Sequential Solver Research Articles

A matrix-free parallel two-level deflation preconditioner for two-dimensional heterogeneous Helmholtz problems

We propose a matrix-free parallel two-level deflation method combined with the Complex Shifted Laplacian Preconditioner (CSLP) for two-dimensional heterogeneous Helmholtz problems encountered in seismic exploration, antennas, and medical imaging. These problems pose challenges in terms of accuracy and convergence due to scalability issues with numerical solvers. Motivated by the limitations imposed by excessive computational time and memory constraints when employing a sequential solver with constructed matrices, we parallelize the two-level deflation method without constructing any matrices. Our approach utilizes preconditioned Krylov subspace methods and approximates the CSLP preconditioner with a parallel geometric multigrid V-cycle. For the two-level deflation, standard inter-grid deflation vectors and further high-order deflation vectors are considered. As another main contribution, the matrix-free Galerkin coarsening approach and a novel re-discretization scheme as well as high-order finite-difference schemes on the coarse grid are studied to obtain wavenumber-independent convergence. The optimal settings for an efficient coarse-grid problem solver are investigated. Numerical experiments of model problems show that the wavenumber independence has been obtained for medium wavenumbers. The matrix-free parallel framework shows satisfactory weak and strong parallel scalability.

Parallel Beam Search Algorithms for Domain-Independent Dynamic Programming

Domain-independent dynamic programming (DIDP), a model-based paradigm based on dynamic programming, has shown promising performance on multiple combinatorial optimization problems compared with mixed integer programming (MIP) and constraint programming (CP). The current DIDP solvers are based on heuristic search, and the state-of-the-art solver, complete anytime beam search (CABS), uses beam search. However, the current DIDP solvers cannot utilize multiple threads, unlike state-of-the-art MIP and CP solvers. In this paper, we propose three parallel beam search algorithms and develop multi-thread implementations of CABS. With 32 threads, our multi-thread DIDP solvers achieve 9 to 39 times speedup on average and significant performance improvement over the sequential solver, finding the new best solutions for two instances of the traveling salesperson problem with time windows. In addition, our solvers outperform multi-thread MIP and CP solvers in four of the six combinatorial optimization problems evaluated.

A parallel grad‐div stabilized finite element algorithm for the Navier–Stokes equations with a nonlinear damping term

AbstractIn this work, we propose a parallel grad‐div stabilized finite element algorithm for the Navier–Stokes equations attached with a nonlinear damping term, using a fully overlapping domain decomposition approach. In the proposed algorithm, we calculate a local solution in a defined subdomain on a global composite mesh which is fine around the defined subdomain and coarse in other regions. The algorithm is simple to carry out on the basis of available sequential solvers. By a local a priori estimate of the finite element solution, we deduce error bounds of the approximations from our presented algorithm. We perform also some numerical experiments to verify the effectiveness of the proposed algorithm.

Scalable High-Quality Hypergraph Partitioning

Balanced hypergraph partitioning is an NP-hard problem with many applications, e.g., optimizing communication in distributed data placement problems. The goal is to place all nodes across k different blocks of bounded size, such that hyperedges span as few parts as possible. This problem is well-studied in sequential and distributed settings, but not in shared-memory. We close this gap by devising efficient and scalable shared-memory algorithms for all components employed in the best sequential solvers without compromises with regards to solution quality. This work presents the scalable and high-quality hypergraph partitioning framework Mt-KaHyPar . Its most important components are parallel improvement algorithms based on the FM algorithm and maximum flows, as well as a parallel clustering algorithm for coarsening – which are used in a multilevel scheme with log (n) levels. As additional components, we parallelize the n -level partitioning scheme, devise a deterministic version of our algorithm, and present optimizations for plain graphs. We evaluate our solver on more than 800 graphs and hypergraphs, and compare it with 25 different algorithms from the literature. Our fastest configuration outperforms almost all existing hypergraph partitioners with regards to both solution quality and running time. Our highest-quality configuration achieves the same solution quality as the best sequential partitioner KaHyPar , while being an order of magnitude faster with ten threads. Thus, two of our configurations occupy all fronts of the Pareto curve for hypergraph partitioning. Furthermore, our solvers exhibit good speedups, e.g., 29.6x in the geometric mean on 64 cores (deterministic), 22.3x (log (n) -level), and 25.9x ( n -level).

