Approximation Problem Research Articles

Basic algorithm for approximation of the boundary trajectory of short-focus electron beam using the root-polynomial functions of the fourth and fifth order

The new iterative method of approximating the boundary trajectory of a short-focus electron beam propagating in a free drift mode in a low-pressure ionized gas under the condition of compensation of the space charge of electrons is considered and discussed in the article. To solve the given approximation task, the root-polynomial functions of the fourth and fifth order were applied, the main features of which are the ravine character and the presence of one global minimum. As an initial approach to solving the approximation problem, the values of the polynomial coefficients are calculated by solving the interpolation problem. After this, the approximation task is solved iteratively. All necessary polynomial coefficients are calculated multiple times, taking into account the values of the function and its derivative at the reference points. The final values of polynomial coefficients of high-order root-polynomial functions are calculated using the dichotomy method. The article also provides examples of the applying fourth-order and fifth-order root-polynomial functions to approximate sets of numerical data that correspond to the description of ravine functions. The obtained theoretical results are interesting and important for the experts who study the physics of electron beams and design modern industrial electron beam technological equipment.

Use statistical analysis to approximate integrated order batching problem

This paper highlights the tight relationship between the picking and packing processes in warehouse management and the need to consider them as an integrated problem. The study describes and models this integrated problem as a mixed-integer programming model, to optimise overall labour costs by determining the assignment of the subsets of orders, i.e. batches, for picking and packing. To address the issue of model complexity, the paper presents a statistical-based framework for generating approximate models and selecting the optimal one through examination. Based on the examination results, a pair-swapping heuristic is additionally proposed to be combined as a hybrid algorithm. Numerical experiments based on a real-world case demonstrate the effectiveness of the framework-proposed and selected hybrid algorithm by comparison with other framework-proposed approximate models, a solver, and existing heuristics. Our findings indicate that the combined usage of integrated picking and packing processes planning and the hybrid algorithm proposed and selected within the statistical-based framework can effectively reduce the cost of warehouse management.

Strong asymptotics of multi-level Hermite-Padé polynomials

We obtain the strong asymptotics of multiple orthogonal polynomials which arise in a mixed type Hermite-Padé approximation problem defined on a Nikishin system of functions. The results obtained allow to give exact estimates of the rate of convergence of the approximating rational functions and the strong asymptotics of Cauchy biorthogonal polynomials.

Sample average approximation method for a class of stochastic vector variational inequalities

In the present paper, the expected-value (EV) reformulation of a class of stochastic vector variational inequalities (SVVI) is investigated. By using the regularized gap function, the EV reformulation of SVVI is transformed into a constrained optimization problem. Then a sample average approximation (SAA) method is proposed for solving the constrained optimization problem. Under suitable assumptions, the limiting behaviors of the optimal values and optimal solutions of the approximation problem are investigated. Finally, the rates of convergence in the different senses of optimal solutions for sample average approximation problem are discussed under the error bound condition.

On the accuracy of de Casteljau-type algorithms and Bernstein representations

This paper summarizes interesting results on systematic backward and forward error analyses performed for corner cutting algorithms providing evaluation of univariate and multivariate functions defined in terms of Bernstein and Bernstein related bases. Relevant results on the conditioning of the bases are also recalled. Finally, the paper surveys important advances, lately obtained, for the design of algorithms adapted to the structure of totally positive matrices, allowing the resolution of interpolation and approximation problems with Bernstein-type bases achieving computations to high relative accuracy.

A Novel Vieta–Fibonacci Projection Method for Solving a System of Fractional Integrodifferential Equations

In this paper, a new approach for numerically solving the system of fractional integrodifferential equations is devised. To approximate the issue, we employ Vieta–Fibonacci polynomials as basis functions and derive the projection method for Caputo fractional order for the first time. An efficient transformation reduces the problem to a system of two independent equations. Solving two algebraic equations yields an approximate solution to the problem. The proposed method’s efficiency and accuracy are validated. We demonstrate the existence of the solution to the approximate problem and conduct an error analysis. Numerical tests reinforce the interpretations of the theory.

Algebraic Solution of Tropical Best Approximation Problems

We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function, by tropical Puiseux polynomial and rational functions. A new solution approach is proposed, which involves the reduction of the problem of polynomial approximation to the best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation, and we evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form, by using an iterative alternating algorithm. To illustrate the new technique developed, we solve example approximation problems in terms of a real semifield, where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which corresponds to the best Chebyshev approximation by piecewise linear functions.

