projecting the residuals on the overlap region. This method is used in conjunction with a domain decomposition solver which only requires, in each iteration step, the solutions of several small local subproblems over finite element blocks. This algorithm is shown to converge to the solution of the corresponding Lagrange multiplier problem for non‐matching grids. The convergence properties of the algorithms are analyzed and numerical examples are presented. When the multigrid and domain decomposition approaches are combined, the method is shown to be reliable and easy to implement. Furthermore the local nature of the solver allows for a straightforward implementation on multiple parallel computers and graphics processing unit (GPU) clusters. [ABSTRACT FROM AUTHOR]

In this paper, a conserved domain decomposition method for solving convection–diffusion equations with variable coefficients is analyzed. The interface fluxes over the subdomains are firstly obtained by the explicit fluxes scheme. Secondly, the interior solutions and fluxes over each subdomains are computed by the modified upwind implicit scheme. Then, the interface fluxes are corrected by the obtained solutions. We prove rigorously that our scheme is mass conservative, unconditionally stable, and of second‐order convergence in spatial step. Numerical examples test the theoretical analysis and efficiencies. Lastly, we extend our scheme to the nonlinear convection–diffusion equations and give the error estimate. [ABSTRACT FROM AUTHOR]

Summary: This paper discusses techniques for computing a few selected eigenvalue–eigenvector pairs of large and sparse symmetric matrices. A recently developed class of techniques to solve this type of problems is based on integrating the matrix resolvent operator along a complex contour that encloses the interval containing the eigenvalues of interest. This paper considers such contour integration techniques from a domain decomposition viewpoint and proposes two schemes. The first scheme can be seen as an extension of domain decomposition linear system solvers in the framework of contour integration methods for eigenvalue problems, such as FEAST. The second scheme focuses on integrating the resolvent operator primarily along the interface region defined by adjacent subdomains. A parallel implementation of the proposed schemes is described, and results on distributed computing environments are reported. These results show that domain decomposition approaches can lead to reduced run times and improved scalability. [ABSTRACT FROM AUTHOR]

In this paper, the unconditional stability and mass‐preserving splitting domain decomposition method (S‐DDM) for solving three‐dimensional parabolic equations is analyzed. At each time step level, three steps (x‐direction, y‐direction, and z‐direction) are proposed to compute the solutions on each sub‐domains. The interface fluxes are first predicted by the semi‐implicit flux schemes. Second, the interior solutions and fluxes are computed by the splitting implicit solution and flux coupled schemes. Last, we recompute the interface fluxes by the explicit schemes. Due to the introduced z‐directional splitting and domain decomposition, the analysis of stability and convergence is scarcely evident and quite difficult. By some mathematical technique and auxiliary lemmas, we prove strictly our scheme meet unconditional stability and give the error estimates in L2‐norm. Numerical experiments are presented to illustrate the theoretical analysis. [ABSTRACT FROM AUTHOR]

Principal component analysis (PCA) is widely used for dimensionality reduction and unsupervised learning. The reconstruction error is sometimes large even when a large number of eigenmode is used. In this paper, we show that this unexpected error source is the pollution effect of a summation operation in the objective function of the PCA algorithm. The summation operator brings together unrelated parts of the data into the same optimization and the result is the reduction of the accuracy of the overall algorithm. We introduce a domain decomposed PCA that improves the accuracy, and surprisingly also increases the parallelism of the algorithm. To demonstrate the accuracy and parallel efficiency of the proposed algorithm, we consider three applications including a face recognition problem, a brain tumor detection problem using two‐ and three‐dimensional MRI images. [ABSTRACT FROM AUTHOR]

The viscous flow of two immiscible fluids in a porous medium on the Darcy scale is governed by a system of nonlinear parabolic equations. If infinite mobility of one phase can be assumed (e.g., in soil layers in contact with the atmosphere) the system can be substituted by the scalar Richards model. Thus, the porous medium domain may be partitioned into disjoint subdomains where either the full two‐phase or the simplified Richards model dynamics are valid. Extending the previously considered one‐model situations we suggest coupling conditions for this hybrid model approach. Based on an Euler implicit discretization, a linear iterative (L‐type) domain decomposition scheme is proposed, and proved to be convergent. The theoretical findings are verified by a comparative numerical study that in particular confirms the efficiency of the hybrid ansatz as compared to full two‐phase model computations. [ABSTRACT FROM AUTHOR]

Summary: This paper proposes a new scheme for the generation of Gaussian random fields over large domains (domain size much larger than the correlation length). The scheme decomposes the simulation domain into overlapping subdomains and essentially generates independent random fields over each of them before merging them on the overlaps. It is naturally suited for simulation over clusters of computers. With this approach, the number of operations for each processor depends only on the number of local degrees of freedom and not on the total number over all processors. Hence, weak scalability is perfectly met. This paper describes the general scheme and introduces two error estimates for comparison with classical sampling schemes. Improvements in terms of scalability are demonstrated both theoretically and through numerical examples. The behavior in the overlap is studied in detail. Simulations using the localized approach were performed in up to 512 processors and allowed to generate in 41 seconds a realization of a random field over a cube of side 300 correlation lengths (close to 2‐billion sampling points), much more efficiently and rapidly than with classical methods. [ABSTRACT FROM AUTHOR]

