Reconstruction Of Numerical Derivatives From Scattered Noisy Data

To problem on reconstruction of numerical derivatives from scattered noisy data, the classic method often makes use of minimizing a smoothing functional to obtain an approximate function. Based on the study of smoothing splines approximation, in the thesis we make use of natural splines to reconstruct approximation from one-dimensional noisy data, and propose an adaptive algorithm to adding base functions for approximation function to improve the accuracy of approximation. We compare the performance of the improved approximation with the original one, and numerical examples show that the improved approximation is more effective and stable than the classic one.The thesis is organized as follows.The first chapter gives an introduction about the problem of numerical derivatives, and the researches on this problem.The second chapter gives the basic knowledge about the splines and the natural splines, and gives several key properties of the natural splines.The third chapter describes the problem of numerical approximation studied in the thesis in detail, and makes use of the classic model and the natural splines to solving the problem of numerical derivatives by adaptive algorithm.The fourth chapter gives two examples to compare the result of the approximation and the one in this paper[6], then improves the approximation in this thesis by the adaptive algorithm and compares the performance of the original approximation and the approximation with adaptive algorithm by two examples.