On the optimal rank-1 approximation of matrices in the Chebyshev norm

The problem of low rank approximation is ubiquitous in science. Traditionally this problem is solved in unitary invariant norms such as Frobenius or spectral norm due to existence of efficient methods for building approximations. However, recent results reveal the potential of low rank approximations in Chebyshev norm, which naturally arises in many applications. In this paper we tackle the problem of building optimal rank-1 approximations in the Chebyshev norm. We investigate the properties of alternating minimization algorithm for building the low rank approximations and demonstrate how to use it to construct optimal rank-1 approximation. As a result we propose an algorithm that is capable of building optimal rank-1 approximations in Chebyshev norm for moderate matrices.

Algebraic properties of projected problems in LSQR

AbstractLSQR represents a standard Krylov projection method for the solution of systems of linear algebraic equations, linear approximation problems or regularization of discrete inverse problem. Its convergence properties (residual norms, error norms, influence of finite precision arithmetic etc.) have been widely studied. It has been observed that the components of the solution of the projected bidiagonal problem typically increase and their sign alternates. This behavior is the core of approximation properties of LSQR and is observed also for hybrid LSQR with inner Tikhonov regularization. Here we provide rigorous analysis of sign changes and monotonicity of individual components of projected solutions and projected residuals in LSQR. The results hold also for Hybrid LSQR with a fixed inner regularization parameter. The derivations do not rely on maintaining orthogonality in Krylov bases determined by the bidiagonalization process. Numerical illustration is included.

Solving high order of non-linear Volterra-fredholm Integro-differential equation by using bou-baker Polynomials approximation Method

In this paper, the Bou-baker polynomial method is used to evaluate an approximate so1ution initia1 value problem of high-order of non1inear Vo1terra-Fredholm Integro-Differential equation of the second kind. Three different examp1es and their graphics are displayed.

Inner Moreau Envelope of Nonsmooth Conic Chance-Constrained Optimization Problems

Optimization problems with uncertainty in the constraints occur in many applications. Particularly, probability functions present a natural form to deal with this situation. Nevertheless, in some cases, the resulting probability functions are nonsmooth, which motivates us to propose a regularization employing the Moreau envelope of a scalar representation of the vector inequality. More precisely, we consider a probability function that covers most of the general classes of probabilistic constraints: [Formula: see text]where [Formula: see text] is a convex cone of a Banach space. The conic inclusion [Formula: see text] represents an abstract system of inequalities, and ξ is a random vector. We propose a regularization by applying the Moreau envelope to the scalarization of the function [Formula: see text]. In this paper, we demonstrate, under mild assumptions, the smoothness of such a regularization and that it satisfies a type of variational convergence to the original probability function. Consequently, when considering an appropriately structured problem involving probabilistic constraints, we can, thus, entail the convergence of the minimizers of the regularized approximate problems to the minimizers of the original problem. Finally, we illustrate our results with examples and applications in the field of (nonsmooth) joint, semidefinite, and probust chance-constrained optimization problems. Funding: P. Pérez-Aros was supported by Centro de Modelamiento Matemático [Grants ACE210010 and FB210005] and BASAL funds for center of excellence and ANID-Chile grant Fondecyt Regular [Grants 1200283 and 1190110] and Fondecyt Exploración [Grant 13220097]. C. Soto was supported by the National Agency for Research and Development (ANID)/Scholarship Program/Doctorado Nacional Chile [Grant 2017-21170428]. E. Vilches was supported by Centro de Modelamiento Matemático [Grants ACE210010 and FB210005] and BASAL funds for center of excellence and Fondecyt Regular [Grant 1200283] and Fondecyt Exploración [Grant 13220097] from ANID-Chile.