We present in this paper an efficient parallelization approach of the finite element domain decomposition method (FEM‐DDM) for solving large‐scale 3D scattering problems with over ten billion unknowns. In this parallel FEM‐DDM, the whole solution domain is decomposed in a two‐level hierarchical way to minimize the inter process communication. A communication‐avoided block diagonal symmetric Gauss‐Seidel (BD‐SGS) preconditioner is specially designed to accelerate the iterative solution of the interface problem from FEM‐DDM and reduce the CPU time, as well as the peak memory. Besides, a communication strategy is carefully designed to make computation and communication overlapped via nonblocking sending and receiving to further improve the parallel scalability. Numerical examples are presented to demonstrate the accuracy, scalability, and capability of the proposed parallel FEM‐DDM algorithm. The solutions of scattering by a practical aircraft model with a three‐layer antenna radome on its nose section is presented, which are discretized with more than ten billion unknowns. To the best of our knowledge, this is the largest number of unknowns solved by parallel FEM‐DDM in computational electromagnetics to date, for this type of simulation. [ABSTRACT FROM AUTHOR]

This paper introduces an efficient surrogate model with the aim of accelerating joint estimation of subsurface geological properties and relative permeability parameters for high‐dimensional inversion problems. We fully replace the high‐fidelity model with a set of subdomain linear models through integrating model linearization with smooth local parameterization where the Gaussian geological parameters and non‐Gaussian facies indicators are locally parameterized. These subdomain linear models with smooth local parameterization, referred to as SLM‐SLP, are constructed in each subdomain individually using only a few high‐fidelity model simulations. An adaptive scheme, that is, weighting smooth local parameterization (WSLP), is introduced as well to mitigate the negative effects of inappropriate domain decomposition schemes by adaptively optimizing the domain decomposition strategy. The computational advantages of the proposed algorithm are demonstrated on a synthetic non‐Gaussian facies model and a real‐world high‐dimensional Gaussian model. The amount of computational cost has been drastically reduced while reasonable accuracy remains. Specifically, SLM‐SLP only needs 220 fidelity simulations to optimize 302 parameters. Compared to ensemble smoother with multiple data assimilation (ES‐MDA), SLM‐SLP effectively and efficiently mitigates the ensemble collapse problem in the course of uncertain quantification. Key Points: Efficient subdomain linear models with smooth local parameterization are produced as an alternative implementation of forward simulationOnly 220 high‐fidelity model simulations are required in a high‐dimensional inversion problem with 320 uncertain parametersEnsemble collapse problem could be effectively and efficiently mitigated on conditioning a large amount of measurements [ABSTRACT FROM AUTHOR]

Domain decomposition methods have been proved to be an effective strategy to reduce the dimension of parametric partial differential equations (PDEs). However, existing domain decomposition methods for parametric PDEs are usually intrusive, which means domain decomposition based solvers need to be implemented from scratch for each target parametric PDE. To address this issue, we develop a new non‐intrusive domain‐decomposition model reduction method for linear steady‐state PDEs with random‐field coefficients. As a variant of our previous work by Mu and Zhang, the new method only needs access to the final linear system, that is, the global stiffness matrix and the right hand side, of a deterministic PDE solver, in order to build a domain‐decomposition‐based reduced model without intrusive implementation from scratch. The key idea is to remove the interface condition between sub‐domains and rely on the correlation between columns of the linear system to couple the sub‐domains. The non‐intrusive feature enables the applicability of the proposed method to a broader class of uncertainty quantification problems, where many legacy codes/solvers can be fully reused by our method. Two numerical examples including diffusion equations with random diffusivity and convection‐dominated transport with random velocity, are provided to demonstrate the effectiveness and efficiency of our method. [ABSTRACT FROM AUTHOR]

A mathematical framework is provided for a substructuring‐based domain decomposition (DD) approach for nonlocal problems that features interactions between points separated by a finite distance. Here, by substructuring it is meant that a traditional geometric configuration for local partial differential equation (PDE) problems is used in which a computational domain is subdivided into non‐overlapping subdomains. In the nonlocal setting, this approach is substructuring‐based in the sense that those subdomains interact with neighboring domains over interface regions having finite volume, in contrast to the local PDE setting in which interfaces are lower dimensional manifolds separating abutting subdomains. Key results include the equivalence between the global, single‐domain nonlocal problem and its multi‐domain reformulation, both at the continuous and discrete levels. These results provide the rigorous foundation necessary for the development of efficient solution strategies for nonlocal DD methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose a composite Laguerre spectral method for the nonlinear Fokker-Planck equations modelling the relaxation of fermion and boson gases. A composite Laguerre spectral scheme is constructed. Its convergence is proved. Numerical results show the efficiency of this approach and coincide well with theoretical analysis. Some results on the Laguerre approximation and techniques used in this paper are also applicable to other nonlinear problems on the whole line. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Summary: This article investigates the multi‐GPU performance of a 3D buoyancy driven cavity solver using MPI and OpenACC directives on multiple platforms. The article shows that decomposing the total problem in different dimensions affects the strong scaling performance significantly for the GPU. Without proper performance optimizations, it is shown that 1D domain decomposition scales poorly on multiple GPUs due to the noncontiguous memory access. The performance using whatever decompositions can be benefited from a series of performance optimizations in the article. Since the buoyancy driven cavity code is communication‐bounded on the clusters examined, a series of optimizations both agnostic and tailored to the platforms are designed to reduce the communication cost and improve memory throughput between hosts and devices efficiently. First, the parallel message packing/unpacking strategy developed for noncontiguous data movement between hosts and devices improves the overall performance by about a factor of 2. Second, transferring different data based on the stencil sizes for different variables further reduces the communication overhead. These two optimizations are general enough to be beneficial to stencil computations having ghost exchanges. Third, GPUDirect is used to improve the communication on clusters which have the hardware and software support for direct communication between GPUs without staging the memory of CPU. Finally, overlapping the communication and computations is shown to be not efficient on multi‐GPUs if only using MPI or MPI+OpenACC. Although we believe our implementation has revealed enough communication and computation overlap, the actual running does not utilize the overlap well due to a lack of enough asynchronous progression. [ABSTRACT FROM AUTHOR]

