Partition of unity method and smooth approximation

Tolstykh V.N. Partition of unity method and smooth approximation. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2024, vol. 24, no. 2, pp. 314–321 (in Russian). doi: 10.17586/2226-1494-2024-24-2-314-321

This work presents a new piecewise polynomial method of smooth analytic approximation for any dimension and variability of experimental data. Alternatives to this method are cubic and bicubic splines which have their advantages and disadvantages. There are many researches in the field of big data flexible approximation; however nothing similar was found to what is presented in the work, especially concerning multivariate dependencies. Experimental data frequently depend on many variables which for the purposes of compression, prediction, and transmission locally expressed by relatively simple analytic functions. It can be local polynomials, either on some intervals in one- dimensional case or polygons — in two-dimensional cases. Presented in the work method of local functions smooth matching extends from the one-dimension piecewise polynomial approximation method to higher dimensions that has a variety of scientific and practical applications. Under this condition, it makes sense to store and transmit coefficients of local polynomials or other local functions rather than use raw data for those purposes, which frequently requires an unacceptably large amount of resources. In the method described, we use cellular subdivision of the area of interest, and define low-degree polynomials or other parametric functions on the cells. At the junctions between cells, there are overlapping transition zones where local functions match to each other. Their amount is defined by the index of the topological compact covering. As a result, the matching obtains a single double-differentiable analytic function on the entire compact. Defined in the work basic functions are second- and third-degree especial polynomials. The values of these functions smoothly transit from one to zero within a closed unit interval. Derivatives on the interval edges both are equal to zero. The matching is performed by the homotopy which maps a unit interval to the space of functions. Efficiency of the method is demonstrated for one-dimensional case by matching a set of approximating parabolas. We extend this method to the two-dimensional case by applying the known unit partitioning technique with topological maps coverage. The computational experiment demonstrates that even in this case local functions smoothly match making a double-differentiable function on the entire compact. First result is a development of a smooth matching method for experimental data approximation by local parametric functions on a large interval. Second result is development of a new method, based on a unit partitioning, for matching two-dimensional local functions making an approximation on the two-dimensional compact. Third result is a theoretical proof of the method extension from dimension of one and two to any dimension. Task of this study consisted in the development of a useful tool for efficient storage and transmission of experimental data.

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