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DOE OSTI2017

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

We present two efficient iterative algorithms for solving the linear response eigen- value problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into a product eigenvalue problem that is self-adjoint with respect to a K-inner product. This product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. However, the other component of the eigenvector can be easily recovered in a postprocessing procedure. Therefore, the algorithms we present here are more efficient than existing algorithms that try to approximate both components of the eigenvectors simultaneously. The efficiency of the new algorithms is demonstrated by numerical examples.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

N 5 -Scaling Excited-State-Specific Perturbation Theory

In this paper, we show that by working in a basis similar to that of the natural transition orbitals and using a modified zeroth-order Hamiltonian, the cost of a recently introduced perturbative correction to excited-state mean field theory can be reduced from seventh to fifth order in the system size. The (occupied) 2 (virtual) 3 asymptotic scaling matches that of ground-state second-order Møller-Plesset theory but with a significantly higher prefactor because the bottleneck is iterative: it appears in the Krylov-subspace-based solution of the linear equation that yields the first-order wave function. Here, we discuss the details of the modified zeroth-order Hamiltonian we use to reduce the cost and the automatic code generation process we used to derive and verify the cost scaling of the different terms. Overall, we find that our modifications have little impact on the method's accuracy, which remains competitive with singles and doubles equation-of-motion coupled cluster.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2018

Global Existence and Decay Rates of the Solutions Near Maxwellian for Non-linear Fokker–Planck Equations

In this paper, we study the global existence and decay rates of the solutions near Maxwellian for non-linear Fokker–Planck equations in the whole space. The global existence is proved by combining uniform-in-time energy estimates with local solution constructed by Picard type iteration sequence. The decay rates of the nonlinear model is obtained by using the precise spectral analysis of the linearized Fokker–Planck operator as well as the energy method. The nonlinearity in the model brings new difficulty to the energy estimates, which is resolved by additional tailored weighted-in-v energy estimates suitable for Fokker–Planck operators.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2018

Pushing memory bandwidth limitations through efficient implementations of Block-Krylov space solvers on GPUs

Lattice quantum chromodynamics simulations in nuclear physics have benefited from a tremendous number of algorithmic advances such as multigrid and eigenvector deflation. These improve the time to solution but do not alleviate the intrinsic memory-bandwidth constraints of the matrix-vector operation dominating iterative solvers. Batching this operation for multiple vectors and exploiting cache and register blocking can yield a super-linear speed up. Block-Krylov solvers can naturally take advantage of such batched matrix-vector operations, further reducing the iterations to solution by sharing the Krylov space between solves. However, practical implementations typically suffer from the quadratic scaling in the number of vector-vector operations. Here, using the QUDA library, we present an implementation of a block-CG solver on NVIDIA GPUs which reduces the memory-bandwidth complexity of vector-vector operations from quadratic to linear. We present results for the HISQ discretization, showing a 5x speedup compared to highly-optimized independent Krylov solves on NVIDIA's SaturnV cluster.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2023

Advanced two-phase subchannel method via non-linear iteration

A fast-running, robust two-phase flow, sub-channel model is presented based on non-linear solution of the steady-state subchannel fluid flow equations. The drift-flux model solves for conservation of liquid and vapor mass, mixture energy, and axial and transverse mixture momentum as part of an efficient planar marching scheme and nonlinear, nested outer and inner iteration. Here, models based on mechanistic subcooled boiling, two-phase turbulent void mixing, and drift are included. Solution verification and mesh convergence studies were performed for modern GE 10 × 10 fuel geometry and are shown to have excellent convergence behavior. Run time performance for a 50 axial mesh model showed 2.2 seconds on a single CPU core to tightly converge all 3D distributions (flow, void, pressure) for the GE 10 × 10 fuel geometry, supporting its efficient use within the Virtual Environment for Reactor Applications boiling water reactor framework.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2018

Fast iterative refinement method for mixed systems of integral and fractional integro-differential equations