A nonlinear coupled mathematical system of two‐phase seepage flow displacement is discussed in this paper including an elliptic equation for the pressure and a convection‐dominated diffusion equation for the saturation. In fact, the boundary of an underground region where the fluid flows through is nonstationary. So a moving boundary should be considered. The saturation equation is convection‐dominated, therefore the method of upwind finite difference is introduced for the accurate computation. The upwind approximation could eliminate numerical oscillation and strong stability is shown. Since the computational work of saturation is larger than the pressure, the authors apply a parallel method, decomposing the whole domain into several nonoverlapping subdomains, to simplify the computation. A domain decomposition method coupled with upwind differences is presented for the saturation. The pressure equation is discretized by a five‐point center finite difference method. By using a transformation and defining new inner products and norms, error estimates in l2 norm is discussed. Finally, two experimental tests are given to illustrate the efficiency and accuracy of the parallel algorithm. [ABSTRACT FROM AUTHOR]

Mathematical modeling and simulation is a promising approach to personalized cancer medicine. Yet, the complexity, heterogeneity and multi‐scale nature of cancer pose significant computational challenges. Coupling discrete cell‐based models with continuous models using hybrid cellular automata (CA) is a powerful approach for mimicking biological complexity and describing the dynamical exchange of information across different scales. However, when clinically relevant cancer portions are taken into account, such models become computationally very expensive. While efficient parallelization techniques for continuous models exist, their coupling with discrete models, particularly CA, necessitates more elaborate solutions. Building upon FEniCS, a popular and powerful scientific computing platform for solving partial differential equations, we developed parallel algorithms to link stochastic CA with differential equations (https://bitbucket.org/HTasken/cansim). The algorithms minimize the communication between processes that share CA neighborhood values while also allowing for reproducibility during stochastic updates. We demonstrated the potential of our solution on a complex hybrid cellular automaton model of breast cancer treated with combination chemotherapy. On a single‐core processor, we obtained nearly linear scaling with an increasing problem size, whereas weak parallel scaling showed moderate growth in solving time relative to increase in problem size. Finally, we applied the algorithm to a problem that is 500 times larger than previous work, allowing us to run personalized therapy simulations based on heterogeneous cell density and tumor perfusion conditions estimated from magnetic resonance imaging data on an unprecedented scale. [ABSTRACT FROM AUTHOR]

In this work, an improved domain decomposition method is developed to address workload imbalance when implementing the parallel computing of a four‐dimensional lattice spring model (4D‐LSM) to solve problems in rock engineering on a large scale. A cubic domain decomposition scheme is adopted and optimized by a simulated annealing algorithm (SAA) to minimize the workload imbalance among subdomains. The improved domain decomposition method is implemented in the parallel computing of the 4D‐LSM. Numerical results indicate that the proposed domain decomposition method can further improve the workload balance among processors, which is helpful to supersede the limit of computational scale when solving large‐scale geotechnical problems and decrease the runtime of the parallel 4D‐LSM by at most 40% compared to the original cubic decomposition method. This shows the practicability of the proposed method in parallel computing. Two types of target functions of SAA are tested, and their influence on the performance of the parallel 4D‐LSM is investigated. Finally, a computational model with one billion particles for one actual engineering application of using 4D‐LSM is realized, and the result shows the advantages of parallel computing. [ABSTRACT FROM AUTHOR]

