Multigrid second-order accurate solution of parabolic control-constrained problems

2010 ◽  
Vol 51 (2) ◽  
pp. 835-866 ◽  
Author(s):  
S. González Andrade ◽  
A. Borzì
Keyword(s):  
Accurate Solution ◽  
Second Order ◽  
Constrained Problems

scholarly journals Steklov eigenvalues of reflection-symmetric nearly circular planar domains

2018 ◽  
Vol 474 (2220) ◽  
pp. 20180072 ◽  
Author(s):  
Robert Viator ◽  
Braxton Osting
Keyword(s):  
Asymptotic Estimate ◽  
Perturbation Parameter ◽  
Isoperimetric Inequalities ◽  
Second Order ◽  
Numerical Results ◽  
Domain Perturbation ◽  
Planar Domains ◽  
Constrained Problems ◽  
Area And Perimeter ◽  
Steklov Eigenvalues

We consider Steklov eigenvalues of reflection-symmetric, nearly circular, planar domains. Treating such domains as perturbations of the disc, we obtain a second-order formal asymptotic estimate in the domain perturbation parameter. We conclude with a discussion of implications for isoperimetric inequalities. Namely, our results corroborate the results of Weinstock and Brock that state, respectively, that the disc is the maximizer for the area and perimeter constrained problems. They also support the result of Hersch, Payne and Schiffer that the product of the first two eigenvalues is maximal among all open planar sets of equal perimeter. In addition, our results imply that the disc is not the maximizer of the area constrained problems for higher even numbered Steklov eigenvalues, as suggested by previous numerical results.

scholarly journals Adomian decomposition method for solving initial value problems in vibration models

2018 ◽  
Vol 159 ◽  
pp. 02007
Author(s):  
Sudi Mungkasi ◽  
I Made Wicaksana Ekaputra
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
External Source ◽  
Accurate Solution ◽  
Second Order ◽  
Initial Value ◽  
Adomian Decomposition

A number of engineering problems have second-order ordinary differential equations as their mathematical models. In practice, we may have a large scale problem with a large number of degrees of freedom, which must be solved accurately. Therefore, treating the mathematical model governing the problems correctly is required in order to get an accurate solution. In this work, we use Adomian decomposition method to solve vibration models in the forms of initial value problems of second-order ordinary differential equations. However, for problems involving an external source, the Adomian decomposition method may not lead to an accurate solution if the external source is not correctly treated. In this paper, we propose a strategy to treat the external source when we implement the Adomian decomposition method to solve initial value problems of second-order ordinary differential equations. Computational results show that our strategy is indeed effective. We obtain accurate solutions to the considered problems. Note that exact solutions are often not available, so they need to be approximated using some methods, such as the Adomian decomposition method.

scholarly journals Second-Order Optimality Conditions for Nonconvex Set-Constrained Optimization Problems

2022 ◽  
Author(s):  
Helmut Gfrerer ◽  
Jane J. Ye ◽  
Jinchuan Zhou
Keyword(s):  
Constrained Optimization ◽  
Optimality Conditions ◽  
Support Function ◽  
Optimization Problems ◽  
Second Order ◽  
Constrained Optimization Problems ◽  
Second Order Optimality Conditions ◽  
Constrained Problems ◽  
Second Order Tangent Set ◽  
Order Tangent

In this paper, we study second-order optimality conditions for nonconvex set-constrained optimization problems. For a convex set-constrained optimization problem, it is well known that second-order optimality conditions involve the support function of the second-order tangent set. In this paper, we propose two approaches for establishing second-order optimality conditions for the nonconvex case. In the first approach, we extend the concept of the support function so that it is applicable to general nonconvex set-constrained problems, whereas in the second approach, we introduce the notion of the directional regular tangent cone and apply classical results of convex duality theory. Besides the second-order optimality conditions, the novelty of our approach lies in the systematic introduction and use, respectively, of directional versions of well-known concepts from variational analysis.

scholarly journals Constructive generalization of classical sufficient second-order optimality conditions

2019 ◽  
Vol 487 (5) ◽  
pp. 493-495
Author(s):  
Yu. G. Evtushenko ◽  
A. A. Tret’yakov
Keyword(s):  
Constrained Optimization ◽  
Optimality Conditions ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Constrained Optimization Problems ◽  
Constrained Problems ◽  
Slack Variables ◽  
Equality Constrained Optimization

In this paper, we consider new sufficient conditions of optimality of the second-order for equality constrained optimization problems, which essentially enhance and complement the classical ones and are constructive. For example, they establish equivalence between sufficient conditions in the equality constrained optimization problems and sufficient conditions for optimality in equality constrained problems by reducing the latter to equalities with the help of introducing slack variables. Previously, when using the classical sufficient optimality conditions, this fact was not considered to be true, that is, the existing classical sufficient conditions were not complete, so the proposed optimality conditions complement the classical ones and close the question of the equivalence of the problems with inequalities and the problems with equalities when reducing the first to the second by introducing slack variables.

Disappearance Voltage Determinations Using a Convergent Beam

1971 ◽  
Vol 29 ◽  
pp. 184-185
Author(s):  
W. L. Bell
Keyword(s):  
Second Order ◽  
Objective Method ◽  
Uniform Thickness ◽  
Rocking Curve ◽  
Crystal Perfection ◽  
Kikuchi Line ◽  
Central Maximum ◽  
Diffraction Patterns ◽  
Convergent Beam ◽  
Beam Technique

Disappearance voltages for second order reflections can be determined experimentally in a variety of ways. The more subjective methods, such as Kikuchi line disappearance and bend contour imaging, involve comparing a series of diffraction patterns or micrographs taken at intervals throughout the disappearance range and selecting that voltage which gives the strongest disappearance effect. The estimated accuracies of these methods are both to within 10 kV, or about 2-4%, of the true disappearance voltage, which is quite sufficient for using these voltages in further calculations. However, it is the necessity of determining this information by comparisons of exposed plates rather than while operating the microscope that detracts from the immediate usefulness of these methods if there is reason to perform experiments at an unknown disappearance voltage.The convergent beam technique for determining the disappearance voltage has been found to be a highly objective method when it is applicable, i.e. when reasonable crystal perfection exists and an area of uniform thickness can be found. The criterion for determining this voltage is that the central maximum disappear from the rocking curve for the second order spot.

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