Lower Operation Count Research Articles

KAM quasi-periodic solutions for the dissipative standard map

We present results towards a constructive approach to show the existence of quasi-periodic solutions in non-perturbative regimes of some dissipative systems, called conformally symplectic systems. Finding a quasi-periodic solution of conformally symplectic systems with fixed frequency requires to choose a parameter, called the drift parameter.The first step of the strategy is to establish a very explicit quantitative theorem in an a-posteriori format as in Calleja et al. (2013). A-posteriori theorems show that if we can find an approximate solution of an invariance equation, which is sufficiently approximate with respect to some condition numbers (algebraic expressions of derivatives of the approximate solution and estimates on the derivatives of the map), then there is a true solution.The second step in the strategy is to produce numerically a very accurate solution of the invariance equation (discretizations with 218 Fourier coefficients, each one computed with 100 digits of precision).The third step is to compute in a concrete example, the dissipative standard map, the condition numbers and verify numerically the conditions of the theorem in the approximate solutions. For some families which have been studied numerically, the results agree with three figures with the best numerical values. We point out however that the numerical methods developed here work also in examples which have not been accessible to other more conventional methods.The verification of the estimates presented here is not completely rigorous, since we do not control the round-off error, nor the truncation error of several operations in Fourier space. We hope that the positive step taken in this paper will stimulate the complete computer-assisted proof. Making explicit the condition numbers and verifying the conditions (even in an incomplete way) will be valuable for the computation close to breakdown.We make available the approximate solutions, the highly efficient algorithm (quadratic convergence, low storage requirements, low operation count per step) to compute them (incorporating high precision based on the MPFR library) and the routines used to verify the applicability of the theorem.

KAM quasi-periodic tori for the dissipative spin–orbit problem

We provide evidence of the existence of KAM quasi-periodic attractors for a dissipative model in Celestial Mechanics. We compute the attractors extremely close to the breakdown threshold.We consider the spin–orbit problem describing the motion of a triaxial satellite around a central planet under the simplifying assumption that the center of mass of the satellite moves on a Keplerian orbit, the spin-axis is perpendicular to the orbit plane and coincides with the shortest physical axis. We also assume that the satellite is non-rigid; as a consequence, the problem is affected by a dissipative tidal torque that can be modeled as a time-dependent friction, which depends linearly upon the velocity.Our goal is to fix a frequency and compute the embedding of a smooth attractor with this frequency. This task requires to adjust a drift parameter.We have shown in Calleja et al. (2020) that it is numerically efficient to study Poincaré maps; the resulting spin–orbit map is conformally symplectic, namely it transforms the symplectic form into a multiple of itself. In Calleja et al. (2020), we have developed an extremely efficient (quadratically convergent, low storage requirements and low operation count per step) algorithm to construct quasi-periodic solutions and we have implemented it in extended precision. Furthermore, in Calleja et al. (2020) we have provided an “a-posteriori” KAM theorem that shows that if we have an embedding and a drift parameter that satisfy the invariance equation up to an error which is small enough with respect to some explicit condition numbers, then there is a true solution of the invariance equation. This a-posteriori result is based on a Nash–Moser hard implicit function theorem, since the Newton method incurs losses of derivatives.The goal of this paper is to provide numerical calculations of the condition numbers and verify that, when they are applied to the numerical solutions, they will lead to the existence of the torus for values of the parameters extremely close to the parameters of breakdown. Computing reliably close to the breakdown allows to discover several interesting phenomena, which we will report in Calleja et al. (2020).The numerical calculations of the condition numbers presented here are not completely rigorous, since we do not use interval arithmetic to estimate the round off error and we do not estimate rigorously the truncation error, but we implement the usual standards in numerical analysis (using extended precision, checking that the results are not affected by the level of precision, truncation, etc.). Hence, we do not claim a computer-assisted proof, but the verification is more convincing than a standard numerics. We hope that our work could stimulate a computer-assisted proof.

