Non-iterative Solver Research Articles

PaScaL_TCS: A versatile solver for large-scale turbulent convective heat transfer problems with temperature-dependent fluid properties

PaScaL_TCS is an efficient and scalable solver for large-scale direct numerical simulations of natural convective flow that considers temperature-dependent fluid properties. For increased scalability on a massive-scale distributed memory system, the solver decomposes the computational domain into a cubic sub-domain. The numerical procedure is based on the monolithic projection method with staggered time discretization [1], which is an implicit and non-iterative solver for wall-bounded domains. The PaScaL_TDMA library is used to solve batched tridiagonal systems, which are partitioned according to domain decomposition. For parallel fast Fourier transform in a Fourier Poisson solver for pressure, two transpose schemes with different communicator sizes are proposed and compared according to the number of cores. An explicit intermediate aggregation scheme for MPI-IO is suggested to reduce the number of processes that simultaneously take part in parallel IO. The overall implementation was evaluated using the Nurion cluster system of the Korea Institute of Science and Technology Information with detailed performance profiling. With the efficient communication, the results showed good strong and weak scalability up to 131,072 cores. The file IO performance improved with the suggested MPI-IO scheme using explicit aggregation, especially when many processes are involved in parallel IO. The solver was verified by conducting large-scale differentially heated vertical convection flow simulations under the Oberbeck–Boussinesq (OB) approximation. The capability of the solver to capture and quantitatively describe the non-OB effect was demonstrated via glycerol Rayleigh–Bénard convection flow simulations. Program summaryProgram Title: PaScaL_TCSCPC Library link to program files:https://doi.org/10.17632/3b2bysrcxm.1Developer's repository link:https://github.com/MPMC-LabLicensing provisions: MITProgramming language: Fortran90Nature of problem: Solving the incompressible Navier-Stokes equations for natural convective flow considering the non-Oberbeck–Boussinesq effect, which considers the temperature dependence of fluids properties, in three-dimensional channel domain.Solution method: PaScaL_TCS uses the monolithic projection-based method with staggered time discretization (MPM-STD) [1]. It employs the Crank-Nicolson scheme along with staggered time which evaluates scalar and vector variables at the (n+1/2) and (n+1) time level, respectively. PaScaL_TDMA [2] is used to solve multiple tridiagonal matrix systems for decoupled energy and momentum equations in a distributed memory system. The Poisson equation solver for the pressure-correction scheme is based on Fourier diagonalization. MPI_Alltoallw is used for parallel FFT. PaScaL_TCS supports a single-file-based parallel IO for efficient data generation in large-scale computing.

Multi-Rate Mixed-Solver for Real-Time Nonlinear Electromagnetic Transient Emulation of AC/DC Networks on FPGA-MPSoC Architecture

Nonlinear phenomena widely exist in AC/DC power systems, which should be accounted for accurately in real-time electromagnetic transient (EMT) simulation for obtaining precise results for hardware-in-the-loop applications. However, iterative solutions such as the Newton-Raphason method that can precisely obtain the results for highly nonlinear elements, are time consuming and computationally onerous. To fully utilize the time space and optimize hardware computation resources without loss of accuracy, this work proposes a novel multi-rate mixed-solver for AC/DC systems, wherein both iterative and non-iterative solvers with different time-steps are applied to the decomposed subsystems, and the linear solvers are reused within each time-step. The proposed solver and the complete real-time emulation system are implemented on FPGA-MPSoC platform. The real-time results are captured by the oscilloscope and verified with PSCAD/EMTDC and SaberRD for system-level and device-level performance evaluation.

Iterative Solvers for Empirical Partial Atomic Charges: Breaking the Curse of Cubic Numerical Complexity.