The authors’ new method of iterative refinement has been successfully applied to the linear fractional Fredholm integro-differential equation. In this work, we adapt our method to obtain approximate analytical solutions for linear Volterra equations of both the first and the second kind of fractional type. A detailed convergence analysis is presented for each kind. The authors also apply their method to the challenging first kind linear Fredholm integral equation. Three different cases are considered to test the efficacy of our method. These include mixed systems of various forms of linear integral and fractional integro-differential equations. We compare our method with six well-established methods: Picard’s successive approximations, an accelerated form of the latter, the Adomian decomposition, the variational iteration, the homotopy perturbation and the homotopy analysis. The results show the versatility of the new method in solving a wide variety of equations, its high convergence speed and accuracy and its capability of directly dealing with equations of the first kind. We also prove that Picard’s method for linear equations is a special case of our method.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Piecewise linear approximation with minimum number of linear segments and minimum error: A fast approach to tighten and warm start the hierarchical mixed integer formulation

In several areas of economics and engineering, it is often necessary to fit discrete data points or approximate nonlinear functions with continuous functions. Piecewise linear (PWL) functions are a convenient way to achieve this. PWL functions can be modeled in mathematical problems using only linear and integer variables. Moreover, there is a computational benefit in using PWL functions that have the least possible number of segments. This work proposes a novel hierarchical mixed integer linear programming (MILP) formulation that identifies a continuous PWL approximation with minimum number of linear segments for a given target maximum error. The proposed MILP formulation also identifies the solution with the least maximum error among the solutions with minimum number of segments. Then, this work proposes a fast iterative algorithm that identifies non necessarily continuous PWL approximations by solving O(S log N) linear programming (LP) problems, where N is the number of data points and S is the minimum number of segments in the non necessarily continuous case. This work demonstrates that tight bounds for the MILP problem can be derived from these approximations. Next, a fast algorithm is introduced to transform a non necessarily continuous PWL approximation into a continuous one. Finally, the tight bounds and the continuous PWL approximations are used to tighten and warm start the MILP problem. The tightened formulation is shown in experimental results to be more efficient, especially for large data sets, with a solution time that is up to two orders of magnitude less than the existing literature.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2017

A new approach to three-dimensional neutron transport solution based on the method of characteristics and linear axial approximation

A new approach based on the method of characteristics (MOC) is proposed to solve the neutron transport equation. A new three-dimensional (3D) spatial discretization is applied to avoid the instability issue of the transverse leakage iteration of the traditional 2D/1D approach. In this new approach, the axial and radial variables are discretized in two different ways: the linear expansion is performed in the axial direction, then, the 3D solution of the angular flux is transformed to be the planar solution of 2D angular expansion moments, which are solved by the planar MOC sweeping. Based on the boundary and interface continuity conditions, the 2D expansion moment solution is equivalently transformed to be the solution of the axially averaged angular flux. Using the piecewise averaged angular flux at the top and bottom surfaces of 3D meshes, the planes are coupled to give the 3D angular flux distribution. The 3D CMFD linear system is established from the surface net current of every 3D pin-mesh to accelerate the convergence of power iteration. The STREAM code is extended to be capable of handling 3D problems based on the new approach. Several benchmarks are tested to verify its feasibility and accuracy, including the 3D homogeneous benchmarks and heterogeneous benchmarks. The computational sensitivity is discussed. The results show good accuracy in all tests. With the CMFD acceleration, the convergence is stable. In addition, a pin-cell problem with void gap is calculated. This shows the advantage compared to the traditional 2D/1D MOC methods.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2019

Multiresolution spatiotemporal mechanical model of the heart as a prior to constrain the solution for 4D models of the heart

In several nuclear cardiac imaging applications (SPECT and PET), images are formed by reconstructing tomographic data using an iterative reconstruction algorithm with corrections for physical factors involved in the imaging detection process and with corrections for cardiac and respiratory motion. The physical factors are modeled as coefficients in the matrix of a system of linear equations and include attenuation, scatter, and spatially varying geometric response. The solution to the tomographic problem involves solving the inverse of this system matrix. This requires the design of an iterative reconstruction algorithm with a statistical model that best fits the data acquisition. The most appropriate model is based on a Poisson distribution. Using Bayes Theorem, an iterative reconstruction algorithm is designed to determine the maximum a posteriori estimate of the reconstructed image with constraints that maximizes the Bayesian likelihood function for the Poisson statistical model. The a priori distribution is formulated as the joint entropy (JE) to measure the similarity between the gated cardiac PET image and the cardiac MRI cine image modeled as a FE mechanical model. In conclusion, the developed algorithm shows the potential of using a FE mechanical model of the heart derived from a cardiac MRI cine scan to constrain solutions of gated cardiac PET images.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2018