Resonant Equilibrium Configurations in Quasi-Periodic Media: KAM Theory

We develop an a posteriori KAM theory for the equilibrium equations for quasi-periodic solutions in a quasi-periodic Frenkel--Kontorova model when the frequency of the solutions resonates with the frequencies of the substratum. The main motivation for the study of these solutions is that they are present when there are constant external forces while the nonresonant solutions disappear. The KAM theory we develop is very different both in the methods and in the conclusions from the more customary KAM theory for Hamiltonian systems or from the KAM theory in quasi-periodic media for solutions with frequencies which are Diophantine with respect to the frequencies of the media. The main difficulty is that we cannot use transformations (as in the Hamiltonian case) nor Ward identities (as in the case of frequencies Diophantine with those of the media). The technique we use is to add an extra equation that ensures the linearization of the equilibrium equation factorizes. To solve the extra equation requires an extra counterterm. We compare this phenomenon with other phenomena in KAM theory. It seems that this technique could be used in several other problems. As a conclusion, we obtain that the perturbation expansions developed in the previous paper [R. de la Llave, X. Su, and L. Zhang, J. Stat. Phys., 162 (2016), pp. 1522--1538] converge when the potentials are in a codimension one manifold in a space of potentials. The method of proof also leads to efficient (low storage requirements and low operation count) algorithms to compute the quasi-periodic solutions.

Investigation and Comparison of Implicit Smoothers Applied in Agglomeration Multigrid

A straightforward implicit smoothing method implemented in several codes solving the Reynolds-averaged Navier–Stokes equations is the lower–upper symmetric Gauss–Seidel method. It was proposed several years ago and is attractive to implement because of its low memory requirements and low operation count. Since, for many examples, often a severe restriction of the Courant–Friedrichs–Lewy number and loss of robustness are observed, it is the goal of this paper to revisit the lower–upper symmetric Gauss–Seidel implementation and to discuss several alternative implicit smoothing strategies used within an agglomeration multigrid for unstructured meshes. The starting point is a full implicit multistage Runge–Kutta method. Based on this method, several additional features and simplifications are developed and suggested, such that the implicit method is applicable to high-Reynolds-number viscous flows; that is, the required matrices fit into the fast memory of the cluster hardware and the arising linear systems can be approximately solved efficiently. To this end, the focus is on simplifications of the Jacobian, as well as efficient iterative approximate solution methods. To significantly improve the approximate linear solution methods, grid anisotropy is taken care of for approximately solving both the linear systems and the agglomeration strategy. The procedure creating coarse-grid meshes is extended by strategies identifying structured parts of the mesh. This seems to improve the quality of coarse-grid meshes in the way that an overall better reliability of the multigrid can be observed. Furthermore, grid information is exploited within the iterative solution methods for the linear systems. Numerical examples demonstrate the gain with respect to reliability and efficiency.

Memory‐based two‐dimensional‐parallel differential matched filter correlator for global navigation satellite system code acquisition

This study presents a code acquisition architecture for a global navigation satellite system (GNSS). The proposed scheme aim to reduce buffer resource required without reducing the number of code, frequency and satellite bins. Therefore the GNSS signal has no information loss. The code acquisition process in a GNSS attempts to find the code phase of a satellite signal, and it includes three procedures: coherent integration (CI), correlation and in CI. The proposed code acquisition architecture provides a lower operation count for correlation with reduced memory consumption. Firstly, the memory in CI and the shift register in the correlator are integrated by changing the operation sequence of CI and correlation. Thus, half of the memory in CI can be removed, and the shift register becomes the sharing buffer for both correlation and CI. Secondly, the addition count in the correlation is reduced by combining a parallel differential matched filter (PDMF) with a two-dimensional (2D) architecture to form a memory-based 2D-PDMF correlator. Finally, changing the sample sequence before integration reduces the memory of in CI. Compared with conventional code acquisition, the proposed design can reduce register consumption by 80% and addition count by 60% operation count for correlation and 50% memory consumption of incoherent integration.