Domain decomposition methods often exhibit very poor performance when applied to engineering problems with large heterogeneities. In particular, for heterogeneities along domain interfaces, the iterative techniques to solve the interface problem are lacking an efficient preconditioner. Recently, a robust approach, named finite element tearing and interconnection (FETI)-generalized eigenvalues in the overlaps (Geneo), was proposed where troublesome modes are precomputed and deflated from the interface problem. The cost of the FETI-Geneo is, however, high. We propose in this paper techniques that share similar ideas with FETI-Geneo but where no preprocessing is needed and that can be easily and efficiently implemented as an alternative to standard domain decomposition methods. In the block iterative approaches presented in this paper, the search space at every iteration on the interface problem contains as many directions as there are domains in the decomposition. Those search directions originate either from the domain-wise preconditioner (in the simultaneous FETI method) or from the block structure of the right-hand side of the interface problem (block FETI). We show on two-dimensional structural examples that both methods are robust and provide good convergence in the presence of high heterogeneities, even when the interface is jagged or when the domains have a bad aspect ratio. The simultaneous FETI was also efficiently implemented in an optimized parallel code and exhibited excellent performance compared with the regular FETI method. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents an assessment of fast parallel pre-conditioners for numerical solution of the pressure Poisson equation arising in large eddy simulation of turbulent incompressible flows. Focus is primarily on the pre-conditioners suitable for domain decomposition based parallel implementation of finite volume solver on non-uniform structured Cartesian grids. Bi-conjugate gradient stabilized method has been adopted as the Krylov solver for the linear algebraic system resulting from the discretization of the pressure Poisson equation. We explore the performance of multigrid pre-conditioner for the non-uniform grid and compare its performance with additive Schwarz pre-conditioner, Jacobi and SOR( k) pre-conditioners. Numerical experiments have been performed to assess the suitability of these pre-conditioners for a wide range of non-uniformity (stretching) of the grid in the context of large eddy simulation of a typical flow problem. It is seen that the multigrid preconditioner shows the best performance. Further, the SOR( k) preconditioner emerges as the next best alternative. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The accurate prediction of transmural stresses in arterial walls requires on the one hand robust and efficient numerical schemes for the solution of boundary value problems including fluid-structure interactions and on the other hand the use of a material model for the vessel wall that is able to capture the relevant features of the material behavior. One of the main contributions of this paper is the application of a highly nonlinear, polyconvex anisotropic structural model for the solid in the context of fluid-structure interaction, together with a suitable discretization. Additionally, the influence of viscoelasticity is investigated. The fluid-structure interaction problem is solved using a monolithic approach; that is, the nonlinear system is solved (after time and space discretizations) as a whole without splitting among its components. The linearized block systems are solved iteratively using parallel domain decomposition preconditioners. A simple - but nonsymmetric - curved geometry is proposed that is demonstrated to be suitable as a benchmark testbed for fluid-structure interaction simulations in biomechanics where nonlinear structural models are used. Based on the curved benchmark geometry, the influence of different material models, spatial discretizations, and meshes of varying refinement is investigated. It turns out that often-used standard displacement elements with linear shape functions are not sufficient to provide good approximations of the arterial wall stresses, whereas for standard displacement elements or F-bar formulations with quadratic shape functions, suitable results are obtained. For the time discretization, a second-order backward differentiation formula scheme is used. It is shown that the curved geometry enables the analysis of non-rotationally symmetric distributions of the mechanical fields. For instance, the maximal shear stresses in the fluid-structure interface are found to be higher in the inner curve that corresponds to clinical observations indicating a high plaque nucleation probability at such locations. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The Eighth‐Shell method for parallelization of molecular dynamics simulations has previously been shown to be the most optimal for efficiency at large process counts. However, in its current formulation only the P1 space group is supported for periodic boundary conditions (PBC) and thus reflection and/or rotational crystal symmetries are not supported. In this work, we outline the development and implementation of the Extended Eighth‐Shell (EES) method that allows rotational symmetry by using an extended import region compared to the ES method. It simulates only the asymmetric unit and communicates coordinates and forces with images that correspond to P21 PBC. The P21 PBC has application in lipid bilayer simulations as it can be used to allow lipids to switch leaftlets, thus rapidly balancing the chemical potential difference between the two layers. Our results show that the EES method scales efficiently over large number of processes and can be used for simulations with P21 symmetry in an orthorhombic crystal. [ABSTRACT FROM AUTHOR]

This work focuses on the development and analysis of a partitioned numerical method for moving domain, fluid–structure interaction problems. We model the fluid using incompressible Navier–Stokes equations, and the structure using linear elasticity equations. We assume that the structure is thick, that is, described in the same dimension as the fluid. We propose a non‐iterative, domain decomposition method where the fluid and the structure subproblems are solved separately. The method is based on generalized Robin boundary conditions, which are used in both fluid and structure subproblems. Using energy estimates, we show that the proposed method applied to a moving domain problem is unconditionally stable. We also analyze the convergence of the method and show OΔt12 convergence in time and optimal convergence in space. Numerical examples are used to demonstrate the performance of the method. In particular, we explore the relation between the combination parameter used in the derivation of the generalized Robin boundary conditions and the accuracy of the scheme. We also compare the performance of the method to a monolithic solver. [ABSTRACT FROM AUTHOR]

Computational hemodynamics is being developed as an alternative approach for assisting clinical diagnosis and treatment planning for liver diseases. The technology is non‐invasive, but the computational time could be high when the full geometry of the blood vessels is taken into account. Existing approaches use either one‐dimensional model of the artery or simplified three‐dimensional tubular geometry in order to reduce the computational time, but the accuracy is sometime compromised, for example, when simulating blood flows in arteries with plaque. In this work, we study a highly parallel method for the transient incompressible Navier–Stokes equations for the simulation of the blood flows in the full three‐dimensional patient‐specific hepatic artery, portal vein and hepatic vein. As applications, we also simulate the flow in a patient with hepatectomy and calculate the S (PPG). One of the advantages of simulating blood flows in all hepatic vessels is that it provides a direct estimate of the PPG, which is a gold standard value to assess the portal hypertension. Moreover, the robustness and scalability of the algorithm are also investigated. A 83% parallel efficiency is achieved for solving a problem with 7 million elements on a supercomputer with more than 1000 processor cores. [ABSTRACT FROM AUTHOR]