Application of hat basis functions for solving two-dimensional stochastic fractional integral equations

This article concerns with a computational scheme to solve two-dimensional stochastic fractional integral equations (2DSFIEs), numerically. In these equations, the fractional integral is considered in the Riemann–Liouville sense. The proposed method is essentially based on two-dimensional hat basis functions and its fractional operational matrices. The fractional-order operational matrices of integration are applied to reduce the solution of 2DSFIEs to the solution of a system of linear equations which can be solved using a direct method or iterative method. Some results concerning the convergence analysis associated with the proposed technique are discussed. In addition, we establish the rate of convergence of this approach for solving 2DSFIEs is O(h{sup 2}). Finally, some examples are solved using present method to indicate the pertinent features of the method.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2017

Chance-Constrained AC Optimal Power Flow: Reformulations and Efficient Algorithms

Higher levels of renewable electricity generation increase uncertainty in power system operation. To ensure secure system operation, new tools that account for this uncertainty are required. Here, in this paper, we adopt a chance-constrained AC optimal power flow formulation, which guarantees that generation, power flows and voltages remain within their bounds with a pre-defined probability. We then discuss different chance-constraint reformulations and solution approaches for the problem. Additionally, we first discuss an analytical reformulation based on partial linearization, which enables us to obtain a tractable representation of the optimization problem. We then provide an efficient algorithm based on an iterative solution scheme which alternates between solving a deterministic AC OPF problem and assessing the impact of uncertainty. This more flexible computational framework enables not only scalable implementations, but also alternative chance-constraint reformulations. In particular, we suggest two sample based reformulations that do not require any approximation or relaxation of the AC power flow equations.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2023

OpenMxP-Opensource Mixed Precision Computing

This is an opensource library for benchmarking the system's GPU mixed precision capabilities. The software calculates solution of the system of linear equation in 64bit accuracy using mixed precision techniques and iterative refinement. Original benchmark designed is done by ICL, and it is name HPL-MxP (HPL-AI)

Lu, Hao↗

DOE OSTI2023

Scaled ILU Smoothers for Navier-Stokes Pressure Projection

Incomplete LU (ILU) smoothers are effective in the algebraic multigrid (AMG) V-cycle for reducing high-frequency components of the error. However, the requisite direct triangular solves are comparatively slow on GPUs. Previous work has demonstrated the advantages of Jacobi iteration as an alternative to direct solution of these systems. Depending on the threshold and fill-level parameters chosen, the factors can be highly nonnormal and Jacobi is unlikely to converge in a low number of iterations. We demonstrate that row scaling can reduce the departure from normality, allowing us to replace the inherently sequential solve with a rapidly converging Richardson iteration. There are several advantages beyond the lower compute time. Scaling is performed locally for a diagonal block of the global matrix because it is applied directly to the factor. Further, an ILUT Schur complement smoother maintains a constant GMRES iteration count as the number of MPI ranks increases, and thus parallel strong-scaling is improved. Our algorithms have been incorporated into hypre, and we demonstrate improved time to solution for linear systems arising in the Nalu-Wind and PeleLM pressure solvers. For large problem sizes, GMRES+AMG executes at least five times faster when using iterative triangular solves compared with direct solves on massively parallel GPUs.

