Polynomial basis conversion made stable by truncated singular value decomposition

Solution of fractional-order differential equations based on the operational matrices of new fractional Bernstein functions

In this article, the Ritz‐Galerkin method in Bernstein polynomial basis is implemented to give an approximate solution of a hyperbolic partial differential equation with an integral condition. We will deal here with a type of nonlocal boundary value problem, that is, the solution of a hyperbolic partial differential equation with a nonlocal boundary specification. The nonlocal conditions arise mainly when the data on the boundary cannot be measured directly. The properties of Bernstein polynomial and Ritz‐Galerkin method are first presented, then Ritz‐Galerkin method is used to reduce the given hyperbolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique presented in this article. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

This paper presents a unified framework for constructing transfinite finite elements with arbitrary node distributions, using either Lagrange or Bernstein polynomial bases. Three distinct classes of elements are considered. The first includes elements with structured internal node layouts and arbitrarily positioned boundary nodes. The second comprises elements with internal nodes arranged to allow smooth transitions in a single direction. The third class consists of elements defined on structured T-meshes with selectively omitted internal nodes, resulting in sparsely populated or incomplete grids. For all three classes, new macro-element (global interpolation) formulations are introduced, enabling flexible node configurations. Each formulation supports representations based on either Lagrange or Bernstein polynomials. In the latter case, two alternative Bernstein-based models are developed as follows: one that is numerically equivalent to its Lagrange counterpart, and another that offers modest improvements in numerical performance.

This manuscript presents a framework for trajectory generation for soft continuum robots using principles from optimal control. The problem is constrained over the partial differential kinematic equations of the Cosserat rod model, capturing all modes of deformation which soft continuum systems can achieve. The derived optimal control problem is transformed to a nonlinear programming problem which can be solved using the Bernstein polynomial basis. Non-unit quaternions are used to discretely apply rotational transformations to values approximated over Bernstein polynomials, allowing individual components of strain to be constrained separately. Included within this manuscript are numerical results as well as validation through experimental results.

In this paper, we propose a framework for online quadrotor motion planning for autonomous navigation in unknown environments. Based on the onboard state estimation and environment perception, we adopt a fast marching-based path searching method to find a path on a velocity field induced by the Euclidean signed distance field (ESDF) of the map, to achieve better time allocation. We generate a flight corridor for the quadrotor to travel through by inflating the path against the environment. We represent the trajectory as piecewise Bezier curves by using Bernstein polynomial basis and formulate the trajectory generation problem as typical convex programs. By using Bezier curves, we are able to bound positions and higher order dynamics of the trajectory entirely within safe regions. The proposed motion planning method is integrated into a customized light-weight quadrotor platform and is validated by presenting fully autonomous navigation in unknown cluttered indoor and outdoor environments. We also release our code for trajectory generation as an open-source package.

In this paper, we introduce a new interpolation method that is easy to use and its interpolating function can be explicitly expressed. This interpolation method can be used for a wide spectral type of functions, since it has the ability to change the form of the interpolating function. This interpolation inspired by the Shepard type interpolation. We examine the ability of this interpolation to piecewise functions and rational functions and compare it with Lagrange interpolation and linear interpolation by using Bernstein polynomials basis. We applied this interpolation for an integral equation, and we presented a linear system of equations to approximate the solution of a Fredholm integral equation by collocation method. In the collocation method, the Reverse Interpolation expresses the approximate solution in the form of a linear combination of some basic functions. Several examples are presented to illustrate the effectiveness of the proposed method and the numerical results confirm the desired accuracy.

An identity for multivariate Bernstein polynomials

This letter deals with linear algebraic equations where the global coefficient matrix and constant vector are given respectively, by the summation of the coefficient matrices and constant vectors of the individual agents. Our approach is based on reformulating the original problem as an unconstrained optimization. Based on this exact reformulation, we first provide a gradient-based, centralized algorithm which serves as a reference for the ensuing design of distributed algorithms. We propose two sets of exponentially stable continuous-time distributed algorithms that do not require the individual agent matrices to be invertible, and are based on estimating non-distributed terms in the centralized algorithm using dynamic average consensus. The first algorithm works for time-varying weight-balanced directed networks, and the second algorithm works for general directed networks for which the communication graphs might not be balanced. Numerical simulations illustrate our results.