Based on overlapping domain decomposition, a new class of parallel split least‐squares (PSLS) mixed finite element methods is presented for solving parabolic problem. The algorithm is fully parallel. In the overlapping domains, the partition of unity is applied to distribute the corrections reasonably, which makes that the new method only needs one or two iteration steps to reach given accuracy at each time step while the classical Schwarz alternating methods need many iteration steps. The dependence of the convergence rate on the spacial mesh size, time increment, iteration times, and subdomains overlapping degree is analyzed. Some numerical results are reported to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper introduces a robust preconditioner for general sparse matrices based on low-rank approximations of the Schur complement in a Domain Decomposition framework. In this 'Schur Low Rank' preconditioning approach, the coefficient matrix is first decoupled by a graph partitioner, and then a low-rank correction is exploited to compute an approximate inverse of the Schur complement associated with the interface unknowns. The method avoids explicit formation of the Schur complement. We show the feasibility of this strategy for a model problem and conduct a detailed spectral analysis for the relation between the low-rank correction and the quality of the preconditioner. We first introduce the SLR preconditioner for symmetric positive definite matrices and symmetric indefinite matrices if the interface matrices are symmetric positive definite. Extensions to general symmetric indefinite matrices as well as to nonsymmetric matrices are also discussed. Numerical experiments on general matrices illustrate the robustness and efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

Parallel semi-implicit method for pressure-linked equations(SIMPLE) algorithm is used to solve the 3-D incompressible pipe flow problem. In this paper, we proposed a novel parallel SIMPLE algorithm that uses the alternate tiling technique. Firstly, a parallel SIMPLE algorithm based on domain decomposition method was established, and the implementation of domain partition and data exchange was presented. Then, we presented serial finite difference stencil algorithm based on alternate tiling. Furthermore, an iteration space parallel two-way finite difference stencil algorithm based on alternate tiling was proposed, introducing the sequence of iterative space tiles as the sequence of execution and using time skewing technique to partition the iteration space, thus to improve the data locality of algorithm. The cache misses and the cost of communication and synchronization are reduced by reordering the tiles of iteration space. Finally, the effectiveness of the two parallel SIMPLE algorithms were compared. The results showed that the parallel SIMPLE algorithm that uses the two-way finite difference stencil algorithm based on alternate tiling has good data locality, performance, and scalability in the Deepcomp7000 cluster computing environment. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Summary: Kalman filter (KF) is one of the most important and common estimation algorithms. We introduce an innovative designing of Kalman filter algorithm based on domain decomposition (we call it DD‐KF). DD‐KF involves decomposition of the whole computational problem, partitioning of the solution and a slight modification of KF algorithm allowing a correction at run‐time of local solutions. The resulted parallel algorithm consists of concurrent copies of KF algorithm, each one requiring the same amount of computations on each subdomain and an exchange of boundary conditions between adjacent subdomains. Main advantage of this approach is that it can be potentially applied in a moderately nonintrusive manner to existing codes for tracking and controlling systems in location, navigation, in computer graphics and in much more state estimation problems. To highlight the capability of DD‐KF of exploiting the computing power provided by future designs of microprocessors based on multi/many‐cores CPU/GPU technologies, we consider DD both at physical core level and at microprocessor level and we discuss scalability of DD‐KF algorithm at coarse and fine grained level. Throughout the present work, we derive and discuss DD‐KF algorithm for solving constrained least square model, which underlies any data sampling and estimation problem. [ABSTRACT FROM AUTHOR]

Domain decomposition methods for finite element problems using a partition based on the underlying finite element mesh have been extensively studied, in this paper, we discuss algebraic extensions of the class of overlapping domain decomposition algorithms for general sparse matrices. The subproblems are created with an overlapping partition of the graph corresponding to the sparsity structure of the matrix. These algebraic domain decomposition methods are especially useful for unstructured mesh problems. We also discuss some difficulties encountered in the algebraic extension, particularly the issues related to the coarse solver. [ABSTRACT FROM AUTHOR]

