CPU Timings Research Articles

AWEbox: An Optimal Control Framework for Single- and Multi-Aircraft Airborne Wind Energy Systems

In this paper, we present AWEbox, a Python toolbox for modeling and optimal control of multi-aircraft systems for airborne wind energy (AWE). AWEbox provides an implementation of optimization-friendly multi-aircraft AWE dynamics for a wide range of system architectures and modeling options. It automatically formulates typical AWE optimal control problems based on these models, and finds a numerical solution in a reliable and efficient fashion. To obtain a high level of reliability and efficiency, the toolbox implements different homotopy methods for initial guess refinement. The first type of method produces a feasible initial guess from an analytic initial guess based on user-provided parameters. The second type implements a warm-start procedure for parametric sweeps. We investigate the software performance in two different case studies. In the first case study, we solve a single-aircraft reference problem for a large number of different initial guesses. The homotopy methods reduce the expected computation time by a factor of 1.7 and the peak computation time by a factor of eight, compared to when no homotopy is applied. Overall, the CPU timings are competitive with the timings reported in the literature. When the user initialization draws on expert a priori knowledge, homotopies do not increase expected performance, but the peak CPU time is still reduced by a factor of 5.5. In the second case study, a power curve for a dual-aircraft lift-mode AWE system is computed using the two different homotopy types for initial guess refinement. On average, the second homotopy type, which is tailored for parametric sweeps, outperforms the first type in terms of CPU time by a factor of three. In conclusion, AWEbox provides an open-source implementation of efficient and reliable optimal control methods that both control experts and non-expert AWE developers can benefit from.

Combining p-multigrid and Multigrid Reduction in Time methods to obtain a scalable solver for Isogeometric Analysis

The use of sequential time integration schemes becomes more and more the bottleneck within large-scale computations due to a stagnation of processor’s clock speeds. In this study, we combine the parallel-in-time Multigrid Reduction in Time method with a p-multigrid method to obtain a scalable solver specifically designed for Isogeometric Analysis. Numerical results obtained for two- and three-dimensional benchmark problems show the overall scalability of the proposed method on modern computer architectures and a significant improvement in terms of CPU timings compared to the use of standard spatial solvers.

On the accuracy, robustness, and performance of high order interpolation schemes for the overset method on unstructured grids

AbstractA comprehensive study on interpolation schemes used in overset grid techniques is here presented. Based on a literature review, numerous schemes are implemented, and their robustness, accuracy, and performance are assessed. Two code verification exercises are performed for this purpose: a 2D analytical solution of a laminar Poiseuille steady flow; and an intricate manufactured solution of a turbulent flow case, characteristic of a boundary layer flow combined with an unsteady separation bubble. For both cases, the influence of grid layouts, grid refinement, and time‐step is investigated. Local and global errors, convergence orders, and mass imbalance are quantified. In terms of computational performance, strong scalability, cpu timings, load imbalancing, and domain connectivity information (DCI) overhead are reported. The effect of the overset‐grid interpolation schemes on the numerical performance of the solver, that is, number of nonlinear iterations, is also scrutinized. The results show that, for a second order finite volume code, once diffusion is dominant (low Reynolds number), interpolation schemes higher than second order, for example, least squares of degree 2, are needed not to increase the total discretization errors. For convection dominated flows (high Reynolds numbers), the results suggest that second order schemes, for example, nearest cell gradient, are sufficient to prevent overset grid schemes to taint the underlying discretization errors. In terms of performance, by single‐process and parallel communication optimization, the total overset‐grid overhead (with DCI done externally to the CFD code) may be less than 4% of the total run time for second‐order schemes and 8% for third‐order ones, therefore empowering higher‐order schemes and more accurate solutions.

