Basic review for hierarchical matrices, including quasiseparable, SSS, HSS, H and H^2 matrices. Hierarchical matrix methods exploit the low-rank structure of off-diagonal or well-separated matrix blocks and use nested bases to compress and reuse interactions across multiple scales.
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Butterfly Algorithm
From complementary low-rank structure to butterfly algorithms, butterfly factorization, and interpolative decomposition butterfly factorization.
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Spherical harmonic transform
Review some efficient methods for spherical harmonic transform.
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Computation on disk and sphere
State of the art methods for approximation on polar and spherical geometries.
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Spectral method on domains bounded by planar curves
A sparse spectral method for solving partial differential equations on a class of two-dimensional geometries bounded by algebraic curves.
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Volterra Integral Equations
Collect some classic numerical method for Volterra integral equations.
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Function Approximation on Squares
Notes about two dimensional function approximation on squares, based on Alex Townsend’s PhD thesis.
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Radial Basis Function Approximation
Adcock, Ben, Daan Huybrechs, and Cecile Piret. Stable and accurate least squares radial basis function approximations on bounded domains. SIAM Journal on Numerical Analysis 62.6 (2024): 2698-2718.
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Spectral Method on Disk
John P. Boyd, Fu Yu, Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan–Shepp ridge polynomials, Chebyshev–Fourier Series, cylindrical Robert functions, Bessel–Fourier expansions, square-to-disk conformal mapping and radial basis functions. Journal of Computational Physics, 230.4 (2011).
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AAA Algorithm
Nakatsukasa, Yuji, Olivier Sète, and Lloyd N. Trefethen. The AAA algorithm for rational approximation. SIAM Journal on Scientific Computing 40.3 (2018).