Firming Technologies to Reach 100% Renewable Energy Production in Australia’s National Electricity Market (NEM)

Australia has committed to reducing its greenhouse gas emissions in a manner consistent with limiting anthropogenic climate change to no more than 2 degrees Celsius. One of the ways in which this commitment is being realised is through a shift towards variable renewable energy (VRE) within Australia’s National Electricity Market (NEM). Substituting existing dispatchable thermal plant with VRE requires consideration of long-term energy resource adequacy given the unpredictability of solar and wind resources. While pumped hydro and battery storage are key technologies for addressing short-term mismatches between resource availability and demand, they may be unable to cost effectively address ‘energy droughts’. In this article, we present a time sequential solver model of the NEM and an optimal firming technology plant mix to allow the system to be supplied by 100% VRE. Our conclusion is that some form of fuel-based technology (most likely hydrogen) will probably be required. This has important implications for Australian energy policy.

On the use of a multigrid-reduction-in-time algorithm for multiscale convergence of turbulence simulations

Simulations of turbulent flow present challenges in terms of accuracy and affordability on modern highly-parallel computer architectures. A multigrid-reduction-in-time algorithm is used to provide a framework for separately evolving different scales of turbulence and for parallelizing the temporal domain, thereby increasing the concurrency. It is hypothesized that the space–time locality of the small scales of turbulence can be used to circumvent difficulties in applying temporal multigrid to flows dominated by inertial physics. For algorithms that fall well short of spectral accuracy (fourth-order is used in this work) attention must be paid to the accuracy of features on scales transferred between multigrid levels. Numerical experiments were performed using implicit large-eddy simulation. Results from applying the approach to an infinite-Reynolds number Taylor–Green flow and a double-shear flow at a Reynolds number of 11650 provide strong evidence that the approach has merit. The multigrid-reduction-in-time framework can be used to parallelize the temporal domain of a high-Reynolds-number turbulent flow and permit independent convergence of different scales. Establishing this foundation allows for future research in reducing the wall-clock time to solve turbulent flows while retaining the same accuracy as sequential solvers. Current performance results from parallelizing the temporal domain are not competitive with those from sequential-in-time methods.

A parallel grad-div stabilized finite element algorithm for the Stokes equations with damping

This work studies a parallel grad-div stabilized finite element algorithm for the damped Stokes equations. In this algorithm, in the light of a fully overlapping domain decomposition technique, we solve a global grad-div stabilized problem to compute a local solution in an intersecting subdomain on a global composite mesh, which is fine in the subdomain and rough elsewhere, making the proposed algorithm easy to implement based on an available sequential solver. We derive error bounds of the approximate solutions from our presented algorithm by the theoretical tool of local a priori estimate for the grad-div stabilized finite element solution. Numerical results verify the validity of the theoretical analysis and demonstrate the benefits of the proposed algorithm. On the one hand, compared with the counterpart one excluding grad-div stabilization, this algorithm can reduce significantly the effect of pressure on the approximate velocities, and hence, yields much better approximate velocities in the case of small viscosities. On the other hand, it takes much less computational time in getting approximate solutions with a comparable accuracy than the standard grad-div stabilization method.

