Multilevel Acceleration Research Articles

NON-NESTED MULTILEVEL ACCELERATION OF MESHLESS SOLUTION OF HEAT CONDUCTION EQUATION IN COMPLEX DOMAINS

In this paper, we developed and demonstrated a non-nested multilevel procedure for solving the heat conduction equation in complex domains using a meshless discretization method. Previous multilevel methods for solving elliptic partial differential equations in complex domains have mostly used one of the four approaches: nested refinement, agglomeration, additive corrections, and algebraic coarsening. Each of these techniques has some issues of generality, robustness, and speed of acceleration. In this paper, we developed a generally applicable multilevel algorithm for partial differential equations discretized on complex domains using unstructured finite volume, finite element, and meshless methods. We applied this multilevel method to accelerate convergence of the set of discrete equations obtained by a meshless technique. The heat conduction equation is discretized at scattered points using a polyharmonic spine (PHS) radial basis function (RBF) interpolation with appended polynomials to achieve exponential convergence of discretization errors. The RBF interpolations are performed over clouds of points, and the partial differential equation is collocated at the scattered points. The multilevel algorithm to solve the set of linear equations utilizes multiple independently generated coarser sets of points. Restriction of residuals and prolongation of the corrections are also performed using the RBF interpolations. The fast convergence of the algorithm is demonstrated for solution of the heat conduction equation in three model complex domains with manufactured solutions. A simple successive over-relaxation point solver is used as the relaxation scheme.

Achieving maximum speedup in multi-level acceleration for massive coronavirus testing

It is well and widely known that sample pooling could provide an effective and efficient way for fast coronavirus testing among massive asymptomatic individuals. The method of multi-level acceleration for asymptomatic COVID-19 screening has been introduced, and for one and two levels, the optimal group sizes have been obtained. However, there are still multiple challenges. First, it is not clear how to find the optimal group sizes for three or more levels. Second, there is lack of closed-form expressions for the optimal group sizes for two or more levels. Third, it is not clear how to determine the optimal number of levels. And last, it is not known what the maximum achievable speedup is. The motivation of this paper is to address all the above challenges. The optimization of a hierarchical pooling strategy includes its number of levels and the group size of each level. In this paper, based on multi-variable optimization and Taylor approximation, we are able to derive closed-form expressions for the optimal number of levels , the optimal group sizes , ,…, , and the maximum possible speedup of a hierarchical pooling strategy of , where is the fraction of infected people. The above speedup is nearly a linear function of the reciprocal of , in the sense that it is asymptotically greater than any sub-linear function of the reciprocal of for any small . Using the results in this paper, we can quickly and easily predict the performance of an optimal hierarchical pooling strategy. For instance, if the fraction of infected people is 0.0001, an 8-level hierarchical pooling strategy can achieve speedup of nearly 400.

A faster than real-time heat transfer model for continuous steel casting

In this paper, a three-dimensional faster than real-time heat transfer model is presented for the continuous casting process. The model features a high computational capability required for real-time prediction, optimization, and control of the casting process. The computational model is implemented with a multi-level acceleration method on GPUs to ultimately shrink the computational time cost. This acceleration method helps to build a proper mapping from numerical models to GPU hardware features and provides two acceleration strategies with slight accuracy loss. With the multi-level acceleration method, a 117× speedup with a max 1.74% relative error compared to a 14-threaded CPU implementation is achieved. Moreover, the accuracy of the model has been verified with the Stefan problem and validated with industry measurement. The relative computational time, which is required to be less than 1 in real-time simulations and significantly far less than 1 in dynamic optimization of the casting process, is adopted to evaluate the real-time ability of the model. The proposed model achieves a relative computational time between 0.0003 for a coarse mesh and 0.015 for a very fine mesh. Our demonstrated high-performance modeling tool herein is envisioned to dominate among the applications of advanced process control and optimization in continuous casting.