Efficient Newton‐multigrid FEM solver for multifield nonlinear coupled problems applied to thixoviscoplastic flows

AbstractThis note is concerned with efficient Newton‐multigrid finite element method (FEM) solver for multifield nonlinear flow problems. In our approach, for efficient FEM solver, we advantageously use the delicate symbiosis aspects of the problem settings for FEM approximations, and the algorithmic tools to obtain the numerical solutions. We concretize our ideas on thixoviscoplastic flow problems. It is a two‐field coupled nonlinear problem. And beside the integrated nonlinearity within momentum and microstructure equations, thixoviscoplastic problems induce a nonlinear two‐way coupling. As far as FEM numerical solutions are concerned, we set the problem in a suitable variational form to use the corresponding wellposedness analysis to develop FEM techniques for the solver. Indeed, the wellposedness study is not an intellectual exercise, rather it is the foundation for the approximate thixoviscoplastic problem as well as for the development of an efficient solver. We base our investigations for the solver on our wellposedness and error analysis results of thixoviscoplastic flow problems published in Proc. Appl. Math. Mech. We continue our series, and proceed to develop a monolithic Newton‐multigrid thixoviscoplastic solver. The solver is based on Newton's method and geometric multigrid techniques to treat the coupling of the problem. So, we use Local Pressure Schur Complement (LPSC) concept to solve the coupled problem on mesh's elements, and proceed with outer blocks Gauss‐Seidel iteration to update the global solutions. Furthermore, we handle the nonlinearity of the problem with the combined adaptive discrete Newton's and multigrid methods. The adaptivity within discrete Newton's method is based on the adaptive step‐length control for the discrete differencing in the Jacobian calculations, while the convergence of linear multigrid solver is made to match the convergence requirement of nonlinear solver, accordingly. And the solver's update parameters are solely dependent on the actual convergence rate of the nonlinear problem. We provide the numerical results of solver performance for thixoviscoplastic lid‐driven cavity flow.

Existence of multi‐dimensional contact discontinuities for the ideal compressible magnetohydrodynamics

AbstractWe establish the local existence and uniqueness of multi‐dimensional contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) in Sobolev spaces, which are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for hyperbolic systems of conservation laws. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversely, which lead to a two‐phase free boundary problem where the pressure, velocity and magnetic field are continuous across the interface whereas the entropy and density may have jumps. To overcome the difficulties of possible nonlinear Rayleigh–Taylor instability and loss of derivatives, here we use crucially the Lagrangian formulation and Cauchy's celebrated integral (1815) for the magnetic field. These motivate us to define two special good unknowns; one enables us to capture the boundary regularizing effect of the transversal magnetic field on the flow map, and the other one allows us to get around the troublesome boundary integrals due to the transversality of the magnetic field. In particular, our result removes the additional assumption of the Rayleigh–Taylor sign condition required by Morando, Trakhinin and Trebeschi (J. Differ. Equ. 258 (2015), no. 7, 2531–2571; Arch. Ration. Mech. Anal. 228 (2018), no. 2, 697–742) and holds for both 2D and 3D and hence gives a complete answer to the two open questions raised therein. Moreover, there is no loss of derivatives in our well‐posedness theory. The solution is constructed as the inviscid limit of solutions to some suitably‐chosen nonlinear approximate problems for the two‐phase compressible viscous non‐resistive MHD.

Adaptive critic learning for approximate optimal event-triggered tracking control of nonlinear systems with prescribed performances

This paper studies the approximate optimal event-triggered tracking control problem for a class of second-order nonlinear systems with prescribed performances. By employing a prescribed performances function, restrictions on tracking errors are removed for prescribed performances including convergence rate, maximum overshoot, and steady-state error. On this basis, a complex optimal tracking control problem can be transformed into an optimal stabilising control problem by constructing an augmented system. Subsequently, an event-triggered mechanism is introduced to reduce communication resources. Then, an adaptive critic learning algorithm is used to solve the Hamilton-Jacobi-Bellman equation, where the weights in the critic networks are tuned through the gradient descent approach and experience replay technology. Through the developed learning method, the persistence of excitation condition is released, and the data efficiency is improved. The Lyapunov stability theory is introduced to verify the stability of the close-loop system and the tracking errors are uniformly ultimately bounded. Finally, simulation results are presented to show the effectiveness of the proposed approach.

