Edit on GitHub

NormalSmoothingSplines.jl package

Multivariate Normal Hermite-Birkhoff Uniform Smoothing Splines in Julia

NormalSmoothingSplines.jl implements the normal splines method for solving the following approximation problem:

Problem: Given points $\{p_i, p_i \in R^n\}_{i=1}^{n_1}$, $\{s_j, s_j \in R^n\}_{j=1}^{n_2}$, and $\{\overline p_r, \overline p_r \in R^n\}_{r=1}^{n_3}$, $\{\overline s_t, \overline s_t \in R^n\}_{t=1}^{n_4}$ and sets of unit vectors $\{e_j, e_j \in R^n\}_{r=1}^{n_2}$, $\{\overline e_t, \overline e_t \in R^n\}_{t=1}^{n_4}$ find a function $f$ such that

\[\tag{1} \begin{aligned} & f(p_i) = u_i \, , \quad i = 1, 2, \dots, n_1 \, , \\ & \frac{ \partial{f} }{ \partial{e_j} }(s_j) = v_j \, , \quad j = 1, 2, \dots, n_2 \, , \\ & \underline u_r \le f(\overline p_r) \le \overline u_r \, , \quad r = 1, 2, \dots, n_3 \, , \\ & \underline v_t \le \frac{ \partial{f} }{ \partial{\overline e_t} } (\overline s_t) \le \overline v_t \, , \quad t = 1, 2, \dots, n_4 \, , \\ & n_1 \ge 0 \, , \ n_2 \ge 0 \, , \ n_3 \ge 0 \, , \ n_4 \ge 0 \, , \end{aligned}\]

where $\frac{ \partial{f} }{ \partial{e} }(s) = \nabla f(s) \cdot e = \sum _{k=1}^{n} \frac{ \partial{f} }{ \partial{x_k} } (s) e_{k}$ is a directional derivative of function $f$ at the point $s$ in the direction of $e$, and points $\{p_i\}_{i=1}^{n_1}$, $\{\overline p_r\}_{r=1}^{n_3}$ as well as points $\{s_j\}_{j=1}^{n_2}$, $\{\overline s_t\}_{t=1}^{n_4}$ are pairwise different.

We assume that function $f$ is an element of the Bessel potential space $H^s_\varepsilon (R^n)$ which is defined as:

\[ H^s_\varepsilon (R^n) = \left\{ \varphi | \varphi \in S' , ( \varepsilon ^2 + | \xi |^2 )^{s/2}{\mathcal F} [\varphi ] \in L_2 (R^n) \right\} , \quad \varepsilon \gt 0 , \ s = n/2 + 1/2 + r \, , \quad r = 1,2,\dots \, .\]

where $| \cdot |$ is the Euclidean norm, $S' (R^n)$ is space of L. Schwartz tempered distributions, parameter $s$ may be treated as a fractional differentiation order and $\mathcal F [\varphi ]$ is a Fourier transform of the $\varphi$. The parameter $\varepsilon$ can be considered as a "scaling parameter", it allows to control approximation properties of the normal spline which usually are getting better with smaller values of $\varepsilon$, also it can be used to reduce the ill-conditioness of the related computational problem (in traditional theory $\varepsilon = 1$).

The Bessel potential space $H^s_\varepsilon (R^n)$ is a Reproducing kernel Hilbert space, an element $f$ of that space can be treated as a $r$-times continuously differentiable function.

The normal splines method consists in finding a solution of system (1) having minimal norm in Hilbert space $H^s_\varepsilon (R^n)$, thus an uniform smoothing normal spline $\sigma$ is defined as follows:

\[\tag{2} \sigma = {\rm arg\,min}\{ \| f \|^2 : (1), \ \forall f \in H^s_\varepsilon (R^n) \} \, .\]

The normal splines method is based on the following functional analysis results:

  • Bessel potential space embedding theorem
  • The Riesz representation theorem for Hilbert spaces
  • Reproducing kernel properties

Using these results it is possible to reduce task (2) to solving a finite-dimensional quadratic programming problem.

The normal splines method for one-dimensional function interpolation and linear ordinary differential and integral equations was proposed in [1]. An idea of the multivariate splines in Sobolev space was initially formulated in [7], 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 [2] and applied for solving a mathematical economics problem in [3]. At the same time an interpolation scheme with Matérn kernels was developed in [8], 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 [4,5,6].

References:

[1] V. Gorbunov, The method of normal spline collocation. USSR Comput.Maths.Math.Phys., Vol. 29, No. 1, 1989

[2] I. Kohanovsky, Normal Splines in Computing Tomography (Нормальные сплайны в вычислительной томографии). Avtometriya, No.2, 1995

[3] 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

[4] I. Kohanovsky, Multidimensional Normal Splines and Problem of Physical Field Approximation, International Conference on Fourier Analysis and its Applications, Kuwait, 1998.

[5] 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.

[6] V. Gorbunov, I. Kohanovsky, Heterogeneous Parallel Method for the Construction of Multi-dimensional Smoothing Splines. ESCO 2014 4th European Seminar on Computing, University of West Bohemia, Plzen, Czech Republic, 2014.

[7] A. Imamov, M. Dzhurabaev, Splines in S.L. Sobolev spaces (Сплайны в пространствах С.Л.Соболева). Deposited manuscript. Dep. UzNIINTI, No 880, 1989.

[8] 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.

Contents