Immersed Interface Finite Element Methods for Elasticity Interface Problems with Non-Homogeneous Jump Conditions

<正>In this paper,a class of new immersed interface finite element methods (IIFEM) is developed to solve elasticity interface problems with homogeneous and non-homogeneous jump conditions in two dimensions.Simple non-body-fitted meshes are used.For homogeneous jump conditions,both non-conforming and conforming basis functions are constructed in such a way that they satisfy the natural jump conditions. For non-homogeneous jump conditions,a pair of functions that satisfy the same non-homogeneous jump conditions are constructed using a level-set representation of the interface.With such a pair of functions,the discontinuities across the interface in the solution and flux are removed;and an equivalent elasticity interface problem with homogeneous jump conditions is formulated.Numerical examples are presented to demonstrate that such methods have second order convergence.     

Battle-outcome prediction for an extended system of Lanchester-type differential equations

Battle-outcome-prediction conditions are given for an extended system of Lanchester-type differential equations for two different types of battle-termination conditions: (a) fixed-force-level-breakpoint battles, and (b) fixed-force-ratio-breakpoint battles. Necessary and sufficient conditions for predicting battle outcome are given in the former case for a fight to the finish, while sufficient conditions are given in the latter case. The former results are equivalent to those for the problem of classical analysis of determining (explicitly as a function of the initial conditions) the occurrence of a zero point for the solution to this extended system, although such results as given here have not appeared previously for nonoscillatory (in the strict sense) solutions.     

Conjugate-point conditions for variational problems with delayed argument

A variational problem with delayed argument is investigated. The existing necessary conditions are first reviewed. New results, two conjugate-point conditions, are then derived for this problem. The method of proof is similar to that used by Bliss for the classical problem. An example shows that the two conditions are not equivalent, and that the first-order necessary conditions, the strengthened Legendre conditions, and the conjugate-point conditions do not in general constitute a set of sufficient conditions for the delay problem. It is shown that a special case, referred to as the separated-integrand problem, leads to considerable simplification of the results for the general problem.     

Eine abstrakte Theorie der Kegel- und Segmentbedingungen

We develop an abstract theory of geometric regularity conditions for subsets of normed linear spaces, of the type which occurs in the theory of Sobolev spaces (cone conditions, …). The theory is to systematize the diversity of existing conditions, to replace complicated conditions by simpler ones, and to clarify the interplay between different conditions. Our main result is a very general decomposition theorem.     

Absorbing boundary conditions for the wave equation and parallel computing

Absorbing boundary conditions have been developed for various types of problems to truncate infinite domains in order to perform computations. But absorbing boundary conditions have a second, recent and important application: parallel computing. We show that absorbing boundary conditions are essential for a good performance of the Schwarz waveform relaxation algorithm applied to the wave equation. In turn this application gives the idea of introducing a layer close to the truncation boundary which leads to a new way of optimizing absorbing boundary conditions for truncating domains. We optimize the conditions in the case of straight boundaries and illustrate our analysis with numerical experiments both for truncating domains and the Schwarz waveform relaxation algorithm.

     

On strong and weak second-order necessary optimality conditions for nonlinear programming

Second-order necessary optimality conditions play an important role in optimization theory. This is explained by the fact that most numerical optimization algorithms reduce to finding stationary points satisfying first-order necessary optimality conditions. As a rule, optimization problems, especially the high dimensional ones, have a lot of stationary points so one has to use second-order necessary optimality conditions to exclude nonoptimal points. These conditions are closely related to second-order constraint qualifications, which guarantee the validity of second-order necessary optimality conditions. In this paper, strong and weak second-order necessary optimality conditions are considered and their validity proved under so-called critical regularity condition at local minimizers.     

On Optimality Conditions in Control of Elliptic Variational Inequalities

In the paper we consider optimal control of a class of strongly monotone variational inequalities, whose solution map is directionally differentiable in the control variable. This property is used to derive sharp pointwise necessary optimality conditions provided we do not impose any control or state constraints. In presence of such constraints we make use of the generalized differential calculus and derive, under a mild constraint qualification, optimality conditions in a “fuzzy” form. For strings, these conditions may serve as an intermediate step toward pointwise conditions of limiting (Mordukhovich) type and in the case of membranes they lead to a variant of Clarke stationarity conditions.     