On Splitting of Separatrices Corresponding to the Working Mode of the Watt Regulator

The nonlinear problem of the Watt regulator dynamics is investigated. It is assumed to be installed on a machine that performs the specific harmonic oscillations of small amplitude along the vertical. Viscous friction forces is believed to arise in the hinges of the regulator, and these forces are small. In the main operating mode of the regulator, its rods, carrying massive weights, are deflected from the downward vertical by a constant acute angle. If friction and vertical oscillations of the machine are neglected, then we obtain an approximate problem in which the dynamics of the regulator is described by an autonomous Hamiltonian system with one degree of freedom. On the phase portrait of the approximate problem, the operating mode corresponds to a singular point of the center type. The trajectories surrounding this point lie inside the separatrix, which is a homoclinic doubly asymptotic trajectory that passes through the equilibrium position corresponding to the vertical position of the rods with weights. In the phase portrait, this position corresponds to a saddle singular point. The Melnikov method is used to obtain the splitting condition for the unperturbed separatrix in the complete perturbed problem, taking into account dissipation in the hinges and vertical vibrations of the machine.

Max-Min Fairness Precoding for Physical Layer Security With Partial Channel Knowledge

We propose secure max-min fairness (MMF) precoding for a downlink multi-antenna system with multiple users and eavesdroppers. Since the problem is non-convex and non-smooth and partial channel knowledge is assumed, it is highly challenging to solve. To resolve the difficulties, we first reformulate the problem with conditional average rates, allowing to exploit the partial channel knowledge. Then, we derive the original problem as an approximate smoothing problem using a LogSumExp approach. We identify the first-order optimality condition and cast it to a generalized eigenvalue problem. Solving the problem via a power iteration offers the best local optimal solution. Simulations validate the proposed secure-MMF precoding method.

A Chebyshev-type alternation theorem for best approximation by a sum of two algebras

AbstractLet X be a compact metric space, C(X) be the space of continuous real-valued functions on X and $A_{1},A_{2}$ be two closed subalgebras of C(X) containing constant functions. We consider the problem of approximation of a function $f\in C(X)$ by elements from $A_{1}+A_{2}$. We prove a Chebyshev-type alternation theorem for a function $u_{0} \in A_{1}+A_{2}$ to be a best approximation to f.

The Existence of Entropy Solutions for a Class of Parabolic Equations

The existence and uniqueness of entropy solutions for a class of parabolic equations involving a p(x)-Laplace operator are investigated. We first prove existence of the global weak solution for the p(x)-Laplacian equations with regular initial data via the difference and variation methods as well as the standard domain expansion technique. Then, by constructing and solving a related approximation problem, the entropy solution for the p(x)-Laplacian equations with irregular initial data in whole space is also obtained.

The polar-Cartesian mild-slope equation

A novel mild-slope equation is derived based on a manipulation of the cylindrical and Cartesian coordinate reference systems. The vertical profile of the velocity field is constructed by solving an approximate problem in cylindrical coordinates. This allows us to address the local derivatives on the bottom profile along a constant-slope line. This formulation is as opposed to the Cartesian-based mild-slope equations in terms of which the profile is constructed by assuming a constant depth. An angular profile is derived for the three-dimensional case on a sloping plane beach. For the two-dimensional case, a mild-slope polar-Cartesian equation is derived, for which an improved linear dispersion relation is reconstructed. This is accomplished due to the inclusion of first-order derivatives of the local bottom profile. The coefficients of the polar-Cartesian mild-slope equation contain the derivatives of the bottom profile up to third order as opposed to second-order derivatives in the Cartesian-based equations. The equation is derived by applying the variational principle to the Cartesian Lagrangian when formulated as a function of the profile in polar coordinates. It is then compared with existing models of the mild-slope equation for simulations of two-dimensional test cases and a quasi-three-dimensional case, which have known analytical solutions. Our modified equation exhibits better matching to the exact solutions for a majority of the investigated cases.

Dissipativity-constrained learning of MPC with guaranteeing closed-loop stability

This paper addresses the data-driven approximation of model predictive control (MPC) designed for nonlinear plant systems. MPC has high ability of handling complex system-specifications and of improving the control performance, while it requires high computational complexity. Aiming at reducing the complexity, this paper addresses the data-driven approximation of MPC. To this end, the control law in MPC is described by the Koopman operator, which is a linear operator defined on the infinite-dimensional lifted state space. Then, the problem of data-driven finite-dimensional approximation of the operator is addressed. The problem is formulated as an optimization problem subject to a specified dissipativity constraint, which guarantees closed-loop stability and is modeled by a set of matrix inequalities. This paper also presents a computationally efficient algorithm of solving the optimization problem. Finally, a numerical simulation of controller construction is performed. The approximated MPC control law shows the stability of the overall control system while demonstrating high control performance.