Rational drug design involves a vast amount of computations to get thermodynamically reliable results and often relies on atomic charges as a means to model electrostatic interactions within the system. Computational inefficiency often hampers the development of new and wider dissemination of the known methods; thus, any source to speed up the calculations without a sacrifice in quality is warranted. At the heart of many empirical methods of calculating atomic charges is the solution of a system of linear algebraic equations (SLAE). The classical method of solving SLAE-the Gauss method-has in general case a cubic computational complexity. It is shown that the use of iterative methods for solving SLAE, characteristic to typical empirical atomic charge calculation methods, makes it possible to significantly reduce the amount of calculations and to obtain a computational complexity approaching a quadratic one. Despite the fact that this phenomenon is well-known in numerical methods, iterative solvers surprisingly do not seem to have been systematically applied to calculation of atomic charges via empirical schemes. Another finding is the relative values of the matrix elements, determined by the physical grounds of the interactions within the empirical system, generally lead to SLAE's with well-defined matrices, suited to use with iterative solvers to fasten computation compared to using the noniterative solvers. This finding broadens the applicability range of atomic charges obtained with empirical methods for such cases as, e.g., account of polarizability via "on-the-fly" recalculation of charges in changing surroundings within the force fields in molecular dynamics settings.

Schur complement solver for Quantum Monte-Carlo simulations of strongly interacting fermions

We present a non-iterative solver based on the Schur complement method for sparse linear systems of special form which appear in Quantum Monte-Carlo (QMC) simulations of strongly interacting fermions on the lattice. While the number of floating-point operations for this solver scales as the cube of the number of lattice sites, for practically relevant lattice sizes it is still significantly faster than iterative solvers such as the Conjugate Gradient method in the regime of strong inter-fermion interactions, for example, in the vicinity of quantum phase transitions. The speed-up is even more dramatic for the solution of multiple linear systems with different right-hand sides. We present benchmark results for QMC simulations of the tight-binding models on the hexagonal graphene lattice with on-site (Hubbard) and non-local (Coulomb) interactions, and demonstrate the potential for further speed-up using GPU.

A reactive transport simulator for variable porosity problems

ReactMiCP, a new reactive transport simulator was developed based on the semismooth speciation solver SpecMiCP. Its main feature is a sequential iterative operator splitting algorithm where macroscopic model parameters are explicitly included in the formulation. Its correctness, robustness, and efficiency are tested against the MoMaS benchmark and two sets of cement paste lab experiments. We show that a robust speciation solver is a key requirement for good performance of the reactive transport simulator. We also demonstrate that a sequential iterative solver should be preferred over non-iterative solvers when using operator splitting. The flexibility and the speed of the simulator are used to test the influence of the database, the initial condition, and the diffusion coefficient model for the cement paste simulations.

GcsDecolor: Gradient Correlation Similarity for Efficient Contrast Preserving Decolorization.

This paper presents a novel gradient correlation similarity (Gcs) measure-based decolorization model for faithfully preserving the appearance of the original color image. Contrary to the conventional data-fidelity term consisting of gradient error-norm-based measures, the newly defined Gcs measure calculates the summation of the gradient correlation between each channel of the color image and the transformed grayscale image. Two efficient algorithms are developed to solve the proposed model. On one hand, due to the highly nonlinear nature of Gcs measure, a solver consisting of the augmented Lagrangian and alternating direction method is adopted to deal with its approximated linear parametric model. The presented algorithm exhibits excellent iterative convergence and attains superior performance. On the other hand, a discrete searching solver is proposed by determining the solution with the minimum function value from the linear parametric model-induced candidate images. The non-iterative solver has advantages in simplicity and speed with only several simple arithmetic operations, leading to real-time computational speed. In addition, it is very robust with respect to the parameter and candidates. Extensive experiments under a variety of test images and a comprehensive evaluation against existing state-of-the-art methods consistently demonstrate the potential of the proposed model and algorithms.

Computer implementations of iterative and non-iterative crystal plasticity solvers on high performance graphics hardware

We present parallel implementations of Newton---Raphson iterative and spectral based non-iterative solvers for single-crystal visco-plasticity models on a specialized computer hardware integrating ...