On rotated grid point iterative method for solving 2D fractional reaction–subdiffusion equation with Caputo–Fabrizio operator

In this paper, the Crank–Nicolson (CN) and rotated four-point Fractional Explicit Decoupled Group (FEDG) methods are introduced to solve the two-dimensional reaction–subdiffusion equation with Caputo–Fabrizio operator. The FEDG method is derived by rotation of CN method around the x and y axes. The restarted GMRES with left preconditioner L, based on incomplete LU factorization is used to solve the discretized system obtained by our proposed methods. The FEDG method shows more superior capability in the term of CPU timings and the number of iteration compared to CN method on the standard grid but with same order of accuracy. The stability and convergence analysis in the approximate schemes are investigated. Some numerical experiments performed to show the efficiency of the presented methods in terms of accuracy and CPU time.

High-order rotated grid point iterative method for solving 2D time fractional telegraph equation and its convergence analysis

In this paper, the compact finite difference (CFD) and rotated four-point compact explicit decoupled group (CEDG) methods are proposed to solve the two-dimensional time-fractional telegraph equation. The CEDG method is derived from a rotated of CFD approximation formula combine with the arranging of the grid points in the form of a group. This method shows superior performance in the term of CPU timings and iteration compared to the CFD method on the standard grid but with the same order of accuracy. We have proved the stability and convergence of the proposed schemes using the Fourier analysis. The convergence order of the proposed methods is $$O\left( \tau +h_{x}^{4}+h_{y}^{4}\right) $$ . Some numerical experiments are performed to demonstrate the effectiveness of the proposed methods.

On skewed grid point iterative method for solving 2D hyperbolic telegraph fractional differential equation

This paper presents the development of a new iterative method for solving the two-dimensional hyperbolic telegraph fractional differential equation (2D-HTFDE) which is central to the mathematical modeling of transmission line satisfying certain relationship between voltage and current waves in specific distance and time. This equation can be obtained from the classical two-dimensional hyperbolic telegraph partial differential equation by replacing the first and second order time derivatives by the Caputo time fractional derivatives of order 2α and α respectively, with 1/2< alpha < 1 . The iterative scheme, called the fractional skewed grid Crank–Nicolson FSkG(C-N), is derived from finite difference approximations discretized on a skewed grid rotated clockwise 450 from the standard grid. The skewed finite difference scheme combined with Crank–Nicolson discretization formula will be shown to be unconditionally stable and convergent by the Fourier analysis. The developed FSkG(C-N) scheme will be compared with the fractional Crank–Nicolson scheme on the standard grid to confirm the effectiveness of the proposed scheme in terms of computational complexities and computing efforts. It will be shown that the new proposed scheme demonstrates more superior capabilities in terms of the number of iterations and CPU timings compared to its counterpart on the standard grid but with the same order of accuracy.

Crystal structure prediction of rigid molecules.

A non-polarizable force field based on atomic multipoles fit to reproduce experimental crystal properties and ab initio gas-phase dimers is described. The Ewald method is used to calculate both long-range electrostatic and 1/r(6) dispersion energies of crystals. The dispersion energy of a crystal calculated by a cutoff method is shown to converge slowly to the exact Ewald result. A method for constraining space-group symmetry during unit-cell optimization is derived. Results for locally optimizing 4427 unit cells including volume, cell parameters, unit-cell r.m.s.d. and CPU timings are given for both flexible and rigid molecule optimization. An algorithm for randomly generating rigid molecule crystals is described. Using the correct experimentally determined space group, the average and maximum number of random crystals needed to find the correct experimental structure is given for 2440 rigid single component crystals. The force field energy rank of the correct experimental structure is presented for the same set of 2440 rigid single component crystals assuming the correct space group. A complete crystal prediction is performed for two rigid molecules by searching over the 32 most probable space groups.

Relative performance of body fitted and fictitious domain simulations of flow through fixed packed beds of spheres