algebraic multigrid↗

DOE OSTI2021

Fuzzified PaCcET for Economic-Emission Scheduling of Microgrids

In this paper, a new approach is proposed to solve a multi-objective economic-emission scheduling problem in microgrids (MGs) by simultaneously minimizing the energy and emission costs of the MG with various distributed energy resources (DERs). The proposed approach is an extension of a computationally effective multiobjective optimization technique, Pareto concavity elimination transformation (PaCcET). The proposed approach, referred to as Fuzzified-PaCcET, employs a fuzzy logic controller to dynamically revise crossover and mutation rates in the original PaCcET leading to the faster convergence of the solution. The proposed approach finds the best Pareto front, also referred to as a Non-dominated set (NDS) of solutions, instead of finding a single optimal solution. In order to find the solutions on concave areas of the Pareto front, an iterative objective space transformation is performed in the PaCcET algorithm to allow a linear combination of objective functions (in the transformed objective space). The proposed Fuzzified-PaCcET-based scheduling is implemented on a MG with various dispatchable and non-dispatchable DERs to find the set of optimal solutions according to the total fuel cost of DERs, as well as the most optimum environmental cost. In order to extract the best compromise solution (BCS) among NDS of solutions, a fuzzy-based method is implemented. The comparison of the simulation results of the Fuzzified-PaCcET with that of PaCcET shows that Fuzzified-PaCcET can generate better solution with less computational burden.

Gautam, Mukesh↗

DOE OSTI2021

Fuzzified PaCcET for Economic-Emission Scheduling of Microgrids

42 ENGINEERING↗

DOE OSTI2021

Volumetric recombination in EMC3-EIRENE: Implementation and first application to the pre-fusion power operation phase in ITER

An upgrade to the 3D plasma boundary model EMC3-EIRENE for detached divertor plasmas is introduced and evaluated. Stabilization of the iterative approximation of a self-consistent plasma–neutral gas solution is required at low divertor plasma temperatures, and this has been achieved by linearization of the energy loss term associated with the electron–neutral gas interaction. An implicit method for volumetric recombination has been implemented, which improves agreement with SOLPS-ITER results for the pre-fusion power operation phase in ITER. Resonant magnetic perturbations for control of edge localized modes are found to result in a partially detached divertor plasma with a significantly broader reduction of particle loads in the traditional strike zone, which is attributed to volumetric recombination. The additional non-axisymmetric far scrape-off layer strike points, on the other hand, remain attached at relatively high temperature so that volumetric recombination is not effective there.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2018

Numerically approximating centrality for graph ranking guarantees

Many real-world datasets can be represented as graphs. Using iterative solvers to approximate graph centrality measures allows us to obtain a ranking vector on the nodes of the graph, consisting of a number for each vertex in the graph identifying its relative importance. In this study the centrality measures we use are Katz Centrality and PageRank. Given an approximate solution, we use the residual to accurately estimate how much of the ranking matches the ranking given by the exact solution. Using probabilistic matrix norms, we obtain bounds on the accuracy of the approximation compared to the exact solution with respect to the highly ranked nodes and apply numerical analysis to the computation of centrality with iterative methods. This relates the numerical accuracy of the linear solver to the data analysis accuracy of finding the correct ranking. In particular, we answer the question of which pairwise rankings are reliable given an approximate solution to the linear system. Experiments on many real-world undirected and directed networks up to several million vertices and several hundred million edges validate our theory and show that we are able to accurately estimate large portions of the approximation. We also analyze the difference between global centrality scores and personalized scores (w.r.t. specific seed vertices). By analyzing convergence error, we develop confidence in the ranking schemes of data mining. We show we are able to accurately guarantee ranking of vertices with an approximation to centrality metrics faster than current methods.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2017

On performance of Krylov smoothing for fully-coupled AMG preconditioners for VMS resistive MHD

Here, this study explores the performance and scaling of a GMRES Krylov method employed as a smoother for an algebraic multigrid (AMG) preconditioned Newton- Krylov solution approach applied to a fully-implicit variational multiscale (VMS) nite element (FE) resistive magnetohydrodynamics (MHD) formulation. In this context a Newton iteration is used for the nonlinear system and a Krylov (GMRES) method is employed for the linear subsystems. The efficiency of this approach is critically dependent on the scalability and performance of the AMG preconditioner for the linear solutions and the performance of the smoothers play a critical role. Krylov smoothers are considered in an attempt to reduce the time and memory requirements of existing robust smoothers based on additive Schwarz domain decomposition (DD) with incomplete LU factorization solves on each subdomain. Three time dependent resistive MHD test cases are considered to evaluate the method. The results demonstrate that the GMRES smoother can be faster due to a decrease in the preconditioner setup time and a reduction in outer GMRESR solver iterations, and requires less memory (typically 35% less memory for global GMRES smoother) than the DD ILU smoother.

97 MATHEMATICS AND COMPUTING↗