Fast Sine Research Articles

Sparsifiable spectral equivalence of DtN mapping and its application to elliptic problems

A sparsifiable spectrally equivalent preconditioner is developed for a special finite element discretization of two-dimensional elliptic problems. We start with a spectrally equivalent preconditioner which is closely related to the discrete Dirichlet-to-Neumann (DtN) mappings in sub-domains. The fullness of discrete DtN mappings renders an essential difficulty for a direct implementation. Based on numerical evidences, we present a particular square root matrix that is spectrally equivalent to the discrete DtN mappings. Unfortunately, this square root matrix is still not sparsifiable. To remedy this problem, we resort to the Chebyshev rational approximation. Incorporated with the fast sine transforms, a sparsifiable and spectrally equivalent preconditioner is finally obtained. Numerical examples are provided to validate efficiency of the proposed preconditioning technique.

A fast solvable operator-splitting scheme for time-dependent advection diffusion equation

It is well known that the implicit central difference discretization for unsteady advection diffusion equation (ADE) suffers from being time-consuming to solve when the advection term dominates. In this paper, we propose an operator-splitting scheme for the unsteady ADE, in which the ADE is firstly discretized by Crank-Nicolson (CN) scheme in time and central difference scheme in space; and then the discrete advection-diffusion problem is split as advection sub-problem and diffusion sub-problem at each time-level. The significance of the new scheme is that these sub-problems can be fast and directly solved within a linearithmic complexity (a linear-times-logarithm complexity) by means of fast sine transforms (FSTs). In particular, the complexity is independent of the dominance of the advection term. Theoretically, we show that proposed scheme is unconditionally stable and of second-order convergence in time and space. Numerical results are reported to show the efficiency of the proposed scheme.

Stiff-cut leap-frog scheme for fractional Laplacian diffusion equations

In this study, we investigate the numerical discretization of two-dimensional fractional Laplacian diffusion equations. After discretizing the space derivative, we split the discrete fractional Laplacian into two parts. One is the τ preconditioner multiplying a constant, solved using the leap-frog scheme, while the other part and the remaining nonlinear term are treated with an explicit scheme. This results in what is termed the stiff-cut leap-frog scheme, which is more efficient than the traditional leap-frog scheme. The main advantage of the proposed scheme is that the coefficient matrices of the generated linear systems belong to τ algebra, which can be diagonalized by fast discrete sine transform with the complexity of O(NlogN), where N is the freedom of unknowns. Hence, no iterative method is required at each time integration. Moreover, we theoretically prove that the proposed scheme is unconditionally stable and converges in second order. Ultimately, numerical experiments are conducted to demonstrate the precision and effectiveness of the stiff-cut leap-frog scheme.

MQGeometry-1.0: a multi-layer quasi-geostrophic solver on non-rectangular geometries

Abstract. This paper presents MQGeometry, a multi-layer quasi-geostrophic (QG) equation solver for non-rectangular geometries. We advect the potential vorticity (PV) with finite volumes to ensure global PV conservation using a staggered discretization of the PV and stream function (SF). Thanks to this staggering, the PV is defined inside the domain, removing the need to define the PV on the domain boundary. We compute PV fluxes with upwind-biased interpolations whose implicit dissipation replaces the usual explicit (hyper-)viscous dissipation. The discretization presented here does not require tuning of any additional parameter, e.g., additional eddy viscosity. We solve the QG elliptic equation with a fast discrete sine transform spectral solver on rectangular geometry. We extend this fast solver to non-rectangular geometries using the capacitance matrix method. Subsequently, we validate our solver on a vortex-shear instability test case in a circular domain, on a vortex–wall interaction test case, and on an idealized wind-driven double-gyre configuration in an octagonal domain at an eddy-permitting resolution. Finally, we release a concise, efficient, and auto-differentiable PyTorch implementation of our method to facilitate future developments on this new discretization, e.g., machine-learning parameterization or data-assimilation techniques.