Computation of whiskered invariant tori and their associated manifolds: New fast algorithms

We present efficient (low storage requirement and low operation count) algorithms for the computation of several invariant objects for Hamiltonian dynamics, namely KAM tori (i.e diffeomorphic copies of tori such that the motion on them is conjugated to a rigid rotation) both Lagrangian tori(of maximal dimension) and whiskered tori (i.e. tori with hyperbolic directions which, together with the tangents to the torus and the symplectic conjugates span the whole tangent space). We also present algorithms to compute the invariant splitting and the invariant manifolds of whiskered tori. We present the algorithms for both discrete-time dynamical systems and differential equations. The algorithms do not require that the system is presented in action-angle variables nor that it is close to integrable and are backed up by rigorous a-posteriori bounds. We will report on the implementation results elsewhere.

The Fast Generalized Gauss Transform

The fast Gauss transform allows for the calculation of the sum of $N$ Gaussians at $M$ points in $\mathcal{O}(N+M)$ time. Here, we extend the algorithm to a wider class of kernels, motivated by quadrature issues that arise in using integral equation methods for solving the heat equation on moving domains. In particular, robust high-order product integration methods require convolution with $\mathcal{O}(q)$ distinct Gaussian-type kernels in order to obtain $q$th-order accuracy in time. The generalized Gauss transform permits the computation of each of these kernels and, thus, the construction of fast solvers with optimal computational complexity. We also develop plane-wave representations of these Gaussian-type fields, permitting the “diagonal translation” version of the Gauss transform to be applied. When the sources and targets lie on a uniform grid, or a hierarchy of uniform grids, we show that the curse of dimensionality (the exponential growth of complexity constants with dimension) can be mitigated. Under these conditions, the algorithm has a lower operation count than the fast Fourier transform even for modest values of $N$ and $M$.

Solving the Indefinite Least Squares Problem by Hyperbolic QR Factorization

The indefinite least squares (ILS) problem involves minimizing a certain type of indefinite quadratic form. We develop perturbation theory for the problem and identify a condition number. We describe and analyze a method for solving the ILS problem based on hyperbolic QR factorization. This method has a lower operation count than one recently proposed by Chandrasekaran, Gu, and Sayed that employs both QR and Cholesky factorizations. We give a rounding error analysis of the new method and use the perturbation theory to show that under a reasonable assumption the method is forward stable. Our analysis is quite general and sheds some light on the stability properties of hyperbolic transformations. In our numerical experiments the new method is just as accurate as the method of Chandrasekaran, Gu, and Sayed.

Analysis of the truncation errors in the fast multipole method for scattering problems

Discretisation of the integral equations of acoustic scattering yields large dense systems of linear equations. Using the fast multipole method, an approximate solution to these systems can be computed with a low operation count. When implementing the method, various infinite sums must be truncated. In this paper, sharp computable bounds on the errors of these truncations are derived, which could form the basis for an automatic selection of truncation length. This choice will guarantee a given solution accuracy whilst minimising the operation count of the fast multipole algorithm.

Comparison of fast integral mesh truncation schemesfor hybrid FE-BI systems

By virtue of their low operation count, the application of fast integral methods such as the fast multipole (FMM) and adaptive integral methods (AIM) result in a substantial quickening of the boundary integral portion of the hybrid finite element-boundary integral (FE-BI) method, independent of the shape of the BI contour. Recently, various versions of the FMM have been proposed, each introducing a different approximation to the implementation of the boundary integral. The main goal of this Letter is to provide a comparison of the effect of these fast integral algorithms on the boundary integral when used in conjunction with the traditional FE-BI method.