The relative performance of two numerical approaches involving body conforming and non-conforming grids for simulating porescale flow in complex configurations of fixed packed beds of spheres at moderate pore Reynolds numbers (12⩽Re⩽600) is examined. In the first approach, an unstructured solver is used with tetrahedral meshes which conform to the boundaries of the porespace. In the second approach, a fictitious domain formulation is used which employs non-body conforming Cartesian grids and enforces the no-slip conditions on the pore boundaries implicitly through a rigidity constraint force. Sphere to sphere contact points, where the fluid gap between solid boundaries becomes infinitesimal, are not resolved by either approach, but this is shown to have a negligible effect on the local flow field at the Reynolds numbers considered. Detailed grid convergence studies of both steady and unsteady flow through simple cubic packings indicate that for a fixed level of uncertainty, significantly lower grid densities may be used with the fictitious domain approach which also does not require complex grid generation techniques. This translates into large savings for simulation of flow through realistic packed beds, which is shown by both analytic estimates and actual CPU timings. The applicability of the fictitious domain approach is demonstrated by simulating unsteady flow through a randomly packed bed of 51 spheres at a pore Reynolds number of 600. The results are used to examine the dominance of helical vortices in the porescale flow field.

Skew-Hermitian based iterations for nine-point approximations of convection–diffusion problems

We study skew-Hermitian based splitting methods for the iterative solution of nonsymmetric linear systems arising from high-order compact (HOC) approximations of two-dimensional convection–diffusion problems. Such discretisations lead to nine-point system matrices with block-tridiagonal structures. The nonsymmetric discretisation matrix for a constant-coefficient grid-aligned flow problem is shown to be positive definite and thus Hermitian and skew-Hermitian splitting (HSS) methods can be considered for the solution of the corresponding linear system. For the solution of this linear system using the HSS iteration, we derive an analytical expression for the optimal value for the upper bound of the contraction factor. Comparison of the HSS iteration with triangular and skew-Hermitian splitting (TSS) methods and their corresponding block variants (BTSS) are carried out for various test problems. Cpu timings for the different splitting methods using experimentally determined optimal acceleration parameters are also given. Our results show that the combination of high-order compact discretisations and skew-Hermitian based iterative methods for solving the corresponding linear systems provide efficient procedures for approximating the solutions of two-dimensional convection–diffusion equations.

Efficient implementation of theGWapproximation within the all-electron FLAPW method

We present an implementation of the GW approximation for the electronic self-energy within the full-potential linearized augmented-plane-wave (FLAPW) method. The algorithm uses an all-electron mixed product basis for the representation of response matrices and related quantities. This basis is derived from the FLAPW basis and is exact for wave-function products. The singularity of the bare and screened interaction potentials gives rise to a numerically important self-energy contribution, which we treat analytically to achieve good convergence with respect to the k-point sampling. As numerical realizations of the GW approximation typically suffer from the high computational expense required for the evaluation of the nonlocal and frequency-dependent self-energy, we demonstrate how the algorithm can be made very efficient by exploiting spatial and time-reversal symmetry as well as by applying an optimization of the mixed product basis that retains only the numerically important contributions of the electron-electron interaction. Furthermore, we demonstrate that one can employ an extrapolar approximation for high-lying states to reduce the number of empty states that must be taken into account explicitly in the construction of the polarization function and the self-energy. We show convergence tests, CPU timings, and results for prototype semiconductors and insulators as well as ferromagnetic nickel.

A Schwarz Preconditioner for the Cubed-Sphere

A spectral element formulation of the atmospheric two-dimensional shallow water equations on the cubed-sphere is described. The equations are written in tensor form using the contravariant and covariant velocity components. A semi-implicit time discretization results in a reduced Schur complement system for the pressure. The Laplacian operator is approximated by the L2 pseudo-Laplacian arising in the PN/PN-2 spectral element formulation of the incompressible Stokes problem. The overlapping Schwarz preconditioner of Fischer, Miller, and Tufo [An overlapping Schwarz method for spectral element simulation of three-dimensional incompressible flows, in Parallel Solution of Partial Differential Equations, IMA Vol. Math. Appl. 120, Springer, New York, 2000, pp. 159--180] based on the fast diagonalization method, is extended to generalized curvilinear coordinates. To obtain a separable operator for the linear finite element tensor-product approximation within each spectral element, extrema of the inverse metric tensor and its determinant are employed. Convergence rates and parallel CPU timings are compared against a block-Jacobi preconditioner.