Second-order sufficient conditions for control problems with mixed control-state constraints

References 1–4 develop second-order sufficient conditions for local minima of optimal control problems with state and control constraints. These second-order conditions tighten the gap between necessary and sufficient conditions by evaluating a positive-definiteness criterion on the tangent space of the active constraints. The purpose of this paper is twofold. First, we extend the methods in Refs. 3, 4 and include general boundary conditions. Then, we relate the approach to the two-norm approach developed in Ref. 5. A direct sufficiency criterion is based on a quadratic function that satisfies a Hamilton-Jacobi inequality. A specific form of such a function is obtained by applying the second-order sufficient conditions to a parametric optimization problem. The resulting second-order positive-definiteness conditions can be verified by solving Riccati equations.The authors wish to thank K. Malanowski for helpful discussions.     

Liberating the Subgradient Optimality Conditions from Constraint Qualifications

In convex optimization the significance of constraint qualifications is evidenced by the simple duality theory, and the elegant subgradient optimality conditions which completely characterize a minimizer. However, the constraint qualifications do not always hold even for finite dimensional optimization problems and frequently fail for infinite dimensional problems. In the present work we take a broader view of the subgradient optimality conditions by allowing them to depend on a sequence of ε-subgradients at a minimizer and then by letting them to hold in the limit. Liberating the optimality conditions in this way permits us to obtain a complete characterization of optimality without a constraint qualification. As an easy consequence of these results we obtain optimality conditions for conic convex optimization problems without a constraint qualification. We derive these conditions by applying a powerful combination of conjugate analysis and ε-subdifferential calculus. Numerical examples are discussed to illustrate the significance of the sequential conditions.     

Regular simple geodesic loops on a tetrahedron

A regular simple geodesic loop on a tetrahedron is a simple geodesic loop which does not pass through any vertex of the tetrahedron. It is evident that such loops meet each face of the tetrahedron. Among these loops, the minimal loops are those which meet each face exactly once. Necessary and sufficient conditions for the existence of minimal loops are obtained. These conditions fall naturally into two categories, conditions in the first category being called coherence conditions and conditions in the second category being called separation conditions. It is shown that for the existence of three distinct minimal loops through any point on the face of a tetrahedron it is necessary and sufficient that the tetrahedron be isosceles, which, in turn, amounts to the tetrahedron satisfying three coherence conditions. All other regular simple geodesic loops on an isosceles tetrahedron are then classified. Finally, coherence conditions for the existence of similar loops on an arbitrary tetrahedron are found.     

Second-order conditions in c1, 1 optimization with applications

Second-order necessary conditions for inequality and equality constrained C^{1, 1} optimization problems are derived. A constraint qualification condition which uses the recent generalized second-order directional derivative is employed to obtain these conditions. Various second-order sufficient conditions are given under appropriate conditions on the generalized second-order directional derivative in a neighborhood of a given point. An application of the secondorder conditions to a new class of nonsmooth C^{1, 1} optimization problems with infinitely many constraints is presented.     

Lagrange multiplier necessary conditions for global optimality for non-convex minimization over a quadratic constraint via S-lemma

In this paper, we present Lagrange multiplier necessary conditions for global optimality that apply to non-convex optimization problems beyond quadratic optimization problems subject to a single quadratic constraint. In particular, we show that our optimality conditions apply to problems where the objective function is the difference of quadratic and convex functions over a quadratic constraint, and to certain class of fractional programming problems. Our necessary conditions become necessary and sufficient conditions for global optimality for quadratic minimization subject to quadratic constraint. As an application, we also obtain global optimality conditions for a class of trust-region problems. Our approach makes use of outer-estimators, and the powerful S-lemma which has played key role in control theory and semidefinite optimization. We discuss numerical examples to illustrate the significance of our optimality conditions. The authors are grateful to the referees for their useful comments which have contributed to the final preparation of the paper.     

NOTE ON THE OSCILLATION OF THE SOLUTION FOR GENERALIZED LIENARD EQUATIONS

Ｃｏｎｓｉｄｅｒｉｎｇｓｙｓｔｅｍ：ｘ＝φ（ｙ）－Ｆ（ｘ），ｙ＝－ｇ（ｘ）｛（Ｅ）ｉｎｗｈｉｃｈＦ（ｘ）＝∫ｘ０ｆ（ｔ）ｄｔ，ｆ（ｘ），ｇ（ｘ），φ（ｙ）ａｒｅｃｏｎｔｉｎｕｏｕｓ，ａｎｄｓａｔｉｓｆｙｔｈｅｃｏｎｄｉｔｉｏｎｓｗｈｉｃｈｅｎｓｕｒ...     

