Chebfun in Three Dimensions

Hashemi, Behnam; Trefethen, Lloyd N.

doi:10.1137/16m1083803

Cited by 45 publications

(47 citation statements)

References 28 publications

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“…high-order moments), etc. This approach can be compared to calculation with functions using Chebyshev polynomials [85], and integrating Chebyshev interpolation together with the tensor cross interpolation seems to be a natural direction for further work, continuing the existing work in two and three-dimensions [84,48].…”

Section: Resultsmentioning

confidence: 99%

Parallel cross interpolation for high-precision calculation of high-dimensional integrals

Dolgov

Savostyanov

2020

Computer Physics Communications

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We propose a parallel version of the cross interpolation algorithm and apply it to calculate high-dimensional integrals motivated by Ising model in quantum physics. In contrast to mainstream approaches, such as Monte Carlo and quasi Monte Carlo, the samples calculated by our algorithm are neither random nor form a regular lattice. Instead we calculate the given function along individual dimensions (modes) and use this data to reconstruct its behaviour in the whole domain. The positions of the calculated univariate fibers are chosen adaptively for the given function. The required evaluations can be executed in parallel both along each mode (variable) and over all modes.To demonstrate the efficiency of the proposed method, we apply it to compute highdimensional Ising susceptibility integrals, arising from asymptotic expansions for the spontaneous magnetisation in two-dimensional Ising model of ferromagnetism. We observe strong superlinear convergence of the proposed method, while the MC and qMC algorithms converge sublinearly. Using multiple precision arithmetic, we also observed exponential convergence of the proposed algorithm. Combining high-order convergence, almost perfect scalability up to hundreds of processes, and the same flexibility as MC and qMC, the proposed algorithm can be a new method of choice for problems involving high-dimensional integration, e.g. in statistics, probability, and quantum physics. ≈ −1 I I J J I I I J J J Figure 1: Cross interpolation for matrices. The full matrix A is approximated by a low-rank decompositionÃ based on a small number of columns and rows computed in A. Note that the approximation (1) is exact in the positions of computed rows and columns. One of the first examples of the latter was the parallel density matrix renormalization group (DMRG) algorithm [81] for ground state computations in quantum physics. In mathematical community this research direction started with dimension-parallel linear solver [30] and cross algorithms in HT format [44]. The main difficulty of parallelisation over dimension is the need of algorithmic modifications, since state of the arts tensor algorithms were designed in intrinsically sequential way. Ideally, such modifications should not compromise numerical stability or convergence for the sake of parallel efficiency. In this paper we develop a parallel version of the TT cross interpolation algorithm [74]. The parallel algorithm is adaptive and converges with the same rate as the sequential version, but involves only local communications with constant loading of processes, and demonstrates almost perfect scaling up to the ultimate partitioning where each process is responsible for a single direction (mode, variable). Moreover, further speedup can be achieved using OpenMP parallelisation of tensor algebra in each process.The rest of the paper is organised as follows. In Sec. 2 we recall the cross interpolation method for matrices and provide necessary definitions. In Sec. 3 we discuss how the matrix interpolation can be applied for high-dimensi...

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Section: Resultsmentioning

confidence: 99%

Parallel cross interpolation for high-precision calculation of high-dimensional integrals

Dolgov

Savostyanov

2020

Computer Physics Communications

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“…where the coefficient array C can be computed by FFT in O(N d log N ) time [14]. Following the practice of Chebfun3t [11], for each j = 1, . .…”

Section: Adaptive Construction Letmentioning

confidence: 99%

“…The construction and manipulation of 2D approximations is suitably fast for a wide range of smooth examples. Most recently, Hashemi and Trefethen created an extension of Chebfun called Chebfun3 for 3D approximations on hyperrectangles using low-rank "slice-Tucker" decompositions [11]. The range of functions that Chebfun3 can cope with in a reasonable interactive computing time is somewhat narrower than for Chebfun2, as one would expect.…”

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confidence: 99%

An Adaptive Partition of Unity Method for Multivariate Chebyshev Polynomial Approximations

Aiton¹,

Driscoll²

2019

SIAM J. Sci. Comput.

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Spectral polynomial approximation of smooth functions allows real-time manipulation of and computation with them, as in the Chebfun system. Extension of the technique to two-dimensional and three-dimensional functions on hyperrectangles has mainly focused on low-rank approximation. While this method is very effective for some functions, it is highly anisotropic and unacceptably slow for many functions of potential interest. A method based on automatic recursive domain splitting, with a partition of unity to define the global approximation, is easy to construct and manipulate. Experiments show it to be as fast as existing software for many low-rank functions, and much faster on other examples, even in serial computation. It is also much less sensitive to alignment with coordinate axes. Some steps are also taken toward approximation of functions on nonrectangular domains, by using least-squares polynomial approximations in a manner similar to Fourier extension methods, with promising results.

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“…However, the representation of the distributed variables in the time-periodic case is not obvious. Chebfun was recently extended to functions of more than one variable [25]- [28]. However, it is not possible to mix Fourier and Chebyshev expansions for multivariable functions as we require, it is either all Chebyshev or all Fourier.…”

Section: B Chebyshev and Fourier Representations Of Variablesmentioning

confidence: 99%

Numerical Analysis of Multidomain Systems: Coupled Nonlinear PDEs and DAEs With Noise

Demir

Hanay

2018

IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.

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We present a numerical modeling and simulation paradigm for multidomain, multiphysics systems with components modeled both in a lumped and distributed manner. The lumped components are modeled with a system of differential-algebraic equations (DAEs), whereas the possibly nonlinear distributed components that may belong to different physical domains are modeled using partial differential equations (PDEs) with associated boundary conditions. We address a comprehensive suite of problems for nonlinear coupled DAE-PDE systems including 1) transient simulation; 2) periodic steady-state (PSS) analysis formulated as a mixed boundary value problem that is solved with a hierarchical spectral collocation technique based on a joint Fourier-Chebyshev representation, for both forced and autonomous systems; 3) Floquet theory and analysis for coupled linear periodically time-varying DAE-PDE systems; 4) phase noise analysis for multidomain oscillators; and 5) efficient parameter sweeps for PSS and noise analyses based on first-order and pseudo-arclength continuation schemes. All of these techniques, implemented in a prototype simulator, are applied to a substantial case study: a multidomain feedback oscillator composed of distributed and lumped components in two physical domains, namely, a nano-mechanical beam resonator operating in the nonlinear regime, an electrical delay line, an electronic amplifier and a sensor-actuator for the transduction between the two physical domains.Index Terms-Chebyshev and Fourier representations and collocation, differential-algebraic equations (DAEs), mixed boundary value problems, multidomain systems, multiphysics simulation, nano electro-mechanical systems (NEMS), noise, oscillators, partial differential equations (PDEs), phase noise, spectral methods.

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Chebfun in Three Dimensions

Cited by 45 publications

References 28 publications

Parallel cross interpolation for high-precision calculation of high-dimensional integrals

Parallel cross interpolation for high-precision calculation of high-dimensional integrals

An Adaptive Partition of Unity Method for Multivariate Chebyshev Polynomial Approximations

Numerical Analysis of Multidomain Systems: Coupled Nonlinear PDEs and DAEs With Noise

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