This special issue contains papers from the Thirteenth Copper Mountain Conference on Multigrid Methods, held in the Colorado Rocky Mountains on March 19–23, 2007, co-chaired by Van Henson and Joel Dendy. The papers address a variety of applications and cover a breadth of topics, ranging from theory to high-performance computing. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY Motivated by atomistic-to-continuum coupling, we consider a fine-scale problem defined on a small region embedded in a much larger coarse-scale domain and propose an efficient solution technique on the basis of the domain decomposition framework. Specifically, we develop a nonoverlapping Schwarz method with two important features: (i) the use of an efficient approximation of the Dirichlet-to-Neumann map for the interface conditions; and (ii) the utilization of the inherent scale separation in the solution. The paper includes a detailed formulation of the proposed interface condition, along with the illustration of its effectiveness by using simple but representative numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Summary: The heat equation is solved by using a finite volume discretization in a domain that consists of a two‐dimensional central node and several one‐dimensional outgoing branches. Several interface connection options to match the submodels set on the node and on the branches, with or without continuity, are looked at. For each of them, a monolithic scheme is defined, and existence and uniqueness of the solution is proved. New schemes are deduced, which are obtained through domain decomposition methods in the form of interface systems, with one or two unknowns per interface. A comparative systematic study is carried out from an algebraic and numerical point of view according to the interface conditions: Dirichlet, Neumann, or Robin. An efficient diagonal preconditioning is proposed. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel domain decomposition finite-difference time domain technique for solving large electromagnetic radiation and scattering problems. In this method the computational domain is divided into manageable, moderate-sized subdomains with overlapping regions. PML layers are used for terminating the subdomains to absorb the outgoing waves. The tangential electric fields are stored on an interface plane to be used as a boundary condition to generate the field in the adjacent subdomain. These computations are applied in a serial manner to solve a problem that is too large to be solved in one step because of limitations in the available computational resources. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 2399–2405, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21962 [ABSTRACT FROM AUTHOR]

In this paper, we consider domain decomposition preconditioners for a system of linear algebraic equations arising from the p-version of the FEM. We analyse several multi-level preconditioners for the Dirichlet problems in the sub-domains in two and three dimensions. It is proved that the condition number of the preconditioned system is bounded by a constant independent of the polynomial degree. Relations between the p-version of the FEM and the h-version are helpful in the interpretations of the results. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper extends the relationship between the author's operator trigonometry and convergence rates and other properties of important iterative methods. A new interesting trigonometric preconditioning lemma is given. The general relationship between domain decomposition methods and the operator trigonometry is established. A new basic conceptual link between sparse approximate inverse algorithms and the operator trigonometry is observed. A new underlying fundamental inherent trigonometry of the classical successive over-relaxation scheme is exposed. Some improved trigonometric interpretations of minimum residual schemes are mentioned. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Additive Schwarz preconditioned GMRES is a powerful method for solving large sparse linear systems of equations on parallel computers. The algorithm is often implemented in the Euclidean norm, or the discrete l2 norm, however, the optimal convergence result is available only in the energy norm (or the equivalent Sobolev H1 norm). Very little progress has been made in the theoretical understanding of the l2 behaviour of this very successful algorithm. To add to the difficulty in developing a full l2 theory, in this note, we construct explicit examples and show that the optimal convergence of additive Schwarz preconditioned GMRES in l2 cannot be obtained using the existing GMRES theory. More precisely speaking, we show that the symmetric part of the preconditioned matrix, which plays a role in the Eisenstat–Elman–Schultz theory, has at least one negative eigenvalue, and we show that the condition number of the best possible eigenmatrix that diagonalizes the preconditioned matrix, key to the Saad–Schultz theory, is bounded from both above and below by constants multiplied by h-1/2. Here h is the finite element mesh size. The results presented in this paper are mostly negative, but we believe that the techniques used in our proofs may have wide applications in the further development of the l2 convergence theory and in other areas of domain decomposition methods. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY This paper studies the application of two-level Schwarz algorithms to several models of computational fluid dynamics. The purpose is to build an algorithm suitable for elliptic and convective models. The subdomain approximated solution relies on the incomplete lower-upper factorization. The algebraic coupling between the coarse grid and the Schwarz preconditioner is discussed. The deflation method and the balancing domain decomposition method are studied for introducing the coarse-grid correction as a preconditioner. Standard coarse grids are built with the characteristic or indicator functions of the subdomains. The building of a set of smooth basis functions (analogous to smoothed-aggregation methods) is considered. A first test problem is the Poisson problem with a discontinuous coefficient. The two options are compared for the standpoint of coarse-grid consistency and for the gain in scalability of the global Schwarz iteration. The advection-diffusion model is then considered as a second test problem. Extensions to compressible flows (together with incompressible flows for comparison) are then proposed. Parallel applications are presented and their performance measured. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY Arterial walls in the healthy physiological regime are characterized by quasi-incompressible, anisotropic, hyperelastic material behavior. Polyconvex material functions representing such materials typically incorporate a penalty function to account for the incompressibility. Unfortunately, the penalty will affect the conditioning of the stiffness matrices. For high penalty parameters, the performance of iterative solvers will degrade, and when direct solvers are used, the quality of the solutions will deteriorate. In this paper, an augmented Lagrange approach is used to cope with the quasi-incompressibility condition. Here, the penalty parameter can be chosen much smaller, and as a consequence, the arising linear systems of equations have better properties. An improved convergence is then observed for the finite element tearing and interconnecting-dual primal domain decomposition method, which is used as an iterative solver. Numerical results for an arterial geometry obtained from ultrasound imaging are presented. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY The paper presents a parallel tetrahedral mesh generation approach based on recursive bidivisions using triangular surfaces. Research was conducted for addressing issues concerning mesh generation reliability and element quality. A novel procedure employing local modification techniques is proposed for repairing the intersecting interdomain mesh instead of directly repeating the bidivision procedure, which improves the robustness of the complete meshing procedure significantly. In addition, a new parallel quality improvement scheme is suggested for optimizing the distributed volume meshes. The scheme is free of any communication cost and highly efficient. Finally, mesh experiments of hundreds of millions of elements are performed to demonstrate the reliability, effectiveness and efficiency of the proposed method and its potential applications to large-scale simulations of complex aerodynamics models. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Summary: This work introduces an innovative parallel fully‐distributed finite element framework for growing geometries and its application to metal additive manufacturing. It is well known that virtual part design and qualification in additive manufacturing requires highly accurate multiscale and multiphysics analyses. Only high performance computing tools are able to handle such complexity in time frames compatible with time‐to‐market. However, efficiency, without loss of accuracy, has rarely held the centre stage in the numerical community. Here, in contrast, the framework is designed to adequately exploit the resources of high‐end distributed‐memory machines. It is grounded on three building blocks: (1) hierarchical adaptive mesh refinement with octree‐based meshes; (2) a parallel strategy to model the growth of the geometry; and (3) state‐of‐the‐art parallel iterative linear solvers. Computational experiments consider the heat transfer analysis at the part scale of the printing process by powder‐bed technologies. After verification against a three‐dimensional (3D) benchmark, a strong‐scaling analysis assesses performance and identifies major sources of parallel overhead. A third numerical example examines the efficiency and robustness of (2) in a curved 3D shape. Unprecedented parallelism and scalability were achieved in this work. Hence, this framework contributes to take on higher complexity and/or accuracy, not only of part‐scale simulations of metal or polymer additive manufacturing but also in welding, sedimentation, atherosclerosis, or any other physical problem where the physical domain of interest grows in time. [ABSTRACT FROM AUTHOR]