High-order splitting schemes for semilinear evolution equations

We first derive necessary and sufficient stiff order conditions, up to order four, for exponential splitting schemes applied to semilinear evolution equations. The main idea is to identify the local splitting error as a sum of quadrature errors. The order conditions of the quadrature rules then yield the stiff order conditions in an explicit fashion, similarly to that of Runge–Kutta schemes. Furthermore, the derived stiff conditions coincide with the classical non-stiff conditions. Secondly, we propose an abstract convergence analysis, where the linear part of the vector field is assumed to generate a group or a semigroup and the nonlinear part is assumed to be smooth and to satisfy a set of compatibility requirements. Concrete applications include nonlinear wave equations and diffusion-reaction processes. The convergence analysis also extends to the case where the nonlinear flows in the exponential splitting scheme are approximated by a sufficiently accurate one-step method.     

The use of possibility theory in the definition of fuzzy Pareto-optimality

Pareto-optimality conditions are crucial when dealing with classic multi-objective optimization problems. Extensions of these conditions to the fuzzy domain have been discussed and addressed in recent literature. This work presents a novel approach based on the definition of a fuzzily ordered set with a view to generating the necessary conditions for the Pareto-optimality of candidate solutions in the fuzzy domain. Making use of the conditions generated, one can characterize fuzzy efficient solutions by means of carefully chosen mono-objective problems and Karush-Kuhn-Tucker conditions to fuzzy non-dominated solutions. The uncertainties are inserted into the formulation of the studied fuzzy multi-objective optimization problem by means of fuzzy coefficients in the objective function. Some numerical examples are analytically solved to illustrate the efficiency of the proposed approach.     

A method of convolution of Fourier expansions as applied to solving boundary value problems with intersecting interface lines

An efficient method of construction of solutions to a set of boundary value problems with additional interface conditions, more complicated boundary conditions, and so on on the basis of known solutions to classical boundary value problems is proposed. The method is based on the representation of solutions to classical and more complicated problems in the form of expansions into Fourier series with subsequent reduction of one series to the other. As a result, formulas directly expressing solutions to more complicated problems in terms of solutions to classical problems are obtained. On the basis of the well-known solution to the Dirichlet problem on a half plane, solutions to boundary value problems with interface conditions (including generalized conditions of the type of a crack and a screen) on intersecting straight lines for boundary conditions of the first and the third kind are obtained.     

On the equivalence of two conditions for the existence of transversals

There are two conditions which are known to be necessary for the existence of a transversal in any family of sets, but both are sufficient only if the family is countable. This paper proves that these conditions are always equivalent to each other. The families which are compatible with these conditions are characterised, in the sense that each of their subfamilies possesses a transversal if it satisfies the conditions. Using this, a conjecture of Podewski and Steffens is proved.     

Solution continuity in variational conditions

We present some results about Lipschitzian behavior of solutions to variational conditions when the sets over which the conditions are posed, as well as the functions appearing in them, may vary. These results rely on calmness and inner semicontinuity, and we describe some conditions under which those conditions hold, especially when the sets involved in the variational conditions are convex and polyhedral. We then apply the results to find error bounds for solutions of a strongly monotone variational inequality in which both the constraining polyhedral multifunction and the monotone operator are perturbed.      

Investigation of regularity conditions in optimal control problems with geometric mixed constraints

We investigate regularity conditions in optimal control problems with mixed constraints of a general geometric type, in which a closed non-convex constraint set appears. A closely related question to this issue concerns the derivation of necessary optimality conditions under some regularity conditions on the constraints. By imposing strong and weak regularity condition on the constraints, we provide necessary optimality conditions in the form of Pontryagin maximum principle for the control problem with mixed constraints. The optimality conditions obtained here turn out to be more general than earlier results even in the case when the constraint set is convex. The proofs of our main results are based on a series of technical lemmas which are gathered in the Appendix.     

On state dependent non-local conditions

We introduce a new type of non-local conditions, which we call state dependent non-local conditions, and we study existence and uniqueness of solutions for abstract differential equation subjected to this class of conditions. The non-local condition proposed generalizes several types of non-local conditions studied in the literature. Some examples are given to illustrate our theory.