Fast L1L_1-CkC^k polynomial spline interpolation algorithm with shape-preserving properties

International audienceIn this article, we address the interpolation problem of data points per regular $L_1$-spline polynomial curve that is invariant under a rotation of the data. We iteratively apply a minimization method on ¯ve data, belonging to a sliding window, in order to obtain this interpolating curve. We even show in the $C^k$-continuous interpolation case that this local minimization method preserves well the linear parts of the data, while a global $L_p$ (p >=1) minimization method does not in general satisfy this property. In addition, the complexity of the calculations of the unknown derivatives is a linear function of the length of the data whatever the order of smoothness of the curve

Trajectory reconstruction for robot programming by demonstration

The reproduction of hand movements by a robot remains difficult and conventional learning methods do not allow us to faithfully recreate these movements because it is very difficult when the number of crossing points is very large. Programming by Demonstration gives a better opportunity for solving this problem by tracking the user’s movements with a motion capture system and creating a robotic program to reproduce the performed tasks. This paper presents a Programming by Demonstration system in a trajectory level for the reproduction of hand/tool movement by a manipulator robot; this was realized by tracking the user’s movement with the ArToolkit and reconstructing the trajectories by using the constrained cubic spline. The results obtained with the constrained cubic spline were compared with cubic spline interpolation. Finally the obtained trajectories have been simulated in a virtual environment on the Puma 600 robot

Monotonicity preserving approximation of multivariate scattered data

This paper describes a new method of monotone interpolation and smoothing of multivariate scattered data. It is based on the assumption that the function to be approximated is Lipschitz continuous. The method provides the optimal approximation in the worst case scenario and tight error bounds. Smoothing of noisy data subject to monotonicity constraints is converted into a quadratic programming problem. Estimation of the unknown Lipschitz constant from the data by sample splitting and cross-validation is described. Extension of the method for locally Lipschitz functions is presented.<br /

Tensor B-spline numerical method for PDEs : a high performance approach

Solutions of Partial Differential Equations (PDEs) form the basis of many mathematical models in physics and medicine. In this work, a novel Tensor B-spline methodology for numerical solutions of linear second-order PDEs is proposed. The methodology applies the B-spline signal processing framework and computational tensor algebra in order to construct high-performance numerical solvers for PDEs. The method allows high-order approximations, is mesh-free, matrix-free and computationally and memory efficient. The first chapter introduces the main ideas of the Tensor B-spline method, depicts the main contributions of the thesis and outlines the thesis structure. The second chapter provides an introduction to PDEs, reviews the numerical methods for solving PDEs, introduces splines and signal processing techniques with B-splines, and describes tensors and the computational tensor algebra. The third chapter describes the principles of the Tensor B-spline methodology. The main aspects are 1) discretization of the PDE variational formulation via B-spline representation of the solution, the coefficients, and the source term, 2) introduction to the tensor B-spline kernels, 3) application of tensors and computational tensor algebra to the discretized variational formulation of the PDE, 4) tensor-based analysis of the problem structure, 5) derivation of the efficient computational techniques, and 6) efficient boundary processing and numerical integration procedures. The fourth chapter describes 1) different computational strategies of the Tensor B-spline solver and an evaluation of their performance, 2) the application of the method to the forward problem of the Optical Diffusion Tomography and an extensive comparison with the state-of-the-art Finite Element Method on synthetic and real medical data, 3) high-performance multicore CPU- and GPU-based implementations, and 4) the solution of large-scale problems on hardware with limited memory resources

Continuous State Dynamic Programming via Nonexpansive Approximation

This paper studies fitted value iteration for continuous state dynamic programming using nonexpansive function approximators. A number of nonexpansive approximation schemes are discussed. The main contribution is to provide error bounds for approximate optimal policies generated by the value iteration algorithm.Dynamic Programming; Approximation

Continuous State Dynamic Programming Via Nonexpansive Approximation

This paper studies fitted value iteration for continuous state dynamic programming using nonexpansive function approximators. A number of nonexpansive approximation schemes are discussed. The main contribution is to provide error bounds for approximate optimal policies generated by the value iteration algorithm.

Efficient Numerical Methods for Pricing American Options under Lévy Models

Two new numerical methods for the valuation of American and Bermudan options are proposed, which admit a large class of asset price models for the underlying. In particular, the methods can be applied with Lévy models that admit jumps in the asset price. These models provide a more realistic description of market prices and lead to better calibration results than the well-known Black-Scholes model. The proposed methods are not based on the indirect approach via partial differential equations, but directly compute option prices as risk-neutral expectation values. The expectation values are approximated by numerical quadrature methods. While this approach is initially limited to European options, the proposed combination with interpolation methods also allows for pricing of Bermudan and American options. Two different interpolation methods are used. These are cubic splines on the one hand and a mesh-free interpolation by radial basis functions on the other hand. The resulting valuation methods allow for an adaptive space discretization and error control. Their numerical properties are analyzed and, finally, the methods are validated and tested against various single-asset and multi-asset options under different market models