Summary: The objective of this contribution is to design a novel methodology to enforce interface conditions preserving higher‐order continuity across the interface. In recent years, isogeometric methods using NURBS as basis functions have gained increasing attention, especially in the context of higher‐order partial differential equations. They require, in general, higher continuity across interfaces, and thus, new methodologies for domain decomposition constraints capable to deal with those requirements have to be developed. In this contribution, we introduce, in a first step, the coupling constraints using a Euclidean norm on the interface and construct new basis functions. A reformulation as saddle point system allows for a comparison with classical mortar approaches and leads finally to an extended mortar method to enforce Cn continuity. [ABSTRACT FROM AUTHOR]

Summary: In this work, we present alternative implementations for the Simultaneous‐FETI (S‐FETI) method. Developed in recent years, this method has shown to be very robust for highly heterogeneous problems. However, the memory cost in S‐FETI is greatly increased and can be a limitation to its use. Our main objective is to reduce this memory usage without losing significant time performance. The algorithm is based on the exploitation of the sparsity patterns found on the block of search directions, allowing to store less vectors per iteration in comparison to a full storage scheme. In addition, different variations for the S‐FETI method are also proposed, including a new treatment for the possible dependencies between directions and the use of the Lumped preconditioner. Several tests are performed in order to establish the impact of the modifications presented in this work compared to the original S‐FETI algorithm. [ABSTRACT FROM AUTHOR]

Summary: We present a parallel fully implicit algorithm for the large eddy simulation (LES) of incompressible turbulent flows on unstructured meshes in three dimensions. The LES governing equations are discretized by a stabilized Galerkin finite element method in space and an implicit second‐order backward differentiation scheme in time. To efficiently solve the resulting large nonlinear systems, we present a highly parallel Newton‐Krylov‐Schwarz algorithm based on domain decomposition techniques. Analytic Jacobian is applied in order to obtain the best achievable performance. Two benchmark problems of lid‐driven cavity and flow passing a square cylinder are employed to validate the proposed algorithm. We then apply the algorithm to the LES of turbulent flows passing a full‐size high‐speed train with realistic geometry and operating conditions. The numerical results show that the algorithm is both accurate and efficient and exhibits a good scalability and parallel efficiency with tens of millions of degrees of freedom on a computer with up to 4096 processors. To understand the numerical behavior of the proposed fully implicit scheme, we study several important issues, including the choices of linear solvers, the overlapping size of the subdomains, and, especially, the accuracy of the Jacobian matrix. The results show that an exact Jacobian is necessary for the efficiency and the robustness of the proposed LES solver. We present a parallel fully implicit algorithm for the large eddy simulation of incompressible turbulent flows on unstructured meshes in three dimensions. To efficiently solve the resulting large nonlinear systems, we present a highly parallel Newton‐Krylov‐Schwarz (NKS) algorithm based on domain decomposition techniques in which an analytic Jacobian is applied in order to obtain the best achievable performance. We then apply the algorithm to the simulation of turbulent flows passing a full‐size high‐speed train with realistic geometry and operating conditions. [ABSTRACT FROM AUTHOR]