3D whole-core neutron transport simulation using 2D/1D method via multi-level generalized equivalence theory based CMFD acceleration

A new 2D/1D method with multi-level generalized equivalence theory (GET) based coarse mesh finite difference (CMFD) acceleration was proposed for the whole core transport calculation in this paper. Fouressential features of this new 2D/1D methods are as follows: (1) Two new defined factors, nodal discontinuity factor (NDF) and modified diffusion coefficient factor (MDF), were defined in this new GET based CMFD method to achieve equivalence between CMFD and the transport solutions and especially insure that only positive coupling coefficients would occur in the CMFD linear system; (2) For accelerating the convergence of the CMFD, a multi-level acceleration scheme was implemented and an innovative self-developed RSILU preconditioned GMRES solver was developed to solve the CMFD linear system; (3) Within the new CMFD framework, 2D MOC and 1D two-node nodal expansion solvers were developed for 3D whole core transport calculation and the sub-plane technique was applied to minimize the nodal error successfully while maintain a reasonable computing time; (4) For implementation, transverse leakage splitting technique was applied to avoid the total source to become negative, and domain decomposition method based on MPI was implemented to take advantages of high performance clusters. The accuracy of this 2D/1D method via multi-level GET based CMFD and the effectiveness and acceleration performance of the multi-level CMFD method were examined for the well-known C5G7 benchmarks problems. The numerical results demonstrate that superior accuracy is achievable and the multi-level acceleration schemeis efficient and enhances the converge speed of both the GET based CMFD acceleration and the MOC calculation.

Multilevel acceleration of scattering-source iterations with application to electron transport

Abstract Acceleration/preconditioning strategies available in the SCEPTRE radiation transport code are described. A flexible transport synthetic acceleration (TSA) algorithm that uses a low-order discrete-ordinates (S N ) or spherical-harmonics (P N ) solve to accelerate convergence of a high-order S N source-iteration (SI) solve is described. Convergence of the low-order solves can be further accelerated by applying off-the-shelf incomplete-factorization or algebraic-multigrid methods. Also available is an algorithm that uses a generalized minimum residual (GMRES) iterative method rather than SI for convergence, using a parallel sweep-based solver to build up a Krylov subspace. TSA has been applied as a preconditioner to accelerate the convergence of the GMRES iterations. The methods are applied to several problems involving electron transport and problems with artificial cross sections with large scattering ratios. These methods were compared and evaluated by considering material discontinuities and scattering anisotropy. Observed accelerations obtained are highly problem dependent, but speedup factors around 10 have been observed in typical applications.

Multi-level acceleration with manifold mapping of strongly coupled partitioned fluid–structure interaction

To accelerate the convergence of strongly coupled partitioned fluid–structure interaction simulations, the manifold mapping algorithm is used which combines a high-fidelity model with a low-fidelity model. A computationally inexpensive low-fidelity fluid–structure interaction (FSI) model is combined with a high-fidelity FSI model in order to accelerate the convergence of the coupling iterations of the high-fidelity FSI model. The manifold mapping algorithm is applied to the fluid–structure interaction problem in order to minimize the fluid–structure interface residual. Originating from multi-fidelity optimization, the manifold mapping algorithm is applied for the first time in a simulation context, instead of an optimization context. The manifold mapping algorithm is applied to an unsteady flow in a one-dimensional tube, incompressible laminar flow over a fixed cylinder with an attached flexible flap, and a wave propagation in a three-dimensional elastic tube problem. A reduction of approximately 50% in terms of high-fidelity iterations is achieved compared to the state-of-the-art Interface Quasi-Newton Inverse Least Squares algorithm. The convergence of the high-fidelity model is accelerated even further when information from previous time steps is reused.

Exact interpolation scheme with approximation vector used as a column of the prolongator

SummaryOur method is a kind of exact interpolation scheme by Achi Brandt et al. In the exact interpolation scheme, for x being the fine‐level approximation of the solution, the coarse‐space V = V(x) is constructed so that x satisfies x∈V. We achieve it simply by adding the vector x as a first column of the prolongator. (The columns of the prolongator P form a computationally relevant basis of the coarse‐space V = Range(P).) The advantages of this construction become obvious when solving non‐linear problems. The cost of enriching the coarse‐space V by the current approximation x is a single dense column of the prolongator that has to be updated each iteration. Our method can be used for multilevel acceleration of virtually any iterative method used for solving both linear and non‐linear systems. Copyright © 2015 John Wiley & Sons, Ltd.

