Multivariate Normal Hermite-Birkhoff Uniform Smoothing Splines in Julia
(This project is under development)
Further examples are given in documentation.
The normal splines method for one-dimensional function interpolation and linear ordinary differential and integral equations was proposed in [2]. An idea of the multivariate splines in Sobolev space was initially formulated in [8], however it was not well-suited to solving real-world problems. Using that idea the multivariate generalization of the normal splines method was developed for two-dimensional problem of low-range computerized tomography in [3] and applied for solving a mathematical economics problem in [4]. At the same time an interpolation scheme with Matérn kernels was developed in [9], this scheme coincides with interpolating normal splines method. Further results related to applications of the normal splines method were reported at the seminars and conferences [5,6,7].
For more information and explanation see Documentation.
References
[1] Halton sequence
[2] V. Gorbunov, The method of normal spline collocation. USSR Computational Mathematics and Mathematical Physics, Vol. 29, No. 1, 1989
[3] I. Kohanovsky, Normal Splines in Computing Tomography (Нормальные сплайны в вычислительной томографии). Avtometriya, No.2, 1995
[4] V. Gorbunov, I. Kohanovsky, K. Makedonsky, Normal splines in reconstruction of multi-dimensional dependencies. Papers of WSEAS International Conference on Applied Mathematics, Numerical Analysis Symposium, Corfu, 2004
[5] I. Kohanovsky, Multidimensional Normal Splines and Problem of Physical Field Approximation, International Conference on Fourier Analysis and its Applications, Kuwait, 1998.
[6] I. Kohanovsky, Inequality-Constrained Multivariate Normal Splines with Some Applications in Finance. 27th GAMM-Seminar on Approximation of Multiparametric functions, Max-Planck-Institute for Mathematics in the Sciences, Leipzig, Germany, 2011.
[7] V. Gorbunov, I. Kohanovsky, Heterogeneous Parallel Method for the Construction of Multi-dimensional Smoothing Splines. ESCO 2014 4th European Seminar on Computing, 2014
[8] A. Imamov, M. Dzhurabaev, Splines in S.L. Sobolev spaces (Сплайны в пространствах С.Л.Соболева). Deposited manuscript. Dep. UzNIINTI, No 880, 1989.
[9] J. Dix, R. Ogden, An Interpolation Scheme with Radial Basis in Sobolev Spaces H^s(R^n). Rocky Mountain J. Math. Vol. 24, No.4, 1994.