The use of professional audio external USB sound cards for building acoustics measurements

CSIRO purchased an open application programming interface for hardware with an ethernet connection from a major acoustical instrument manufacturer. Using this interface, the author developed software for fractional octave band filtering with linear and exponential averaging. He then extended this software to automatically measure reverberation time. Fast Fourier Transform and Stepped Sine software was also developed. He then realized that CSIRO also owned two professional audio external USB sound cards. One of these had eight analogue inputs and ten analogue outputs. The other had two analogue inputs and two analogue outputs. Both could supply 48 V phantom power on their analogue inputs. It is possible to purchase phantom power preamplifiers for professional measurement microphones, but CSIRO decided to purchase phantom power to IEPE pre-amplifier adaptors. This enabled the use of professional pre-polarized microphones with IEPE microphone preamplifiers and accelerometers with built-in IEPE preamplifiers. CSIRO also had external USB hardware from another acoustical instrument manufacturer. Upon opening this instrument, it consisted of a professional audio external USB sound card and an IEPE front end for the two analogue input channels. The author decided to produce a version of his software that could be used with these three external USB sound cards.

Application of the Method of Fast Expansions to Construction of a Trajectory of Movement of a Body with Variable Mass from Its Initial Position in a Gained Final Position in a Gravitational Field

An analytical solution of the problem of the movement of a spacecraft from the starting point to the final point in a certain time is given. First, the method of fast sine expansions is used. The space problem considered here is essentially non-linear, what necessitates the use of trigonometric interpolation methods, which surpass all known interpolations in accuracy and simplicity. In this case, the problem of calculating Fourier coefficients by integral formulas is replaced by the solution of an orthogonal interpolation system. In this regard, two cases are considered on the segment \(\left[ {0,a} \right]\): universal interpolation and trigonometric sine and cosine interpolations. A theorem on the rapid decrease of expansion coefficients is proved, and a compact formula for calculating the interpolation coefficients is obtained. A general theory of fast expansions is given. It is shown that in this case, the Fourier coefficients decrease significantly faster with the growth of the ordinal number compared to the Fourier coefficients in the classical case. This property makes it possible to significantly reduce the number of terms taken into account in the Fourier series, significantly increase the accuracy of calculations and reduce the amount of calculations on a computer. The analysis of the obtained solutions of the spacecraft motion problem is carried out and their comparison with the exact solution of the test problem is proposed. An approximate solution by the method of fast expansions can be taken as an exact one, since the input data of the problem used from reference books have a higher error.

Matrix splitting preconditioning based on sine transform for solving two-dimensional space-fractional diffusion equations

Finite difference discretization of the two-dimensional space-fractional diffusion equations derives a complicated linear system consisting of identity matrix and four scaled block-Toeplitz with Toeplitz block (BTTB) matrices resulted from the left and right Riemann–Liouville fractional derivatives in different directions. Incorporating with the diffusion coefficients and the symmetric parts of the BTTB matrices, we construct a diagonal and symmetric splitting (DSS) iteration method, which is demonstrated to be convergent conditionally when the considered space-fractional diffusion equations have sufficiently close diffusion coefficients. By further replacing the symmetric Toeplitz matrices involved in the BTTB matrices with τ matrices, an approximated DSS (ADSS) preconditioner based on two-dimensional fast sine transform is designed to accelerate the convergence rates of the Krylov subspace iteration methods. In this way, the total computational complexity of the ADSS-preconditioned GMRES method will be of O(n2logn), where n2 represents the dimension of the corresponding discrete linear system. In addition, theoretical analysis demonstrates that the eigenvalues of the ADSS-preconditioned matrix are weakly clustered around a complex disk centered at 1 with the radius less than 1. Numerical experiments show that the ADSS-preconditioned GMRES method is much more efficient than the other existing methods, and can show h-independent convergence behavior.