A multiscale finite element method for coupled heat and water transfer in heterogeneous soils

Large-scale modeling of coupled heat and water transfer (CHWT) is challenging due to the spatial variabilities of soil hydraulic and thermal properties. A multi-scale finite element method (MsFEM) has been designed for simulating liquid water fluxes in unsaturated soils. In this study, the MsFEM approach is expanded as a new scheme that can handle CHWT in soils. Two groups of MsFEM basis functions are constructed to incorporate the heterogeneities in soil hydraulic conductivity and thermal conductivity, and a Petrov-Galerkin formulation is adopted to implement the proposed MsFEM scheme. The MsFEM scheme is also exploited as a sequential solver when heat transfer and water transfer are expressed in a partially coupled formulation (Wang et al., 2022a). Numerical examples illustrate that, without essentially increasing the computing load, the MsFEM scheme can improve the accuracy by up-to 30% compared to the standard finite element method (FEM), especially for thermally driven water transfer. Some fine scale spatial variations in soil temperature can only be revealed with the MsFEM scheme. If the MsFEM sequential solver is applied, the coupled heat and water transfer model can be rewritten into a sequence of modules, where liquid water transfer, heat transfer and vapor transfer are solved step-by-step, but the thermally driven liquid water is omitted due to its relatively small value (Wang et al., 2022a). With the MsFEM sequential solver, a flexible modeling architecture can be achieved at the cost of a relatively small increase of error (<5%). Therefore, the MsFEM scheme presented in this study is an effective numerical approach to simulating CHWT in heterogeneous soils.

Faster Support Vector Machines

The time complexity of support vector machines (SVMs) prohibits training on huge datasets with millions of data points. Recently, multilevel approaches to train SVMs have been developed to allow for time-efficient training on huge datasets. While regular SVMs perform the entire training in one—time-consuming—optimization step, multilevel SVMs first build a hierarchy of problems decreasing in size that resemble the original problem and then train an SVM model for each hierarchy level, benefiting from the solved models of previous levels. We present a faster multilevel support vector machine that uses a label propagation algorithm to construct the problem hierarchy. Extensive experiments indicate that our approach is up to orders of magnitude faster than the previous fastest algorithm while having comparable classification quality. For example, already one of our sequential solvers is on average a factor 15 faster than the parallel ThunderSVM algorithm, while having similar classification quality. 1

Using an Algorithm Portfolio to Solve Sokoban

The game of Sokoban is an interesting platform for algorithm research. It is hard for humans and computers alike. Even small levels can take a lot of computation for all known algorithms. In this paper we will describe how a search based Sokoban solver can be structured and which algorithms can be used to realize each critical part. We implement a variety of those, construct a number of different solvers and combine them into an algorithm portfolio. The solver we construct this way can outperform existing solvers when run in parallel, that is, our solver with 16 processors outperforms the previous sequential solvers.

Adaptive Parallelization for Constraint Satisfaction Search

This paper introduces two adaptive paradigms that parallelize search for solutions to constraint satisfaction problems. Both are intended for any sequential solver that uses contention-oriented variable-ordering heuristics and restart strategies. Empirical results demonstrate that both paradigms improve the search performance of an underlying sequential solver, and also solve challenging problems left open after recent solver competitions.

Local and parallel stabilized finite element methods based on full domain decomposition for the stationary Stokes equations

Abstract Based upon full domain decomposition, local and parallel stabilized finite element methods for the stationary Stokes equations are proposed and analysed, where the quadratic equal-order finite elements are employed for the velocity and pressure approximations, and a stabilized term based on two local Gauss integrations is used to offset the discrete pressure space to circumvent the discrete inf-sup condition. In the proposed parallel method, all of the computations are performed on the locally refined global grids that are fine around the interested subdomain and coarse elsewhere, making the method easy to implement based on a sequential solver with low communication cost. Stability and optimal error estimates of the present methods are deduced. Numerical results on examples including a problem with known analytic solution, lid-driven cavity flow, backward-facing step flow and flow around a cylinder are given to verify the theoretical predictions and demonstrate the high efficiency of the method. Results show that our parallel method can provide an approximate solution with the convergence rate of the same order as the solution computed by the standard stabilized finite element method, with a substantial reduction in computational time.

