scholarly journals Modifications of the Charged Balls Method

2020 ◽  
Vol 10 (1) ◽  
pp. 30-32
Author(s):  
Majid Abbasov ◽  
Faramoz Aliev
Keyword(s):  
Computational Geometry ◽  
Orthogonal Projection ◽  
Numerical Experiments ◽  
Minimum Distance ◽  
Smooth Boundary ◽  
Optimization Methods ◽  
Closed Set ◽  
Convex Closed Set ◽  
Charged Ball

AbstractThe Charged Balls Method is based on physical ideas. It allows one to solve problem of finding the minimum distance from a point to a convex closed set with a smooth boundary, finding the minimum distance between two such sets and other problems of computational geometry. This paper proposes several new quick modifications of the method. These modifications are compared with the original Charged Ball Method as well as other optimization methods on a large number of randomly generated model problems.We consider the problem of orthogonal projection of the origin onto an ellipsoid. The main aim is to illustrate the results of numerical experiments of Charged Balls Method and its modifications in comparison with other classical and special methods for the studied problem.

A Maxmin Distance Problem

1972 ◽  
Vol 94 (2) ◽  
pp. 155-158 ◽  
Author(s):  
R. Aggarwal ◽  
G. Leitmann
Keyword(s):  
Minimum Distance ◽  
Terminal State ◽  
Initial State ◽  
Closed Set ◽  
Distance Problem

The problem of maximizing the minimum distance of a dynamical system’s state from a given closed set, while transferring the system from a given initial state to a given terminal state, is considered. Two different methods of solution of this problem are given.

scholarly journals Smoothing spline in a convex closed set of Hilbert space

2002 ◽  
Vol 66 (3) ◽  
pp. 359-368
Author(s):  
Natasha Dicheva
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Linear Operators ◽  
Smoothing Spline ◽  
Dual Cone ◽  
Closed Set ◽  
Convex Closed Set ◽  
Operator Trace

A characterisation of a smoothing spline is sought in a convex closed set C of Hilbert space: , T and A are linear operators. A representation of the solution is obtained in the terms of the kernels of the above operators, of the dual operators T*, A* and of the dual cone C0. A particular case is considered when T is the differential operator and A is the operator-trace of a function.

Accuracy estimates of regularization methods and conditional well-posedness of nonlinear optimization problems

2018 ◽  
Vol 26 (6) ◽  
pp. 789-797
Author(s):  
Mikhail Y. Kokurin
Keyword(s):  
Optimization Problems ◽  
Lower Semicontinuous ◽  
Well Posedness ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Closed Set ◽  
Nonlinear Minimization ◽  
Convex Closed Set ◽  
Ill Posed ◽  
Necessary And Sufficient

Abstract We investigate the nonlinear minimization problem on a convex closed set in a Hilbert space. It is shown that the uniform conditional well-posedness of a class of problems with weakly lower semicontinuous functionals is the necessary and sufficient condition for existence of regularization procedures with accuracy estimates uniform on this class. We also establish a necessary and sufficient condition for the existence of regularizing operators which do not use information on the error level in input data. Similar results were previously known for regularization procedures of solving ill-posed inverse problems.

scholarly journals Numerical methods preserving multiple Hamiltonians for stochastic Poisson systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lijin Wang ◽  
Pengjun Wang ◽  
Yanzhao Cao
Keyword(s):  
Vector Field ◽  
Numerical Methods ◽  
Orthogonal Projection ◽  
Numerical Experiments ◽  
Numerical Schemes ◽  
Projection Technique ◽  
Discrete Gradient ◽  
Casimir Functions ◽  
Convergence Orders ◽  
Theoretical Results

<p style='text-indent:20px;'>In this paper, we propose a class of numerical schemes for stochastic Poisson systems with multiple invariant Hamiltonians. The method is based on the average vector field discrete gradient and an orthogonal projection technique. The proposed schemes preserve all the invariant Hamiltonians of the stochastic Poisson systems simultaneously, with possibility of achieving high convergence orders in the meantime. We also prove that our numerical schemes preserve the Casimir functions of the systems under certain conditions. Numerical experiments verify the theoretical results and illustrate the effectiveness of our schemes.</p>

scholarly journals Correspondence between Topological and Discrete Connectivities in Hausdorff Discretization

2019 ◽  
Vol 3 (1) ◽  
pp. 1-28
Author(s):  
Christian Ronse ◽  
Loic Mazo ◽  
Mohamed Tajine
Keyword(s):  
Lower Bound ◽  
Metric Space ◽  
Hausdorff Distance ◽  
Closed Subset ◽  
Minimum Distance ◽  
Topological Properties ◽  
Closed Set ◽  
Adjacency Graph ◽  
Discrete Subspace ◽  
Partial Connection

Abstract We consider Hausdorff discretization from a metric space E to a discrete subspace D, which associates to a closed subset F of E any subset S of D minimizing the Hausdorff distance between F and S; this minimum distance, called the Hausdorff radius of F and written rH(F), is bounded by the resolution of D. We call a closed set F separated if it can be partitioned into two non-empty closed subsets F1 and F2 whose mutual distances have a strictly positive lower bound. Assuming some minimal topological properties of E and D (satisfied in ℝn and ℤn), we show that given a non-separated closed subset F of E, for any r > rH(F), every Hausdorff discretization of F is connected for the graph with edges linking pairs of points of D at distance at most 2r. When F is connected, this holds for r = rH(F), and its greatest Hausdorff discretization belongs to the partial connection generated by the traces on D of the balls of radius rH(F). However, when the closed set F is separated, the Hausdorff discretizations are disconnected whenever the resolution of D is small enough. In the particular case where E = ℝn and D = ℤn with norm-based distances, we generalize our previous results for n = 2. For a norm invariant under changes of signs of coordinates, the greatest Hausdorff discretization of a connected closed set is axially connected. For the so-called coordinate-homogeneous norms, which include the Lp norms, we give an adjacency graph for which all Hausdorff discretizations of a connected closed set are connected.

Implicitly Restarted Refined Partially Orthogonal Projection Method with Deflation

2017 ◽  
Vol 7 (1) ◽  
pp. 1-20
Author(s):  
Wei Wei ◽  
Hua Dai
Keyword(s):  
Eigenvalue Problem ◽  
Projection Method ◽  
Orthogonal Projection ◽  
Numerical Experiments ◽  
Large Scale ◽  
Low Rank ◽  
Polynomial Eigenvalue Problem ◽  
Structures And Properties ◽  
Implicit Restarting

AbstractIn this paper we consider the computation of some eigenpairs with smallest eigenvalues in modulus of large-scale polynomial eigenvalue problem. Recently, a partially orthogonal projection method and its refinement scheme were presented for solving the polynomial eigenvalue problem. The methods preserve the structures and properties of the original polynomial eigenvalue problem. Implicitly updating the starting vector and constructing better projection subspace, we develop an implicitly restarted version of the partially orthogonal projection method. Combining the implicit restarting strategy with the refinement scheme, we present an implicitly restarted refined partially orthogonal projection method. In order to avoid the situation that the converged eigenvalues converge repeatedly in the later iterations, we propose a novel explicit non-equivalence low-rank deflation technique. Finally some numerical experiments show that the implicitly restarted refined partially orthogonal projection method with the explicit non-equivalence low-rank deflation technique is efficient and robust.

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