scholarly journals Convergence analysis of time-discretisation schemes for rate-independent systems

2019 ◽  
Vol 25 ◽  
pp. 65 ◽  
Author(s):  
Dorothee Knees
Keyword(s):  
Convergence Analysis ◽  
Numerical Schemes ◽  
Solution Concepts ◽  
Continuous Solutions ◽  
Energy Functionals ◽  
Convergence Of Solutions ◽  
Time Discretisation ◽  
Energetic Solutions ◽  
Nonconvex Energy ◽  
The Given

It is well known that rate-independent systems involving nonconvex energy functionals in general do not allow for time-continuous solutions even if the given data are smooth. In the last years, several solution concepts were proposed that include discontinuities in the notion of solution, among them the class of global energetic solutions and the class of BV-solutions. In general, these solution concepts are not equivalent and numerical schemes are needed that reliably approximate that type of solutions one is interested in. In this paper, we analyse the convergence of solutions of three time-discretisation schemes, namely an approach based on local minimisation, a relaxed version of it and an alternate minimisation scheme. For all three cases, we show that under suitable conditions on the discretisation parameters discrete solutions converge to limit functions that belong to the class of BV-solutions. The proofs rely on a reparametrisation argument. We illustrate the different schemes with a toy example.

scholarly journals Diagonally scaled memoryless quasi–Newton methods with application to compressed sensing

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zohre Aminifard ◽  
Saman Babaie-Kafaki
Keyword(s):  
Compressed Sensing ◽  
Convergence Analysis ◽  
Newton Methods ◽  
The Given ◽  
Quasi Newton

<p style='text-indent:20px;'>Memoryless quasi–Newton updating formulas of BFGS (Broyden–Fletcher–Goldfarb–Shanno) and DFP (Davidon–Fletcher–Powell) are scaled using well-structured diagonal matrices. In the scaling approach, diagonal elements as well as eigenvalues of the scaled memoryless quasi–Newton updating formulas play significant roles. Convergence analysis of the given diagonally scaled quasi–Newton methods is discussed. At last, performance of the methods is numerically tested on a set of CUTEr problems as well as the compressed sensing problem.</p>

scholarly journals Convergence Analysis of Two Numerical Schemes Applied to a Nonlinear Elliptic Problem

2016 ◽  
Vol 71 (1) ◽  
pp. 329-347 ◽  
Author(s):  
Christine Bernardi ◽  
Jad Dakroub ◽  
Gihane Mansour ◽  
Farah Rafei ◽  
Toni Sayah
Keyword(s):  
Convergence Analysis ◽  
Elliptic Problem ◽  
Nonlinear Elliptic Problem ◽  
Numerical Schemes ◽  
Nonlinear Elliptic

scholarly journals Unified Convergence Analysis of Numerical Schemes for a Miscible Displacement Problem

2018 ◽  
Vol 19 (2) ◽  
pp. 333-374 ◽  
Author(s):  
Jérôme Droniou ◽  
Robert Eymard ◽  
Alain Prignet ◽  
Kyle S. Talbot
Keyword(s):  
Convergence Analysis ◽  
Miscible Displacement ◽  
Numerical Schemes ◽  
Displacement Problem

scholarly journals Fair Division Under Cardinality Constraints

2018 ◽  
Author(s):  
Arpita Biswas ◽  
Siddharth Barman
Keyword(s):  
Resource Allocation ◽  
Fair Division ◽  
Division Problem ◽  
Fair Allocation ◽  
Indivisible Goods ◽  
Solution Concepts ◽  
Cardinality Constraints ◽  
The Given

We consider the problem of fairly allocating indivisible goods, among agents, under cardinality constraints and additive valuations. In this setting, we are given a partition of the entire set of goods---i.e., the goods are categorized---and a limit is specified on the number of goods that can be allocated from each category to any agent. The objective here is to find a fair allocation in which the subset of goods assigned to any agent satisfies the given cardinality constraints. This problem naturally captures a number of resource-allocation applications, and is a generalization of the well-studied unconstrained fair division problem.  The two central notions of fairness, in the context of fair division of indivisible goods, are envy freeness up to one good (EF1) and the (approximate) maximin share guarantee (MMS). We show that the existence and algorithmic guarantees established for these solution concepts in the unconstrained setting can essentially be achieved under cardinality constraints. Furthermore, focusing on the case wherein all the agents have the same additive valuation, we establish that EF1 allocations exist even under matroid constraints.

scholarly journals Linearly constrained evolutions of critical points and an application to cohesive fractures

2017 ◽  
Vol 27 (02) ◽  
pp. 231-290 ◽  
Author(s):  
Marco Artina ◽  
Filippo Cagnetti ◽  
Massimo Fornasier ◽  
Francesco Solombrino
Keyword(s):  
Critical Points ◽  
Energy Functional ◽  
Two Dimensions ◽  
Infinite Dimensional ◽  
Energy Functionals ◽  
Finite Dimensional ◽  
Nonconvex Energy ◽  
Linearly Constrained ◽  
Cohesive Fracture Model ◽  
Crack Initiation Criterion

