Multivariate Normal Hermite-Birkhoff Interpolating Splines in Julia This announces the v0.4.0 release of NormalHermiteSplines.jl package that implements the normal splines method for solving the following interpolation problem: [problem] Documentation Hermite-Birkhoff Interpolation of Scattered…

Sounds great! What are the complexities of constructing and evaluating the splines, respectively, as a function of n_1,n_2,n?

It is necessary to factorize a dense symmetric positive definite matrix of size n_1 (n_1 + n_2) in order to construct an interpolating (Hermite) normal spline. In the package it is done by means of Cholesky decomposition which complexity is about 1/3 n_1^3 (1/3 (n_1 + n_2)^3).

as for evaluating the spline in a node that complexity is O(n(n_1 + n_2)).

I’m curious about something similar to @stevengj : do you need a Cholesky factor, or would some more exotic symmetric factorization that also admitted nice solves be feasible? Several H-matrix formats don’t easily admit a triangular symmetric factor, but they have other ways of getting nice fast solv…

Thank you, @ stevengj and @ cgeoga for your ideas, it could work. There are a few approaches of improving performance of the RBF-based interpolation methods, e.g. partition of unity, multilevel and fast multi-poles ones. However, my next goal is implementing code of the Multivariate Normal Smoothing …

That sounds like a very interesting next goal! All cool stuff. Thanks for sharing and again for writing up the nice docs and references.

Yesterday I found a paper that fully described the interpolating normal splines method. Here we are: DIX, JULIO G., and ROBERT D. OGDEN. “AN INTERPOLATION SCHEME WITH RADIAL BASIS IN SOBOLEV SPACES H s. The Rocky Mountain Journal of Mathematics, vol. 24, no. 4, 1994, pp. 1319–1337. JSTOR, AN INTERP…

[ANN] NormalHermiteSplines.jl v.0.4.0

Package Announcements

stevengj March 19, 2021, 9:19pm 5

Is it possible to use something like a fast Gauss transform to accelerate this?

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[ANN] NormalHermiteSplines.jl v.0.4.0

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