Efficient energy-preserving eighth-order compact finite difference schemes for the sine-Gordon equation

In this paper, three efficient energy conserving compact finite difference schemes are developed to numerically solve the sine-Gordon equation with homogeneous Dirichlet boundary conditions. To be specific, the spatial discretization is carried out by a novel eighth-order accurate compact difference scheme in which the fast discrete sine transform can be utilized for efficient implementation. The second-order conservative Crank–Nicolson scheme is considered in the temporal direction. Then the conservative property and convergence of the first scheme in two-dimensional space are discussed. A linearized iteration based on the fast discrete sine transform technique is devised to solve the nonlinear system effectively. Since the resultant algorithm does not use matrix inversion, it is computationally efficient in long-time calculations. Furthermore, the two other schemes are constructed based on improved scalar auxiliary variable approaches by converting the sine-Gordon equation into an equivalent new system which involves solving linear systems with constant coefficients at each time step. Finally, numerical experiments are presented to validate the correctness of the theoretical findings and demonstrate the excellent performance in long-time conservation of the schemes.

A fast implicit difference scheme for solving high-dimensional time-space fractional nonlinear Schrödinger equation

In this work, an efficient implicit difference scheme is developed for solving the high-dimensional time-space fractional nonlinear Schrödinger equation. The derived scheme is constructed by utilizing a fast evaluation of Caputo fractional derivative based on the - formula; meanwhile, the compact finite difference with matrix transfer technique is adopted for the spatial discretization. Moreover, a linearized iteration method based on the fast discrete sine transform technique is considered to solve the nonlinear system effectively. Because the resultant algorithm does not use matrix inversion, it is computationally efficient in long-time calculations. The stability, truncation error and convergence analysis of the discrete scheme are discussed in detail. Furthermore, a fast iterative algorithm is provided. Finally, several numerical examples are given to verify the efficiency and accuracy of the derived scheme, and a comparison with similar work is presented.

A block Toeplitz preconditioner for all-at-once systems from linear wave equations

In this work, we propose a novel parallel-in-time preconditioner for an all-at-once system, arising from the numerical solution of linear wave equations. Namely, our main result concerns a block tridiagonal Toeplitz preconditioner that can be diagonalized via fast sine transforms, whose effectiveness is theoretically shown for the nonsymmetric block Toeplitz system resulting from discretizing the concerned wave equation. Our approach is to first transform the original linear system into a symmetric one and subsequently develop the desired preconditioning strategy based on the spectral symbol of the modified matrix. Various Krylov subspace methods are considered. That is, we show that the minimal polynomial of the preconditioned matrix is of low degree, which leads to fast convergence when the generalized minimal residual method is used. To fully utilize the symmetry of the modified matrix, we additionally construct an absolute-value preconditioner which is symmetric positive definite. Then, we show that the eigenvalues of the preconditioned matrix are clustered around $\pm 1$, which gives a convergence guarantee when the minimal residual method is employed. Numerical examples are given to support the effectiveness of our preconditioner. Our block Toeplitz preconditioner provides an alternative to the existing block circulant preconditioner proposed by McDonald, Pestana, and Wathenin [SIAM J. Sci. Comput., 40 (2018), pp. A1012–A1033],advancing the symmetrization preconditioning theory that originated from the same work.

Fast Compact Difference Scheme for Solving the Two-Dimensional Time-Fractional Cattaneo Equation

The time-fractional Cattaneo equation is an equation where the fractional order α∈(1,2) has the capacity to model the anomalous dynamics of physical diffusion processes. In this paper, we consider an efficient scheme for solving such an equation in two space dimensions. First, we obtain the space’s semi-discrete numerical scheme by using the compact difference operator in the spatial direction. Then, the semi-discrete scheme is converted to a low-order system by means of order reduction, and the fully discrete compact difference scheme is presented by applying the L2-1σ formula. To improve the computational efficiency, we adopt the fast discrete Sine transform and sum-of-exponentials techniques for the compact difference operator and L2-1σ difference operator, respectively, and derive the improved scheme with fast computations in both time and space. That aside, we also consider the graded meshes in the time direction to efficiently handle the weak singularity of the solution at the initial time. The stability and convergence of the numerical scheme under the uniform meshes are rigorously proven, and it is shown that the scheme has second-order and fourth-order accuracy in time and in space, respectively. Finally, numerical examples with high-dimensional problems are demonstrated to verify the accuracy and computational efficiency of the derived scheme.

