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Kernel matrices are popular in machine learning and scientific computing, but they are limited by their quadratic complexity in both construction and storage. It is well-known that as one varies the kernel parameter, e.g., the width parameter in radial basis function kernels, the kernel matrix changes from a smooth low-rank kernel to a diagonally-dominant… (More)

This paper presents an efficient multiscale butterfly algorithm for computing Fourier integral operators (FIOs) of the form (Lf)(x) = R d a(x, ξ)e 2πıΦ(x,ξ) f (ξ)dξ, where Φ(x, ξ) is a phase function, a(x, ξ) is an amplitude function, and f (x) is a given input. The frequency domain is hierarchically decomposed into a union of Cartesian coronas. The… (More)

The paper introduces the butterfly factorization as a data-sparse approximation for the matrices that satisfy a complementary low-rank property. The factorization can be constructed efficiently if either fast algorithms for applying the matrix and its adjoint are available or the entries of the matrix can be sampled individually. For an N × N matrix, the… (More)

The hierarchical interpolative factorization (HIF) offers an efficient way for solving or precon-ditioning elliptic partial differential equations. By exploiting locality and low-rank properties of the operators, the HIF achieves quasi-linear complexity for factorizing the discrete positive definite elliptic operator and linear complexity for solving the… (More)

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