Parallel iterative stabilized finite element algorithms for the Navier–Stokes equations with nonlinear slip boundary conditions

AbstractBased on full domain partition technique, some parallel iterative pressure projection stabilized finite element algorithms for the Navier–Stokes equations with nonlinear slip boundary conditions are designed and analyzed. In these algorithms, the lowest equal‐order P1 − P1 elements are used for finite element discretization and a local pressure projection stabilized method is used to counteract the invalidness of the discrete inf‐sup condition. Each subproblem is solved on a global composite mesh with the vast majority of the degrees of freedom associated with the particular subdomain that it is responsible for and hence can be solved in parallel with other subproblems by using an existing sequential solver without extensive recoding. All of the subproblems are nonlinear and are independently solved by three kinds of iterative methods. We estimate the optimal error bounds of the approximate solutions with the use of some (strong) uniqueness conditions. Numerical results are also given to demonstrate the effectiveness of the parallel algorithms.

Parallel iterative finite-element algorithms for the Navier–Stokes equations with nonlinear slip boundary conditions

Abstract Based on full domain partition, three parallel iterative finite-element algorithms are proposed and analyzed for the Navier–Stokes equations with nonlinear slip boundary conditions. Since the nonlinear slip boundary conditions include the subdifferential property, the variational formulation of these equations is variational inequalities of the second kind. In these parallel algorithms, each subproblem is defined on a global composite mesh that is fine with size h on its subdomain and coarse with size H (H ≫ h) far away from the subdomain, and then we can solve it in parallel with other subproblems by using an existing sequential solver without extensive recoding. All of the subproblems are nonlinear and are independently solved by three kinds of iterative methods. Compared with the corresponding serial iterative finite-element algorithms, the parallel algorithms proposed in this paper can yield an approximate solution with a comparable accuracy and a substantial decrease in computational time. Contributions of this paper are as follows: (1) new parallel algorithms based on full domain partition are proposed for the Navier–Stokes equations with nonlinear slip boundary conditions; (2) nonlinear iterative methods are studied in the parallel algorithms; (3) new theoretical results about the stability, convergence and error estimates of the developed algorithms are obtained; (4) some numerical results are given to illustrate the promise of the developed algorithms.

A parallel stabilized finite element variational multiscale method based on fully overlapping domain decomposition for the incompressible Navier-Stokes equations

Based on a fully overlapping domain decomposition approach, a parallel stabilized finite element variational multiscale method for the incompressible Navier-Stokes equations is proposed, where the stabilizations both for the velocity and pressure are based on two local Gauss integrations at the element level. The basic idea of the method is to use a locally refined global mesh to compute a stabilized solution in the given subdomain of interest. The proposed method only requires the application of an existing Navier-Stokes sequential solver on the locally refined global mesh associated with each subdomain, and thus can reuse the existing sequential solver without substantial recoding. Error bound of the approximate solutions from the proposed method is estimated with the use of local a priori error estimate for the stabilized solution. Algorithmic parameter scalings of the method are also derived. Some numerical simulations are presented to demonstrate the effectiveness of the method.

Interactive simulation of realistic flexible and tearable membrane using virtual reality and haptic force-feedback interface

The paper deals with the description of a methodology for addressing the real-time simulation of a soft membrane that can be deformed and torn by the interactive two-way action of the user. The methodology makes use of the combination of an efficient real-time solver, a haptic interface with force feedback and it is implemented in a virtual reality environment in order to achieve a very high level of immersion. The elastic properties of the membrane and the corresponding haptic feedback are modelled using an accurate structural model based on a flexible-rigid (f-rigid) multibody model. The f-rigid approach is a trade-off between the accuracy of a full finite element model and the computational efficiency of a multibody model able to be solved using high efficiency sequential impulse solver. The f-rigid model is verified by dedicated experimental tests about nonlinear deformation and rupture. The proposed methodology can be the base of the development of interactive training environments, ergonomic and usability studies and for the assessment and optimization of product design.

