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Register a new userComputation of highly oscillatory Bessel transforms with algebraic singularities
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In this paper, we consider the Clenshaw-Curtis-Filon method for the highly oscillatory Bessel transform $int_0^1x^alpha (1-x)^beta f(x) J_{ u}(omega x)dx$, where $f$ is a smooth function on $[0, 1]$, and $ ugeq0.$ The method is based on Fast Fourier Transform (FFT) and fast computation of the modified moments. We give a recurrence relation for the modified moments and present an efficient method for the evaluation of modified moments by using recurrence relation. Moreover, the corresponding error bound in inverse powers of $N$ for this method for the integral is presented. Numerical examples are provided to support our analysis and show the efficiency and accuracy of the method.
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We develop two classes of composite moment-free numerical quadratures for computing highly oscillatory integrals having integrable singularities and stationary points. The first class of the quadrature rules has a polynomial order of convergence and the second class has an exponential order of convergence. We first modify the moment-free Filon-type method for the oscillatory integrals without a singularity or a stationary point to accelerate their convergence. We then consider the oscillatory integrals without a singularity or a stationary point and then those with singularities and stationary points. The composite quadrature rules are developed based on partitioning the integration domain according to the wave number and the singularity of the integrand. The integral defined on a subinterval has either a weak singularity without rapid oscillation or oscillation without a singularity. The classical quadrature rules for weakly singular integrals using graded points are employed for the singular integral without rapid oscillation and the modified moment-free Filon-type method is used for the oscillatory integrals without a singularity. Unlike the existing methods, the proposed methods do not have to compute the inverse of the oscillator. Numerical experiments are presented to demonstrate the approximation accuracy and the computational efficiency of the proposed methods. Numerical results show that the proposed methods outperform methods published most recently.
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This work settles the problem of constructing entropy stable non-oscillatory (ESNO) fluxes by framing it as a least square optimization problem. A flux sign stability condition is introduced and utilized to construct arbitrary order entropy stable flux as a convex combination of entropy conservative and non-oscillatory flux. This simple approach is robust which does not explicitly requires the computation of costly dissipation operator and high order reconstruction of scaled entropy variable for constructing the diffusion term. The numerical diffusion is optimized in the sense that entropy stable flux reduces to the underlying non-oscillatory flux. Different non-oscillatory entropy stable fluxes are constructed and used to compute the numerical solution of various standard scalar and systems test problems. Computational results show that entropy stable schemes are comparable in term of non-oscillatory nature of schemes using only the underlying non-oscillatory fluxes. Moreover, these entropy stable schemes maintains the formal order of accuracy of the lower order flux used in the convex combination.
This paper focuses on finite-time in-network computation of linear transforms of distributed graph data. Finite-time transform computation problems are of interest in graph-based computing and signal processing applications in which the objective is to compute, by means of distributed iterative methods, various (linear) transforms of the data distributed at the agents or nodes of the graph. While finite-time computation of consensus-type or more generally rank-one transforms have been studied, systematic approaches toward scalable computing of general linear transforms, specifically in the case of heterogeneous agent objectives in which each agent is interested in obtaining a different linear combination of the network data, are relatively less explored. In this paper, by employing ideas from algebraic geometry, we develop a systematic characterization of linear transforms that are amenable to distributed in-network computation in finite-time using linear iterations. Further, we consider the general case of directed inter-agent communication graphs. Specifically, it is shown that emph{almost all} linear transformations of data distributed on the nodes of a digraph containing a Hamiltonian cycle may be computed using at most $N$ linear distributed iterations. Finally, by studying an associated matrix factorization based reformulation of the transform computation problem, we obtain, as a by-product, certain results and characterizations on sparsity-constrained matrix factorization that are of independent interest.
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