TOA-Based Self-Calibration of Dual-Microphone Array

In this paper, we study the time-of-arrival (TOA) based self-calibration problem of dual-microphone array for known and unknown rack distance, and also for different combinations of dimension for the affine spaces spanned by the receivers and by the senders. Particularly, we analyze the minimum cases and present minimum solvers for the case of microphones and speakers in 3-D/3-D, in 2-D/3-D, and in 3-D/2-D, with given or unknown rack length. We identify for each of these minimal problems the number of solutions in general and develop efficient and numerically stable, non-iterative solvers. Solving these problems are of both theoretical and practical interest. This includes understanding what the minimal problems are and how and when they can be solved. The solvers can be used to initialize local optimization algorithms for finding the maximum likelihood estimate of the parameters. The solvers can also be used for robust estimation of the parameters in the presence of outliers, using, e.g., RANSAC algorithms. We demonstrate that the proposed solvers are numerically stable in synthetic experiments. We also demonstrate how the solvers can be used with the RANSAC paradigm. We also apply our method for several real data experiments, using ultra-wide-band measurements and using acoustic data.

Spectral database solutions to elasto-viscoplasticity within finite elements: Application to a cobalt-based FCC superalloy

We present a computationally efficient, multi-scale model based on crystal plasticity theory for simulations of heterogeneous plastic deformation of metallic components in commercial finite element (FE) codes. Although the model can handle a single crystal, the primary purpose with the present model is to embed a meso-scale polycrystal homogenization at a FE integration point, with the meso-scale homogenization being a Taylor-type model. The responses of single crystals are obtained using a recently developed non-iterative solver, which is based on databases of discrete Fourier transforms allowing for fast retrieval of pre-computed crystal plasticity solutions. We have shown that, when this non-iterative solver is used in place of the Newton–Raphson iterative solver, substantial wall-clock speedups can be achieved. Additionally, the implementation presented here takes the advantage of calculations of the elastic properties based on the spectral representation. To calibrate and validate the new FE elasto-viscoplastic model, we use stress-strain curves and texture data of the cobalt-based face-centered cubic superalloy Haynes 25. For this purpose, the material was deformed monotonically in compression over a wider range of strain rates and temperatures. The model is subsequently applied to simulate the macro-scale mechanical response of the material in compression and rolling. We show that the predictions of the model compare favorably with experimental measurements in terms of the mechanical response and texture evolution. Finally, the evolution of the underlying crystallographic texture in rolling predicted by the new model is compared against the corresponding predictions from the finite element visco-plastic self-consistent (FE-VPSC) model. It is observed that both implementations produce similar predictions, but the model presented here is substantially faster.

2.5D forward and inverse modeling for interpreting low-frequency electromagnetic measurements

We present 2.5D fast and rigorous forward and inversion algorithms for deep electromagnetic (EM) applications that include crosswell and controlled-source EM measurements. The forward algorithm is based on a finite-difference approach in which a multifrontal LU decomposition algorithm simulates multisource experiments at nearly the cost of simulating one single-source experiment for each frequency of operation. When the size of the linear system of equations is large, the use of this noniterative solver is impractical. Hence, we use the optimal grid technique to limit the number of unknowns in the forward problem. The inversion algorithm employs a regularized Gauss-Newton minimization approach with a multiplicative cost function. By using this multiplicative cost function, we do not need a priori data to determine the so-called regularization parameter in the optimization process, making the algorithm fully automated. The algorithm is equipped with two regularization cost functions that allow us to reconstruct either a smooth or a sharp conductivity image. To increase the robustness of the algorithm, we also constrain the minimization and use a line-search approach to guarantee the reduction of the cost function after each iteration. To demonstrate the pros and cons of the algorithm, we present synthetic and field data inversion results for crosswell and controlled-source EM measurements.