A stabilized fully-discrete scheme for phase field crystal equation

A finite difference scheme is developed for solving the phase field crystal equation with the Dirichlet boundary condition. The second-order stabilized semi-implicit scheme is applied to discretize the time variable. The mass conservation, unique solvability of the semi-discrete scheme are proved. Then, the spatial discretization is attained by the fourth-order compact difference scheme. The unconditional energy stability and convergence analysis of the fully-discrete scheme are presented. In the implementation, a fast sine transform technique is utilized to reduce the computational cost. Several numerical examples are presented to verify the effectiveness of the proposed scheme.

Design of the Processors for Fast Cosine and Sine Fourier Transforms

Design of the Processors for Fast Cosine and Sine Fourier Transforms

Application of the Fast Expansion Method in Space–Related Problems

In the paper, numerical and approximate analytical solutions for the problem of the motion of a spacecraft from a starting point to a final point during a certain time are obtained. The unpowered and powered portions of the flight are considered. For a numerical solution, a finite-difference scheme of the second order of accuracy is constructed. The space-related problem considered in the study is essentially nonlinear, which necessitates the use of trigonometric interpolation methods to replace the task of calculating the Fourier coefficients with the integral formulas by solving the interpolation system. One of the simplest options for trigonometric sine interpolation on a semi-closed segment [–a, a), where the right end is not included in the general system of interpolation points, is considered. In order to maintain the conditions of orthogonality of sines, an even number of 2M calculation points is uniformly applied to the segment. The sine interpolation theorem is proved and a compact formula is given for calculating the interpolation coefficients. A general theory of fast sine expansion is given. It is shown that in this case, the Fourier coefficients decrease much faster with the increase in serial number compared to the Fourier coefficients in the classical case. This property allows reducing the number of terms taken into account in the Fourier series, as well as the amount of computer calculations, and increasing the accuracy of calculations. The analysis of the obtained solutions is carried out, and their comparison with the exact solution of the test problem is proposed. With the same calculation error, the time spent on a computer using the fast expansion method is hundreds of times less than the time spent on classical finite-difference method.

Regularity-Constrained Fast Sine Transforms

This letter proposes a fast implementation of the regularity-constrained discrete sine transform (R-DST). The original DST \textit{leaks} the lowest frequency (DC: direct current) components of signals into high frequency (AC: alternating current) subbands. This property is not desired in many applications, particularly image processing, since most of the frequency components in natural images concentrate in DC subband. The characteristic of filter banks whereby they do not leak DC components into the AC subbands is called \textit{regularity}. While an R-DST has been proposed, it has no fast implementation because of the singular value decomposition (SVD) in its internal algorithm. In contrast, the proposed regularity-constrained fast sine transform (R-FST) is obtained by just appending a regularity constraint matrix as a postprocessing of the original DST. When the DST size is $M\times M$ ($M=2^\ell$, $\ell\in\mathbb{N}_{\geq 1}$), the regularity constraint matrix is constructed from only $M/2-1$ rotation matrices with the angles derived from the output of the DST for the constant-valued signal (i.e., the DC signal). Since it does not require SVD, the computation is simpler and faster than the R-DST while keeping all of its beneficial properties. An image processing example shows that the R-FST has fine frequency selectivity with no DC leakage and higher coding gain than the original DST. Also, in the case of $M=8$, the R-FST saved approximately $0.126$ seconds in a 2-D transformation of $512\times 512$ signals compared with the R-DST because of fewer extra operations.

Fast associated classical orthogonal polynomial transforms

We discuss a fast approximate solution to the associated classical–classical orthogonal polynomial connection problem. We first show that associated classical orthogonal polynomials are solutions to a fourth-order quadratic eigenvalue problem with polynomial coefficients such that the differential operator is degree-preserving. Upon linearization, the discretization of this quadratic eigenvalue problem is block upper-triangular and banded. After a perfect shuffle, we extend a divide-and-conquer approach to the upper-triangular and banded generalized eigenvalue problem to the blocked case, which may be accelerated by one of a few different algorithms. Associated orthogonal polynomials arise from iterated Stieltjes transforms of orthogonal polynomials; hence, fast approximate conversion to classical cases combined with fast discrete sine and cosine transforms provides a modular mechanism for synthesis of singular integral transforms of classical orthogonal polynomial expansions.