Efficient computer implementation of the renormalized coupled-cluster methods: The R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) approaches

The recently proposed renormalized (R) and completely renormalized (CR) coupled-cluster (CC) methods of the CCSD[T] and CCSD(T) types have been implemented using recursively generated intermediates and fast matrix multiplication routines. The details of this implementation, including the complete set of equations that have been used in writing efficient computer codes, memory requirements, and typical CPU timings, are discussed. The R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) computer codes and similar codes for the standard CC methods, including the LCCD, CCD, CCSD, CCSD[T], and CCSD(T) approaches, have been incorporated into the gamess package. Information about the main features of this new set of CC programs is provided.

Accuracy, robustness and efficiency comparison in iterative computation of convection diffusion equation with boundary layers

Nine-point fourth-order compact finite difference scheme, central difference scheme, and upwind difference scheme are compared for solving the two-dimensional convection diffusion equations with boundary layers. The domain is discretized with a stretched nonuniform grid. A grid transformation technique maps the nonuniform grid to a uniform one, on which the difference schemes are applied. A multigrid method and a multilevel preconditioning technique are used to solve the resulting sparse linear systems. We compare the accuracy of the computed solutions from different discretization schemes, and demonstrate the relative efficiency of each scheme. Comparisons of maximum absolute errors, iteration counts, CPU timings, and memory cost are made with respect to the two solution strategies. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 379–394, 2000

The tautomers of uracil: A local correlation treatment

AbstractThe relative energies of all six uracil tautomers have been determined at the MP4(SDQ)/6‐31G** level, using both conventional correlation theory and the Local Correlation method. Geometries were optimized at the SCF/6‐31G* level with offset forces. Comparison of our energies with energies from structures optimized at the SCF level supports the conclusion that offset forces are an advantageous alternative to correlated geometry optimization. The Local Correlation method compares very well with conventional Møller–Plesset theory, recovering at least 98.4% of the conventional correlated energy in all cases. More importantly, the relative energies also show good agreement with the conventional results, even for these delocalized systems. CPU timings show a substantial computational savings for the Local Correlation method over the conventional method. The results of the local method using Boys localization are compared with those using Pipek–Mezey localization. The dioxo tautomer (1) is predicted to be the most stable. The (1)–(3) and (1)–(4) energy differences are found to be within the bounds estimated from experimental work. © 1993 John Wiley &amp; Sons, Inc.

An improved criterion for evaluating the efficiency of two-electron integral algorithms

We present a general criterion for theoretical performance assessment of algorithms for two-electron integral computation which is appropriate for most modern computers. The new prescription is to minimize the total number of memory references in the algorithm, as opposed to the traditional approach of minimizing the total number of floating-point operations. CPU timings on a range of machines demonstrate that memory operations are better correlated to machine cycles than are floating-point operations.

Computing molecular electrostatic potentials with the PRISM algorithm

The PRISM integral algorithm has been applied to the computation of the ab initio molecular electrostatic potential and its derivatives. Implementational details which are relevant to the additional efficiency of the algorithm in the electrostatic case are discussed. On a range of machines, CPU timings of the PRISM electrostatic properties program, which is included in the GAUSSIAN 92 quantum chemistry package, reveal a dramatic performance increase (in some cases more than two orders of magnitude) over other commonly used electrostatic programs (GAUSSIAN 90, GAMESS, MOPAC ESP, CHELPG). In addition, timings are reported for a particularly large electrostatic potential evaluation job on the six base-pair oligonucleotide CTCGAG (C 116H 138N 46O 68P 10 10−.

The efficient transformation of ( m0 | n0 ) to ( ab| cd) two-electron repulsion integrals

A new method is presented for transforming ( m0 | n0 ) integrals to ( ab | cd ) integrals efficiently. A key feature of the new approach is the optimization of the total number of memory operations (mops) rather than floating-point operations (flops). Solutions to the inherent tree-search problem were obtained which are radically different than those following when flops are taken as the cost function. These involve new, multi-unit angular momentum transfer relations which belong to a general family having the familiar one-unit transfer relation of Head—Gordon and Pople as its simplest member. CPU timings demonstrate the new method to be clearly superior to two others considered, including optimal use of the one-unit transfer relation alone.