Chapter 2 - Simultaneous linear algebraic equations

The paper presents a new digital image hiding algorithm based on wavelet packets transform and singular value decomposition. The low-frequency sub-band of wavelet packets transform has strong anti-jamming capacity and the singular value has very strong stability. The presented algorithm implements bit plane decomposition on the secret image and wavelet packet decomposition on the carrier image. Then, it hides the bit planes with important information into the singular value matrix of the low frequency coefficient matrix, and also hides the bit planes with secondary information into the remainder sub-band matrix with higher entropy energy. The hiding location is adaptively determined by the carrier image. The experimental results indicate that, the proposed image hiding algorithm has strong robustness and anti-attack, and it also has good invisibility and big capability.

Conversion from NURBS to Bézier representation

Application of the conjugate projected gradient method to large-scale contact problems

The two papers in this issue have to do with matrices and sparsity—but from different points of view. Sparsity, in the first paper, means many zero elements in the matrix, while in the second paper it refers to many zero singular values, i.e., low rank. The context of the first paper, “On the Block Triangular Form of Symmetric Matrices” by Iain Duff and Bora Uçar, is the solution of linear systems of equations $Ax = b$ whose coefficient matrix A is sparse, i.e., has many zero elements. A direct method, such as Gaussian elimination, becomes more efficient if one can first permute the rows and columns of A into block triangular form. The classical method for permuting a matrix to block triangular form is due to Dulmage and Mendelsohn and dates back to 1963. The idea is to represent the matrix A as a bipartite graph whose nodes are columns and rows of A, and whose edges correspond to nonzero elements of A, and then to determine a matching of maximum cardinality in this graph. That means determining a maximal number of nonzero elements no two of which belong to the same row or column. Duff and Uçar analyze the block triangular form for a particular class of square matrices A: These matrices are structurally symmetric, i.e., their zero/nonzero structure is symmetric; and they can be structurally rank deficient, i.e., any rank deficiency is evident from the zero/nonzero structure of A. This paper illustrates nicely how graph theory can contribute to improving the efficiency of sparse matrix methods. The paper should appeal to those who need to solve large sparse linear systems, as well as those interested in graph theory. The second paper, “Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization” by Benjamin Recht, Maryam Fazel, and Pablo Parrilo, has applications to model reduction and system identification, machine learning, and image compression, just to name a few. Given a set of affine constraints for a matrix, the problem is to find a matrix of minimum rank that satisfies these constraints. The authors attack this hard nonconvex optimization problem by replacing it with a convex approximation: Instead of minimizing the rank, they minimize the sum of the singular values, the so-called nuclear norm. Nuclear norm minimization thus extends the compressed sensing framework from finding sparse vectors via $\ell_1$ minimization to finding low-rank matrices via $\ell_1$ minimization of the (vector of) singular values. The purpose of this paper is to justify mathematically why nuclear norm minimization does so well in practice. In addition to extending the restricted isometry property to matrices, the authors also discuss algorithms, computational performance, and numerical issues. This is a fascinating and well-written paper that combines results from compressed sensing, matrix theory, optimization, and probability.

In learning systems, high operation precision is often a desirable objective for the algorithm design. Though centralized algorithms are generally adopted, they are subjected to restrictive hypotheses on the learning systems. To overcome this challenging problem, we aim to propose some distributed learning algorithms that focus specifically on achieving the perfect tracking tasks for iterative learning control (ILC) systems. By noting the equivalent relation between the perfect tracking problem of ILC systems and the solving problem of linear algebraic equations (LAEs), we first present an observer-based distributed learning algorithm to solve LAEs, where a multiagent system is constructed with every agent being only required to access some partial information for LAEs. The distributed learning algorithm benefits from integrating both observer-based design and consensus-based design ideas such that for any solvable LAE, all agents can agree on a common solution of it under any initial conditions of agents, regardless of whether it has a unique solution or not. Then, with the distributed learning algorithm for LAEs, we further develop two classes of distributed learning control algorithms for ILC systems, which establish the perfect tracking objective in the presence of the trackable desired output even without using the basic relative degree condition that is generally imposed for conventional ILC.

This paper addresses some of the issues concerning the use of variable accuracy optical processors to improve the processing time required to obtain a high accuracy solution to a set of Linear Algebraic Equations (LAEs). We begin with a standard error analysis of the Steepest Descent iterative algorithm used to find the solution to the LAEs. This results in an expression relating the accuracy of the solution to the number of iterations and the inherent system accuracy in each mode of operation, along with the eigen-structure of the matrix describing the LAE. The accuracy at any iteration is a combination of terms representing the ideal algorithmic improvement plus the degradation due to processor inaccuracies. An evaluation of the proper number of iterations for processing in both low and high accuracy modes can be inferred through an examination of the tradeoffs in accuracy between these two terms. The expression is evaluated for several sample problems obtained from the Adaptive Phased Array Radar field. These results are then interpreted with respect to specific optical processing architectures.