A multi-level surface rebalancing approach for efficient convergence acceleration of 3D full core multi-group fine grid nodal diffusion iterations

A new multi-level surface rebalancing (MLSR) approach has been developed, aimed at enabling an improved non-linear acceleration of nodal flux iteration convergence in 3D steady-state and transient reactor simulation. This development is meant specifically for anticipating computational needs for solving envisaged multi-group diffusion-like SPN calculations with enhanced mesh resolution (i.e. 3D multi-box up to 3D pin-by-pin grid). For the latter grid refinement regime, the previously available multi-level coarse mesh rebalancing (MLCMR) strategy has been observed to become increasingly inefficient with increasing 3D mesh resolution. Furthermore, for very fine 3D grids that feature a very fine axial mesh as well, non-convergence phenomena have been observed to emerge. In the verifications pursued up to now, these problems have been resolved by the new approach. The novelty arises from taking the interface current balance equations defined over all Cartesian box edges, instead of the nodal volume-integrated process-rate balance equation, as an appropriate restriction basis for setting up multi-level acceleration of fine grid interface current iterations. The new restriction strategy calls for the use of a newly derived set of adjoint spectral equations that are needed for computing a limited set of spectral response vectors per node. This enables a straightforward determination of group-condensed interface current spectral coupling operators that are of crucial relevance in the new rebalancing setup. Another novelty in the approach is a new variational method for computing the neutronic eigenvalue. Within this context, the latter is treated as a control parameter for driving another, newly defined and numerically more fundamental eigenvalue to unity. This paper presents a rigorous derivation of the new approach, followed by a comparison on convergence efficiencies, for a number of 3D full core nodal grid resolution regimes, between the previously available multi-level rebalancing setup and the new multi-level surface rebalancing concept. The surface rebalancing methodology and a number of related concepts are covered in the patents EP2091049, EP2287855, EP2287854 and EP2287853 that were granted in 2012.

Research on N-Body Problem with FMM Algorithm Based on FPGA Acceleration

FMM is an efficient algorithm in computing N-body problem. This paper firstly partitions the FMM into 8 procedures. Based on the analysis the running time and computing type of each procedure, we choose key procedures accelerated on FPGA, and then we present the speedup ratio of each accelerated procedure through experiments. According to the FPGA acceleration effect on the procedures, a multi-level acceleration project is proposed. Finally we put forward a hardware architecture of FPGA accelerated high performance computer.

Numerical Solution of the Hartree–Fock Equation in Multilevel Tensor-Structured Format

In this paper, we describe a novel method for a robust and accurate iterative solution of the self-consistent Hartree–Fock equation in $\mathbb{R}^3$ based on the idea of tensor-structured computation of the electron density and the nonlinear Hartree and (nonlocal) exchange operators at all steps of the iterative process. We apply the self-consistent field (SCF) iteration to the Galerkin discretization in a set of low separation rank basis functions that are solely specified by the respective values on a three-dimensional Cartesian grid. The approximation error is estimated by $O(h^3)$, where $h=O(n^{-1})$ is the mesh size of an $n\times n\times n$ tensor grid, while the numerical complexity to compute the Galerkin matrices scales linearly in $n\log n$. We propose the tensor-truncated version of the SCF iteration using the traditional direct inversion in the iterative subspace scheme enhanced by the multilevel acceleration with the grid-dependent termination criteria at each discretization level. This implies that the overall computational cost scales almost linearly in the univariate problem size n. Numerical illustrations are presented for the all electron case of H$_2$O and the pseudopotential case of CH$_4$ and CH$_3$OH molecules. The proposed scheme is not restricted to a priori given rank-1 basis sets, allowing analytically integrable convolution transform with the Newton kernel that opens further perspectives for promotion of the tensor-structured methods in computational quantum chemistry.