SUMMARY This paper presents mathematical derivation of enrichment functions in the extended finite element method for numerical modeling of strong and weak discontinuities. The proposed approach consists in combining the level set method with characteristic functions as well as domain decomposition and reproduction technique. We start with the simple case of a triangular linear element cut by one interface across which displacement field suffers a jump. The main steps towards the derivation of enrichment functions are as follows: (1) extension of the subfields separated by the interface to the whole element domain and definition of complementary nodal variables; (2) construction of characteristic functions for describing the geometry and physical field; (3) determination of the sets of basic nodal variables; (4) domain decompositions according to Step 3 and then reproduction of the physical field in terms of characteristic functions and nodal variables; and (5) comparison of the piecewise interpolations formulated at Steps 3 and 4 with the standard extended finite element method form, which yields enrichment functions. In this process, the physical meanings of both the basic and complementary nodal variables are clarified, which helps to impose Dirichlet boundary conditions. Enrichment functions for weak discontinuities are constructed from deeper insights into the structure of the functions for strong discontinuities. Relationships between the two classes of functions are naturally established. Improvements upon basic enrichment functions for weak discontinuities are performed so as to achieve satisfactory convergence and accuracy. From numerical viewpoints, a simple and efficient treatment on the issue of blending elements is also proposed with implementation details. For validation purposes, applications of the derived functions to heterogeneous problems with imperfect interfaces are presented. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Summary: A multilevel generalization of the dual‐primal finite element tearing and interconnecting (FETI‐DP) domain decomposition method is proposed for very large‐scale discrete problems to address the bottleneck associated with the solution of the coarse problems at such scales. This bottleneck destroys the parallel scalability of the original, two‐level FETI‐DP method when using more than a few thousand processor cores. In the multilevel formulation proposed here, the FETI‐DP method is applied recursively to solve all coarse problems but the smallest one. Crucially, this recursive application of the method is enabled by utilizing a new primal formulation of the augmentation/enrichment process of the coarse problems. The efficiency and scalability of the proposed approach are demonstrated for up to 32 768 processor cores, and large‐scale real‐world and benchmark problems with more than 21 billion degrees of freedom. The obtained performance results show that the three‐ and four‐level FETI‐DP methods exhibit a better scalability than the original two‐level FETI‐DP method. [ABSTRACT FROM AUTHOR]

Summary: Factorized sparse approximate inverse (FSAI) preconditioners are robust algorithms for symmetric positive matrices, which are particularly attractive in a parallel computational environment because of their inherent and almost perfect scalability. Their parallel degree is even redundant with respect to the actual capabilities of the current computational architectures. In this work, we present two new approaches for FSAI preconditioners with the aim of improving the algorithm effectiveness by adding some sequentiality to the native formulation. The first one, denoted as block tridiagonal FSAI, is based on a block tridiagonal factorization strategy, whereas the second one, domain decomposition FSAI, is built by reordering the matrix graph according to a multilevel k‐way partitioning method followed by a bandwidth minimization algorithm. We test these preconditioners by solving a set of symmetric positive definite problems arising from different engineering applications. The results are evaluated in terms of performance, scalability, and robustness, showing that both strategies lead to faster convergent schemes regarding the number of iterations and total computational time in comparison with the native FSAI with no significant loss in the algorithmic parallel degree. [ABSTRACT FROM AUTHOR]

Estimations for the values on interface lines are necessary in a domain decomposition method. However, the accuracy of the estimations is of the first order for most of unconditionally stable domain decomposition schemes. In this paper, a second order of accuracy for the estimations on interface lines is presented. With the new scheme, the optimal number of decomposed subdomains has been observed and proposed. Moreover when the SOR iterative method is used, the optimal over-relaxation parameter is also studied. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper provides a proof of the robustness of the overlapping domain decomposition preconditioners for the linear finite element approximation of second-order elliptic boundary value problems with strongly discontinuous coefficients. By analyzing the eigenvalue distribution of the domain decomposition preconditioned system, we prove that only a small number of eigenvalues may deteriorate with respect to the discontinuous jump or mesh size, and all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and mesh size. As a result, we prove that the convergence rate of the preconditioned conjugate gradient methods is nearly uniform with respect to the large jump and mesh size. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We describe and analyse the parallel implementation of a novel domain decomposition preconditioner for the fast iterative solution of linear systems of algebraic equations arising from the discretization of elliptic partial differential equations (PDEs) in three dimensions. In previous theoretical work, this preconditioner has been proved to be optimal for symmetric positive-definite (SPD) linear systems. In this paper, we provide details of our three-dimensional parallel implementation and demonstrate that the technique may be generalized to the solution of non-symmetric algebraic systems, such as those arising when convection–diffusion problems are discretized using either Galerkin or stabilized finite-element methods (FEMs). Furthermore, we illustrate the potential of the preconditioner for use within an adaptive finite-element framework by successfully solving convection-dominated problems on locally, rather than globally, refined meshes. Copyright © 2003 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a method is presented for analyzing very large finite arrays of tapered-slot antenna elements. The Toeplitz matrix property of regularly spaced arrays, combined with a parametrically truncated coupling radius, generates fixed matrix storage based on the coupling radius rather than array size, thus allowing very large problem analysis. © 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 38: 323–328, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.11050 [ABSTRACT FROM AUTHOR]

This paper shows the ability of a domain decomposition method with a Neumann-Neumann preconditioner to solve industrial and biomechanical problems. For a mid-size biomedical simulation involving soft tissues the results obtained by using this algorithm are comapred to those obtained by more classical methods. The present algorithm thus appears to be more reliable than standard iterative methods and more efficient than a direct method. Copyright © 1999 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]