Comparison of three FMM techniques for solving hybrid FE-BI systems

By virtue of its low operation count, the application of the fast multipole method (FMM) results in a substantial speed-up of the boundary-integral (BI) portion of the hybrid finite-element/boundary-integral technique, independent of the shape of the BI contour. Previously, various versions of the fast multipole method have been proposed, each introducing a different approximation to the implementation of the boundary integral. The main goal of this paper is to provide a comparison of the various FMM approaches on the basis of implementation, CPU time, and accuracy. To gain an appreciation of the differences among the various FMM methodologies, a large portion of the paper is devoted to a discussion of the algorithms at a tutorial level. Flow charts and pseudo-code are also given, at sufficient detail to facilitate their implementation. We present quantitative CPU and memory requirements, using the scattering by a groove as the basis for comparison, and conclude that the FMM can accelerate the BI computation without any significant deterioration in accuracy. A simpler FMM-based algorithm results in a much smaller execution time but has a larger error. However, it turns out that a third algorithm, designated the windowed FMM, provides a very good compromise with respect to error and execution time. The paper concludes with the presentation of some three-dimensional applications for which a hybrid FE-BI technique, in conjunction with a fast-integral algorithm, is well suited.

A new triangular and tetrahedral basis for high‐order (hp) finite element methods

AbstractIn this paper we describe the foundations of a new hierarchical modal basis suitable for high‐order (hp) finite element discretizations on unstructured meshes. It is based on a generalized tensor product of mixed‐weight Jacobi polynomials. The generalized tensor product property leads to a low operation count with the use of sum factorization techniques. Variable p‐order expansions in each element are readily implemented which is a crucial property for efficient adaptive discretizations. Numerical examples demonstrate the exponential convergence for smooth solutions and the ability of this formulation to handle easily very complex two‐ and three‐dmensional computational domains employing standard meshes.

Algorithms for LU decomposition on a shared memory multiprocessor

In this paper we propose an improved algorithm for the parallel LU decomposition of an ( m + 1)-banded upper Hessenberg matrix on a shared memory multi-processor, which requires O(2 nm 2/ p) parallel operations, where n is the dimension of the matrix and p is the number of processors. We show that for the special case of tridiagonal matrices this algorithms has a lower operation count than those in the literature and yields the best existing algorithm for the solution of tridiagonal systems of equations.

A Generalized Prime Factor FFT Algorithm for any $N = 2^p 3^q 5^r $

Prime factor fast Fourier transform (FFT) algorithms have two important advantages: they can be simultaneously self-sorting and in-place, and they have a lower operation count than conventional FFT algorithms. The major disadvantage of the prime factor FFT has been that it was only applicable to a limited set of values of the transform length N. This paper presents a generalized prime factor FFT, which is applicable for any $N = 2^p 3^q 5^r $, while maintaining both the self-sorting in-place capability and the lower operation count. Timing experiments on the Cray Y-MP demonstrate the advantages of the new algorithm.

Transonic Navier-Stokes computations for an oscillating wing using zonal grids

Modern jet transports and maneuvering tactical fighters operating in the transonic regime often give rise to time-dependent fluid physics that interact with flexible structural components, e.g., vortical flow, shocks, and separation. Efficient computational fluid dynamic methods are required to study such computationally intensive problems. A numerical method is presented to address this problem. Time-dependent, compressible, NavierStokes equations are used to simulate unsteady transonic flow about a three-dimensional rigid wing undergoing a forced periodic motion in angle of attack. An efficient, implicit, diagonal algorithm is utilized because of its low operation count per time step compared to other methods that solve systems of block matrix equations. The formal time accuracy is theoretically addressed and numerically demonstrated by comparison of computational results with experimental data. A zonal grid approach, capable of treating complex geometries, is presented and its time accuracy is demonstrated by comparing two- and three-zone computations with a single grid computation and experimental data.

All possible subset regressions using the QR decomposition

Existing algorithms for producing all possible subset regressions with the QR decomposition are compared, and a new algorithm is proposed with lower operation count and more accuracy.