The problem of polynomial regression in which the usual monomial basis is replaced by the Bernstein basis is considered. The coefficient matrix A of the overdetermined system to be solved in the least squares sense is then a rectangular Bernstein-Vandermonde matrix. In order to use the method based on the QR decomposition of A, the first stage consists of computing the bidiagonal decomposition of the coefficient matrix A. Starting from that bidiagonal decomposition, an algorithm for obtaining the QR decomposition of A is the applied. Finally, a triangular system is solved by using the bidiagonal decomposition of the R-factor of A. Some numerical experiments showing the behavior of this approach are included.
In this paper, we present a progressive and iterative approximation method with memory for least square fitting(MLSPIA). It adjusts the control points and the weighted sums iteratively to construct a series of fitting curves (surfaces) with three weights. For any normalized totally positive basis even when the collocation matrix is of deficient column rank, we obtain a condition to guarantee that these curves (surfaces) converge to the least square fitting curve (surface) to the given data points. It is proved that the theoretical convergence rate of the method is faster than the one of the pr…
Stretched Gaussian distribution is the fundamental solution of the Hausdorff derivative diffusion equation and its corresponding stretched Gaussian noise is a widely encountered non-Gaussian noise in science and engineering. The least square method is a standard regression approach to fit Gaussian noisy data, but has distinct limits for non-Gaussian noise. Based on the Hausdorff calculus, this study develops a stretched least square method to fit stretched Gaussian noise by using the Hausdorff fractal distance as the horizontal coordinate. To better compare with the least square method, differ…
The investigation of samples with a spatial resolution in the nanometer range relies on the precise and stable positioning of the sample. Due to inherent mechanical instabilities of typical sample stages in optical microscopes, it is usually required to control and/or monitor the sample position during the acquisition. The tracking of sparsely distributed fiducial markers at high speed allows stabilizing the sample position at millisecond time scales. For this purpose, we present a scalable fitting algorithm with significantly improved performance for two-dimensional Gaussian fits as compared …
We implement a trust region method on the GPU for nonlinear least squares curve fitting problems using a new deep learning Python library called JAX. Our open source package, JAXFit, works for both unconstrained and constrained curve fitting problems and allows the fit functions to be defined in Python alone -- without any specialized knowledge of either the GPU or CUDA programming. Since JAXFit runs on the GPU, it is much faster than CPU based libraries and even other GPU based libraries, despite being very easy to use. Additionally, due to JAX's deep learning foundations, the Jacobian in JAX…
Weighted polynomial approximation of rational Bézier curves
Institute of Computer Science, University of Wrocław（弗罗茨瓦夫大学计算机研究所）；Faculty of Mathematics and Information Science, Warsaw University of Technology（华沙理工大学数学与信息科学学院）
This study focuses on Multi-Channel Transcranial Electrical Stimulation, a non-invasive brain method for stimulating neuronal activity under the influence of low-intensity currents. We introduce mathematical formulation for finding a current pattern which optimizes a L1-norm fit between a given focal target distribution and volume current density inside the brain. L1-norm is well-known to favor well-localized or sparse distributions compared to L2-norm (least-squares) fitted estimates. We present a linear programming approach which performs L1-norm fitting and penalization of the current patte…
Total least squares (TLS) methods have been widely used in data fitting. Compared with the least squares method, for TLS problem we takes into account not only the observation errors, but also the errors in the measurement matrix. This is more realistic in practical applications. For the large-scale discrete ill-posed problem $Ax \approx b$, we introduce the quantum-inspired techniques to approximate the truncated total least squares (TTLS) solution. We analyze the accuracy of the quantum-inspired truncated total least squares algorithm and perform numerical experiments to demonstrate the effi…
In this paper, we present a nonlinear least-squares fitting algorithm using B-splines with free knots. Since its performance strongly depends on the initial estimation of the free parameters (i.e. the knots), we also propose a fast and efficient knot-prediction algorithm that utilizes numerical properties of first-order B-splines. Using $\ell_p\;(p=1,2,\infty)$ norm solutions, we also provide three different strategies for properly selecting the free knots. Our initial predictions are then iteratively refined by means of a gradient-based variable projection optimization. Our method is general …