A Hierarchical 3-D Direct Helmholtz Solver by Domain Decomposition and Modified Fourier Method

The paper contains a noniterative solver for the Helmholtz and the modified Helmholtz equations in a hexahedron. The solver is based on domain decomposition. The solution domain is divided into mostly parallelepiped subdomains. In each subdomain a particular solution of the nonhomogeneous Helmholtz equation is first computed by a fast spectral 3-D method which was developed in our earlier papers (see, for example, SIAM J. Sci. Comput., 20 (1999), pp. 2237--2260). This method is based on the application of the discrete Fourier transform accompanied by a subtraction technique. For high accuracy the subdomain boundary conditions must be compatible with the specified inhomogeneous right-hand side at the edges of all the interfaces. In the following steps the partial solutions are hierarchically matched. At each step pairs of adjacent subdomains are merged into larger units. The paper describes in detail the matching algorithm for two boxes which is a basis for the domain decomposition scheme. The hierarchical approach is convenient for parallelization and can minimize the global communication. The algorithm requires O(N3, log,N) operations, where N is the number of grid points in each direction.

Fast solvers for one dimensional light scattering in the discrete dipole approximation

In this paper we propose new algorithms for solution of light scattering on non-spherical particles using one-dimensional variant of discrete dipole approximation. We discuss recent advances in algorithms for matrices with structures in context of the discrete dipole approximation and show that it is possible to apply these advances to form non-iterative solvers and improve algorithmic complexity in case of many incoming plane parallel waves.

A Hierarchical 3-D Poisson Modified Fourier Solver by Domain Decomposition

We present a Domain Decomposition non-iterative solver for the Poisson equation in a 3-D rectangular box. The solution domain is divided into mostly parallelepiped subdomains. In each subdomain a particular solution of the non-homogeneous equation is first computed by a fast spectral method. This method is based on the application of the discrete Fourier transform accompanied by a subtraction technique. For high accuracy the subdomain boundary conditions must be compatible with the specified inhomogeneous right hand side at the edges of all the interfaces. In the following steps the partial solutions are hierarchically matched. At each step pairs of adjacent subdomains are merged into larger units. In this paper we present the matching algorithm for two boxes which is a basis of the domain decomposition scheme. The hierarchical approach is convenient for parallelization and minimizes the global communication. The algorithm requires O(N3elogeN) operations, where N is the number of grid points in each direction.

Numerical simulation of two‐phase flow in pipes using Godunov method

The partial differential equations that describe two-phase flows in pipes are highly non-linear due to the strong dependence between pressure and wave celerity. The possible appearance of shock waves makes Godunov schemes very attractive, for they can handle such discontinuities automatically. Fluxes at the cell interfaces are computed by solving a Riemann problem. To do so, an approximate state, non-iterative solver designed in a previous study is used. The treatment of boundary conditions uses an iterative procedure. Existing approximations for the sound celerity under the isothermal assumption are generalized to other situations. A comparison between the isothermal and the adiabatic assumptions shows that such assumptions play an important role in the behaviour of the solution. Finally, numerical results obtained using the first-order Godunov method on representative test cases are presented and the need for higher-order reconstruction techniques is acknowledged. Copyright © 2001 John Wiley & Sons, Ltd.

Riemann solvers for water hammer simulations by Godunov method

The water hammer phenomenon can be described by a 2×2 system of hyperbolic partial differential equations (PDEs). Numerical solution of these PDEs using finite-volume schemes is investigated herein. The underlying concept of the Godunov scheme is the Riemann problem, that must be solved to provide fluxes between the computational cells. The presence of the kinetic terms in the momentum equation determines the existence of shock and rarefaction waves, which influence the design of the Riemann solver. Approximation of the expressions for the Riemann invariants and jump relationships can be used to derive first- and second-order approximate, non-iterative solvers. The first-order approximate solver is almost 2000 times faster than the exact one, but gives inaccurate predictions when the densities and celerities are low. The second-order approximate solver gives very accurate solutions, and is 300 times faster than the exact, iterative one. Detailed indications are provided in the appendices for the practical implementation of the Riemann solvers described herein. Copyright © 2000 John Wiley & Sons, Ltd.