We introduce a novel constructive approach to define time evolution of critical points of an energy functional. Our procedure, which is different from other more established approaches based on viscosity approximations in infinite-dimension, is prone to efficient and consistent numerical implementations, and allows for an existence proof under very general assumptions. We consider in particular rather nonsmooth and nonconvex energy functionals, provided the domain of the energy is finite-dimensional. Nevertheless, in the infinite-dimensional case study of a cohesive fracture model, we prove a consistency theorem of a discrete-to-continuum limit. We show that a quasistatic evolution can be indeed recovered as a limit of evolutions of critical points of finite-dimensional discretizations of the energy, constructed according to our scheme. To illustrate the results, we provide several numerical experiments both in one- and two-dimensions. These agree with the crack initiation criterion, which states that a fracture appears only when the stress overcomes a certain threshold, depending on the material.

scholarly journals Convergence Analysis of Gradient Descent for Eigenvector Computation

2018 ◽  
Author(s):  
Zhiqiang Xu ◽  
Xin Cao ◽  
Xin Gao
Keyword(s):  
Convergence Analysis ◽  
Gradient Descent ◽  
Symmetric Matrix ◽  
Linear Convergence ◽  
Learning Problems ◽  
Principal Angle ◽  
Global Rate ◽  
Convex Problem ◽  
Initial Iterate ◽  
The Given

We present a novel, simple and systematic convergence analysis of gradient descent for eigenvector computation. As a popular, practical, and provable approach to numerous machine learning problems, gradient descent has found successful applications to eigenvector computation as well. However, surprisingly, it lacks a thorough theoretical analysis for the underlying geodesically non-convex problem. In this work, the convergence of the gradient descent solver for the leading eigenvector computation is shown to be at a global rate O(min{ (lambda_1/Delta_p)^2 log(1/epsilon), 1/epsilon }), where Delta_p=lambda_p-lambda_p+1>0 represents the generalized positive eigengap and always exists without loss of generality with lambda_i being the i-th largest eigenvalue of the given real symmetric matrix and p being the multiplicity of lambda_1. The rate is linear at (lambda_1/Delta_p)^2 log(1/epsilon) if (lambda_1/Delta_p)^2=O(1), otherwise sub-linear at O(1/epsilon). We also show that the convergence only logarithmically instead of quadratically depends on the initial iterate. Particularly, this is the first time the linear convergence for the case that the conventionally considered eigengap Delta_1= lambda_1 - lambda_2=0 but the generalized eigengap Delta_p satisfies (lambda_1/Delta_p)^2=O(1), as well as the logarithmic dependence on the initial iterate are established for the gradient descent solver. We are also the first to leverage for analysis the log principal angle between the iterate and the space of globally optimal solutions. Theoretical properties are verified in experiments.

scholarly journals Long time behavior of higher-order delay differential equation with vanishing proportional delay and its convergence analysis using spectral method

2021 ◽  
Vol 7 (4) ◽  
pp. 4946-4959
Author(s):  
Ishtiaq Ali ◽  
Keyword(s):  
Differential Equations ◽  
Convergence Analysis ◽  
Higher Order ◽  
Proportional Delay ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
Delay Differential ◽  
The Given ◽  
Past State

<abstract> <p>Delay differential equations (DDEs) are used to model some realistic systems as they provide some information about the past state of the systems in addition to the current state. These DDEs are used to analyze the long-time behavior of the system at both present and past state of such systems. Due to the oscillatory nature of DDEs their explicit solution is not possible and therefore one need to use some numerical approaches. In this article, we developed a higher-order numerical scheme for the approximate solution of higher-order functional differential equations of pantograph type with vanishing proportional delays. Some linear and functional transformations are used to change the given interval [0, T] into standard interval [-1, 1] in order to fully use the properties of orthogonal polynomials. It is assumed that the solution of the equation is smooth on the entire domain of interval of integration. The proposed scheme is employed to the equivalent integrated form of the given equation. A Legendre spectral collocation method relative to Gauss-Legendre quadrature formula is used to evaluate the integral term efficiently. A detail theoretical convergence analysis in L<sub>∞</sub> norm is provided. Several numerical experiments were performed to confirm the theoretical results.</p> </abstract>

ERROR ANALYSIS OF THE MINIMUM DISTANCE ENERGY OF A POLYGONAL KNOT AND THE MÖBIUS ENERGY OF AN APPROXIMATING CURVE

2010 ◽  
Vol 19 (08) ◽  
pp. 975-1000 ◽  
Author(s):  
ERIC J. RAWDON ◽  
JOSEPH WORTHINGTON
Keyword(s):  
Error Analysis ◽  
Upper Bound ◽  
Minimum Distance ◽  
Energy Functionals ◽  
Explicit Bound ◽  
Knot Energy ◽  
The Given

Energy minimizing smooth knot configurations have long been approximated by finding knotted polygons that minimize discretized versions of the given energy. However, for most knot energy functionals, the question remains open on whether the minimum polygonal energies are "close" to the minimum smooth energies. In this paper, we determine an explicit bound between the Minimum-Distance Energy of a polygon and the Möbius Energy of a piecewise-C2 knot inscribed in the polygon. This bound is written in terms of the ropelength and the number of edges and can be used to determine an upper bound for the minimum Möbius Energy for different knot types.

Energy Functionals and Complex Monge–Ampère Equations

2010 ◽  
Vol 9 (3) ◽  
pp. 463-476 ◽  
Author(s):  
Zuoliang Hou ◽  
Qi Li
Keyword(s):  
Boundary Conditions ◽  
Bounded Domains ◽  
Energy Functionals ◽  
Convergence Of Solutions ◽  
Monge Ampere

AbstractWe introduce certain energy functionals to complex Monge–Ampère equations over bounded domains with inhomogeneous boundary conditions, and use these functionals to show the convergence of solutions to certain parabolic Monge–Ampère equations.

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