Multidimensional spline integration of scattered data

Abstract

We introduce a numerical method for reconstructing a multidimensional surface using the gradient of the surface measured at some values of the coordinates. The method consists of defining a multidimensional spline function and minimizing the deviation between its derivatives and the measured gradient. Unlike a multidimensional integration along some path, the present method results in a continuous, smooth surface, furthermore, it also applies to input data that are non-equidistant and not aligned on a rectangular grid. Function values, first and second derivatives and integrals are easy to calculate. The proper estimation of the statistical and systematical errors is also incorporated in the method.

Introduction

Experiments usually result in a discrete set of datapoints $F_i$ measured at a discrete set of coordinates ${\mathbf{q}}_i$. It is in general useful to interpolate this discrete set to obtain a smooth surface $S(\mathbf{q})$ as a function of the coordinates. A more complicated situation is when derivatives $\partial F/\partial q_i$ are measured, and the surface $S(\mathbf{q})$ which fits on this grid of gradients the best is sought for.

The latter case is commonly encountered in statistical physical problems, e.g. in lattice field theory, where using standard Monte Carlo methods one is not able to measure the (logarithm of the) partition function $\log \mathcal{Z}$ itself, but only its derivatives with respect to the parameters of the theory. These parameters play the role of the above coordinates q, and the free energy $-\log \mathcal{Z}$ as a function of the parameters is what one is after. A multidimensional integration is usually carried out in such situations, which, however, can only be applied when the measurements reside on a rectangular grid. Furthermore, the result will also depend on the integration path and is only guaranteed to be smooth along the path itself.

The smooth interpolation throughout the whole grid can be obtained by e.g. a multidimensional spline function. For a detailed discussion of splines and their application see e.g. Refs. [1], [2] or [3]. Methods to determine a two-dimensional spline surface upon a rectangular grid given the values of the surface at the grid points have been known for a long time (see e.g. Ref. [4]), along with algorithms for nonrectangular input datasets (see e.g. Refs. [5], [6]), using once again the values of the surface at given coordinates. Since then spline approximation has received quite a lot of attention and the mathematical basis was studied in detail. For recent reviews on these approaches see e.g. Refs. [7], [8].

Further methods for multivariate approximation have also been developed, e.g. using rational basis functions, B-splines, tensor product splines, Powell–Sabin splines, triangulations, genetic algorithms or modified spline techniques. Moreover, based on such methods, various algorithms for spline fitting are also present in the literature, see Refs. [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26]. Techniques for handling discontinuities (e.g. Ref. [27]) and imposing constraints on the fitted function can also be found (e.g. Refs. [28], [29]).

In this paper we present a method which determines the smooth surface $S(\mathbf{q})$ using the gradient of the surface measured at some scattered values of the coordinates, and thus corresponds to a multidimensional integration scheme. The algorithm includes an introduction of a set of nodepoints, upon which the multidimensional spline is determined. This determination is linear and thus straightforward to compute. By a systematical variation of the number and position of the nodepoints the method is adapted to the particular problem, which is highly important for interpolation in more than one dimensions, as pointed out in Ref. [5]. Other interpolation schemes for which the parametrization is adapted to the data (like the rational basis function approach) are also well suited for the present problem and may be applied in this scope.

The paper is structured as follows. In Section 2 we remind the reader how a spline function in an arbitrary number of dimensions D is defined and then in Section 3 we show how the fit to the measurements is carried out. Since in practice $D>2$ is seldom necessary, in order not to complicate the notation we present the method in detail for the case of two dimensions. Nevertheless, $D>2$ is also straightforward to implement. Then we present the algorithm for the systematical placement of the nodepoints, and finally show several results on mock data.