The p-Version of the Finite Element Method for the Solution of Two-Dimensional Neutron Diffusion Equations

Based on the hierarchical structure, two types of the p-version of the finite element code are developed for solving the two-dimensional neutron diffusion equations. One is the conventional p-type FEM, and the reactor domain is discretized into finite elements, and each element matrices are assembled and solved. In the other, with the domain decomposition approach, the reactor domain is decomposed into subdomains and each subdomain is solved independently. They are coupled by use of incoming and outgoing partial currents along the interfaces, which are obtained analytically from fluxes. Based on the multi-p V-cycle method, multilevel acceleration is implemented into both of the methods. The methods were tested on the IAEA-2D benchmark problem, the initial core of Ulchin Unit 1, and a MOX-fuel loaded core. It turns out that the methods can generate accurate core multiplication factor and assembly average quantities.

Prediction of turbulent oscillatory flows in complex systems

Publisher Summary The chapter discusses oscillatory flow in a complex geometry at transitional Reynolds numbers. A stochastically based time history reconstruction technique is used to enable comparisons with unfiltered turbulence intensity measurements. The performance of the standard k-e. and low and high Reynolds number k-l model variants is considered. Because of near wall grid distribution sensitivity, the k-e model is not found suitable for complex geometry, transitional Reynolds number flows such as those found in electronic systems. The high Reynolds number k-l model's constrained length scale diminishes the near wall grid distribution sensitivity giving similar results to those for the low Reynolds number variant. Predicted amplitudes and frequencies show significant differences when compared to measurements. If the predicted flow unsteadiness amplitudes are scaled to match the measurements and is incorporated into the stochastic reconstruction technique, an encouraging agreement is found with total intensity measurements. A multilevel convergence acceleration is found to give significant time savings for steady flows, and a novel differential equation-based wall distance algorithm is shown to be effective.

Numerical modeling of two-dimensional device structures using Brandt's multilevel acceleration scheme: application to Poisson's equation

In expanding numerical modeling for electronic and optoelectronic devices from a single dimension to multiple dimensions, a large increase in machine storage space is required. Solution approaches based on relaxation techniques are typically used to minimize this increase, but they can be slow to converge. Presented is an adaption of Brandt's multilevel acceleration scheme for control volume discretizations coupled with solvers based on either Stone's strongly implicit method or the Gauss-Siedel (G-S) method to overcome this speed and storage space problem. This approach is demonstrated by solving Poisson's equation in a two-dimensional amorphous silicon thin-film transistor structure. The structure has a generalized density of states function whose occupancy is computed using nonzero degree Kelvin Fermi-Dirac statistics. It is shown that the use of the multilevel acceleration algorithm gives more than an order of magnitude increase in the asymptotic rate of convergence for the potential distribution in this thin-film transistor. Numerical results of the analysis are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A MULTILEVEL-MULTIGRID TECHNIQUE FOR RECIRCULATING FLOWS

A solution algorithm has been developed for the prediction of recirculating flows. Brandt's multilevel acceleration technique is used with Leonard's QUICK differencing scheme and a modified pressure implicit operator splitting scheme. Intermediate calculations enable a tau error distribution to be used for the identification of regions for local grid refinement, i.e., multigrid. The algorithm was tested for the prediction of laminar flow in a shear-driven and a buoyancy-driven cavity.

MULTIGRID TECHNIQUES FOR THE SOLUTION OF THE PASSIVE SCALAR ADVECTION-DIFFUSION EQUATION

The solution of elliptic passive scalar advection-diffusion equations is required in the analysis of many turbulent flow and convective heat transfer problems. The accuracy of the solution may be affected by the presence of regions containing large gradients of the dependent variables. The multigrid concept of local grid refinement is a method for improving the accuracy of the calculations in these problems. In combination with the multilevel acceleration techniques, an accurate and efficient computational procedure is developed. In addition, a robust implementation of the QUICK finite-difference scheme is described. Calculations of a test problem are presented to quantitatively demonstrate the advantages of the multilevel-multigrid method.

A Multilevel-Multigrid Technique for Recirculating Flows

A solution algorithm has been developed for the prediction of recirculating flows. Brandt's multilevel acceleration technique is used with Leonard's QUICK differencing scheme and a modified pressure implicit operator splitting scheme. Intermediate calculations enable a tau error distribution to be used for the identification of regions for local grid refinement, i.e., multigrid. The algorithm was tested for the prediction of laminar flow in a shear-driven and a buoyancy-driven cavity.