Tests on combined projection/forward differencing integration for stiff photochemical family systems at long time step

Accurate estimates of final tracer concentrations guide adjustment of individual species levels so that forward integration of stiff families can proceed at time steps an order of magnitude larger than inherent loss constants. Maintenance of a unified concentration vector in the solver ensures mass conservation. The sequence was perfected by raising the step size upward from 100 s in a standard 25 family 45 species stratospheric box model with explicit Euler numerics. Projections were inserted as instabilities arose. Ultimately, a uniform Δ t of 1 h was achieved from 20–50 km at mid-latitudes. Results were compared over five equinoctial diurnal cycles with runs at 0.1 and 1 s steps in which every constituent other than the individual atoms, CHO and CH 3O, was handled separately. Agreement was better than 10 −2 for the major reservoirs in almost all cases, and was equally close for radicals between 6 a.m. and 6 p.m., provided that photolysis constants were time averaged. The few exceptions were due to experimentation with a linearized implicit concentration predictor, or to the approximations underlying family partitioning recipes. CPU timings extrapolated to a hypothetical GCM grid suggest that 3-D modeling will be possible at the level of chemical resolution in the programs.

Parallel variable-band Choleski solvers for computational structural analysis applications on vector multiprocessor supercomputers

A Choleski method used to solve linear systems of equations that arise in large scale structural analyses is described. The method uses a novel variable-band stroage scheme and is structured to exploit fast local memory caches while minimizing data access delays between main memory and vector registers. Several parallel implementations of this method are described for the CRAY-2 and CRAY Y-MP computers demonstrating the use of microtasking and autotasking directives. A portable parallel language, FORCE, is also used for two different parallel implementations, demonstrating the use of CRAY macrotasking. Results are presented comparing the matrix factorization times for three representative structural analysis problems from runs made in both dedicated and multi-user modes on both the CRAY-2 and CRAY Y-MP computers. CPU and wall clock timings are given for the various parallel methods and are compared to single processor timings of the same algorithm. Computation rates over 1 GIGAFLOP (1 billion floating point operations per second) on a four processor CRAY-2 and over 2 GIGAFLOPS on an eight processor CRAY Y-MP are demonstrated as measured by wall clock time in a dedicated environment. Reduced wall clock times for the parallel methods relative to the single processor implementation of the same Choleski algorithm are also demonstrated for runs made in multi-user mode.

Comparison of Lanczos and subspace iterations for hyperspherical reaction path calculations

In an accurate treatment of 3-dimensional reactive scattering using Adiabatically-adjusting, Principal axes Hyperspherical (APH) coordinates, we obtain a 2-dimensional Schrödinger equation defined on the surface of a hypersphere. The surface Hamiltonian, which depends parametrically on the sector hyperradius ϱ h , must be diagonalized for many (100–200) values of ϱ h . The surface (eigen)functions are expanded in a finite element basis, where a non-uniform finite element mesh is allowed to adjust for each ϱ h . Projection of the finite element basis onto the surface Hamiltonian yields a generalized eigenvalue problem. Typically the lowest 50–100 eigenvalues and eigenfunctions are required at each ϱ h . Since the calculation of the surface functions is currently the expensive step for our reactive scattering procedure, it is necessary to calculate these functions as efficiently as possible. In this paper, we use both the subspace iteration and a particular implementation of the block Lanczos (with selective orthogonalization) method to calculate the surface functions. Using both solution methods, we diagonalize the finite element matrices (of order = 1729 and of half bandwidth = 109) constructed for the LiH + F ⇌ Li + HF system. CPU timings for both methods are examined as a function of the number of converged eigenvalues. Since we can generate a good initial subspace from the eigenvectors calculated at the previous value of ϱ h , we find that the subspace iteration is competitive with the Block Lanczos method when many (90) eigenvalues are calculated. We also consider a larger problem (matrices of order ≈ 3300 and of half bandwidth ≈ 135) constructed for the F + H 2 ⇌ HF + H system. For this problem, we present the CPU timings required to calculate 100 eigenvalues for a sequence of five ϱ h values using the subspace iteration method.