Suboptimal and Approximate Solutions to Extremal Problems

2011 ◽  
pp. 113-132
Author(s):  
Peter I. Kogut ◽  
Günter R. Leugering
Keyword(s):  
Approximate Solutions ◽  
Extremal Problems

scholarly journals About the estimation of the convergence rate of projection-iteration processes of conditional minimization of a functional

2018 ◽  
Author(s):  
L. L. Gart
Keyword(s):  
Hilbert Space ◽  
Gradient Method ◽  
Iteration Method ◽  
Initial Approximation ◽  
Approximate Solutions ◽  
Extremal Problems ◽  
Conditional Gradient Method ◽  
Original Space ◽  
Conditional Minimization ◽  
Convergence Of Sequences

We study projection-iterative processes based on the conditional gradient method to solve the problem of minimizing a functional in a real separable Hilbert space. To solve extremal problems, methods of approximate (projection) type are often used, which make it possible to replace the initial problem by a sequence of auxiliary approximating extremal problems. The work of many authors is devoted to the problems of approximating various classes of extremal problems. Investigations of projection and projection-iteration methods for solving extremal problems with constraints in Hilbert and reflexive Banach spaces were carried out, in particular, in the works of S.D. Balashova, in which the general conditions for approximation and convergence of sequences of exact and approximate solutions of approximating extremal problems considered both in subspaces of the original space and in certain spaces isomorphic to them were proposed. The projection-iterative approach to the approximate solution of an extremal problem is based on the possibility of applying iterative methods to the solution of approximating problems. Moreover, for each of the "approximate" extremal problems, only a few approximations are obtained with the help of a certain iteration method and the last of them as the initial approximation for the next "approximate" problem is used. This paper, in continuation of the author's past work to solve the problem of minimizing a functional on a convex set of Hilbert space, is devoted to obtaining theoretical estimates of the rate of convergence of the projection-iteration method based on the conditional gradient method (for different ways of specifying a step multiplier) of minimization of approximating functionals in certain spaces isomorphic to subspaces of the original space. We prove theorems on the convergence of a projection-iteration method and obtain estimates of error and convergence degree

Interactive Graphics for the Analysis of Tires

1985 ◽  
Vol 13 (3) ◽  
pp. 127-146 ◽  
Author(s):  
R. Prabhakaran
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Approximate Solutions ◽  
Interactive Graphics ◽  
Element Analysis ◽  
Analysis Process ◽  
Physical Problems ◽  
The Finite Element Analysis ◽  
Analysis Computer ◽  
Discretization Technique

Abstract The finite element method, which is a numerical discretization technique for obtaining approximate solutions to complex physical problems, is accepted in many industries as the primary tool for structural analysis. Computer graphics is an essential ingredient of the finite element analysis process. The use of interactive graphics techniques for analysis of tires is discussed in this presentation. The features and capabilities of the program used for pre- and post-processing for finite element analysis at GenCorp are included.

Some Extremal Problems in (D, D+1)-Regular Graphs

2016 ◽  
Vol 6 (2) ◽  
pp. 105
Author(s):  
N. Murugesan ◽  
R. Anitha
Keyword(s):  
Extremal Problems ◽  
Regular Graphs

A new algorithm for finding CRM-model coefficients

2019 ◽  
Vol 5 (3) ◽  
pp. 164-185 ◽  
Author(s):  
Alexander D. Bekman ◽  
Sergey V. Stepanov ◽  
Alexander A. Ruchkin ◽  
Dmitry V. Zelenin
Keyword(s):  
Approximate Solution ◽  
Optimization Problem ◽  
Optimal Solution ◽  
Complex Form ◽  
Approximate Solutions ◽  
The Other ◽  
Programming Algorithm ◽  
Stochastic Algorithms ◽  
Large Set ◽  
Flow Rates

The quantitative evaluation of producer and injector well interference based on well operation data (profiles of flow rates/injectivities and bottomhole/reservoir pressures) with the help of CRM (Capacitance-Resistive Models) is an optimization problem with large set of variables and constraints. The analytical solution cannot be found because of the complex form of the objective function for this problem. Attempts to find the solution with stochastic algorithms take unacceptable time and the result may be far from the optimal solution. Besides, the use of universal (commercial) optimizers hides the details of step by step solution from the user, for example&nbsp;— the ambiguity of the solution as the result of data inaccuracy.<br> The present article concerns two variants of CRM problem. The authors present a new algorithm of solving the problems with the help of “General Quadratic Programming Algorithm”. The main advantage of the new algorithm is the greater performance in comparison with the other known algorithms. Its other advantage is the possibility of an ambiguity analysis. This article studies the conditions which guarantee that the first variant of problem has a unique solution, which can be found with the presented algorithm. Another algorithm for finding the approximate solution for the second variant of the problem is also considered. The method of visualization of approximate solutions set is presented. The results of experiments comparing the new algorithm with some previously known are given.

scholarly journals Approximate Solutions of Time Fractional Diffusion Wave Models

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 923 ◽  
Author(s):  
Abdul Ghafoor ◽  
Sirajul Haq ◽  
Manzoor Hussain ◽  
Poom Kumam ◽  
Muhammad Asif Jan
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Quadrature Rule ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Haar Wavelets ◽  
Test Problems ◽  
Algebraic Equations ◽  
Diffusion Wave ◽  
Wave Models

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

Sign in / Sign up

Export Citation Format

Share Document