Local and parallel stabilized finite element algorithms based on the lowest equal-order elements for the steady Navier–Stokes equations

Based on a fully overlapping domain decomposition approach, local and parallel stabilized finite element algorithms are proposed and investigated for the steady incompressible Navier–Stokes equations, where the inf−sup unstable lowest equal-order P1−P1 finite element pairs are used and the stabilized term is based on two local Gauss integrations defined by the difference between a consistent and under-integrated matrix of pressure interpolants. In these algorithms, each processor computes a local stabilized solution in its own subdomain using a global grid that is locally refined around its own subdomain, making the algorithms have low communication cost and easy to implement based on a sequential solver. Using the technical tool of the local a priori estimate for the stabilized solution, error bounds of the proposed algorithms are derived. Theoretical and numerical results show that, the algorithms can yield an approximate solution with an accuracy comparable to that of the standard stabilized finite element solution with a substantial decrease in CPU time.

The effect of hydrostatic pressure on corrosion electric field considering the mechanochemical coupling effect

In order to study the effect of hydrostatic pressure on the corrosion electric field of ship, the method of filling gas pressure with seamless steel cylinder was used to simulate the hydrostatic pressure on the submarine. The mechanochemical coupling model of corrosion electric field was established by using the solid mechanics and the current distribution (boundary element method, BEM) module in the COMSOL simulation software. The two physical fields were solved by using the sequential solver setup, and the structural stress simulated by the solid mechanics module was coupled to the expressions of equilibrium potential and exchange current density of electrode reaction, which were taken as the boundary conditions of the current distribution simulation module. Finally, the simulation results of the mechanochemical coupling model were compared with the measured corrosion electric field. The results show that the distribution characteristics of corrosion electric field and the variation trend of electric field with gas pressure obtained by simulation have a good agreement with the experimental results, moreover, the errors of peak-to-peak value of electric field obtained by experiment and simulation are within 20%, which demonstrates that the mechanochemical coupling model of corrosion electric field established in this paper can effectively predict the distribution characteristics of electric field under the synergistic action of stress and corrosion medium. When the gas pressure is 0 MPa, the maximum value of electric field modulus is 0.262 mV/m. When the gas pressure increases to 3 MPa, 6 MPa, and 9 MPa, respectively, the maximum value of electric field modulus correspondingly increases to 0.288 mV/m (9.92% increase), 0.317 mV/m (20.10% increase), and 0.348 mV/m (32.82% increase), therefore, the effect of stress on corrosion electric field cannot be neglected.

Sequential Time-Optimal Algorithm for Extended Path Tracking Problem

Abstract This paper presents a fast and easily implementable path tracking algorithm for robots. Usually, for a path tracking problem, the goal is to move the robot on a predefined path, while the joint velocities and accelerations are kept within their limits. This paper deals with the extended case, constraining the forces applied to the objects at the manipulator. First, a problem with a special set of constraints is presented, and a sequential solver method is formulated. The presented sequential solver algorithm has significant computational benefits compared to the direct transcription approach. Then, a practical example is introduced where the proposed algorithm can be applied. At last, the algorithm is validated by real-life experimental results with a six degrees-of-freedom robotic arm.

A parallel stabilized finite element method based on the lowest equal-order elements for incompressible flows

Based on a fully overlapping domain decomposition technique, a parallel stabilized equal-order finite element method for the steady Stokes equations is presented and studied. In this method, each processor computes a local stabilized finite element solution in its own subdomain by solving a global problem on a global mesh that is locally refined around its subdomain, where the lowest equal-order finite element pairs (continuous piecewise linear, bilinear or trilinear velocity and pressure) are used for the finite element discretization and a pressure-projection-based stabilization method is employed to circumvent the discrete inf–sup condition that is invalid for the used finite element pairs. The parallel stabilized method is unconditionally stable, free of parameter and calculation of derivatives, and is easy to implement based on an existing sequential solver. Optimal error estimates are obtained by the theoretical tool of local a priori error estimates for finite element solutions. Numerical results are also given to verify the theoretical predictions and illustrate the effectiveness of the method.