Fast High-Order Difference Scheme for the Modified Anomalous Subdiffusion Equation Based on Fast Discrete Sine Transform

The modified anomalous subdiffusion equation plays an important role in the modeling of the processes that become less anomalous as time evolves. In this paper, we consider the efficient difference scheme for solving such time-fractional equation in two space dimensions. By using the modified L1 method and the compact difference operator with fast discrete sine transform technique, we develop a fast Crank-Nicolson compact difference scheme which is proved to be stable with the accuracy of O τ min 1 + α , 1 + β + h 4 . Here, α and β are the fractional orders which both range from 0 to 1, and τ and h are, respectively, the temporal and spatial stepsizes. We also consider the method of adding correction terms to efficiently deal with the nonsmooth problems. Numerical examples are provided to verify the effectiveness of the proposed scheme.

Fast Computation of Orthogonal Systems with a Skew‐Symmetric Differentiation Matrix

AbstractOrthogonal systems in L2(ℝ), once implemented in spectral methods, enjoy a number of important advantages if their differentiation matrix is skew‐symmetric and highly structured. Such systems, where the differentiation matrix is skew‐symmetric, tridiagonal, and irreducible, have been recently fully characterised. In this paper we go a step further, imposing the extra requirement of fast computation: specifically, that the first N coefficients of the expansion can be computed to high accuracy in operations. We consider two settings, one approximating a function f directly in (−∞, ∞) and the other approximating [f(x) + f(−x)]/2 and [f(x) − f(−x)]/2 separately in [0, ∞). In each setting we prove that there is a single family, parametrised by α, β > − 1, of orthogonal systems with a skew‐symmetric, tridiagonal, irreducible differentiation matrix and whose coefficients can be computed as Jacobi polynomial coefficients of a modified function. The four special cases where α, β = ± 1/2 are of particular interest, since coefficients can be computed using fast sine and cosine transforms. Banded, Toeplitz‐plus‐Hankel multiplication operators are also possible for representing variable coefficients in a spectral method. In Fourier space these orthogonal systems are related to an apparently new generalisation of the Carlitz polynomials. © 2020 Wiley Periodicals, Inc.

Fast Crank-Nicolson compact difference scheme for the two-dimensional time-fractional mobile/immobile transport equation

In this paper, we consider the efficient numerical scheme for solving time-fractional mobile/immobile transport equation. By utilizing the compact difference operator to approximate the Laplacian, we develop an efficient Crank-Nicolson compact difference scheme based on the modified L1 method. It is proved that the proposed scheme is stable with the accuracy of $ O(\tau^{2-\alpha}+h^4) $, where $ \tau $ and $ h $ are respectively the temporal and spatial stepsizes, and the fractional order $ \alpha\in(0, 1) $. In addition, we improve the computational performance for the non-smooth issue by the fast discrete sine transform technology and the method of adding correction terms. Finally, numerical examples are provided to verify the effectiveness of the proposed scheme.

Splitting preconditioning based on sine transform for time-dependent Riesz space fractional diffusion equations

We study the sine-transform-based splitting preconditioning technique for the linear systems arising in the numerical discretization of time-dependent one dimensional and two dimensional Riesz space fractional diffusion equations. Those linear systems are Toeplitz-like. By making use of diagonal-plus-Toeplitz splitting iteration technique, a sine-transform-based splitting preconditioner is proposed to accelerate the convergence rate efficiently when the Krylov subspace method is implemented. Theoretically, we prove that the spectrum of the preconditioned matrix of the proposed method is clustering around 1. In practical computations, by the fast sine transform the computational complexity at each time level can be done in $${{\mathcal {O}}}(n\log n)$$ operations where n is the matrix size. Numerical examples are presented to illustrate the effectiveness of the proposed algorithm.