composite finite elements for the approximation of pdes on domains with complicated micro structures usually the minimal dimension of a finite element space is closely related to the geometry of the physical object of interest this means that sometimes the resolution of small micro structures in the domain requires an inadequately fine finite element grid from the viewpoint of the desired accuracy this fact limits also the application of multi grid methods to practical situations because the condition that the coarsest grid should resolve the physical object often leads to a huge number of unknowns on the coarsest level we present here a strategy for coarsening finite element spaces independently of the shape of the object this technique can be used to resolve complicated domains with only few degrees of freedom and to apply multi grid methods efficiently to pdes on domains with complex boundary in this paper we will prove the approximation property of these generalized fe spaces

optimality conditions for simultaneous topology and shape optimization new optimality conditions are derived for a class of shape optimization problems the conditions are established on the boundary by an application of the boundary variations technique and in the interior of an optimal domain by exploiting the topological derivative method an example is provided for which the classical second order sufficient optimality conditions are verified for an optimal simply connected domain however the value of the cost can be improved by the topology variations and therefore the optimal solution can be substantially changed by applying the topology optimization

a new algorithm for topology optimization using a level set method the level set method has been recently introduced in the field of shape optimization enabling a smooth representation of the boundaries on a fixed mesh and therefore leading to fast numerical algorithms however most of these algorithms use a hamilton jacobi equation to connect the evolution of the level set function with the deformation of the contours and consequently they can hardly create new holes in the domain at least in 2d in this work we propose an evolution equation for the level set function based on a generalization of the concept of topological gradient this results in a new algorithm allowing for all kinds of topology changes

on the hill determinant method the conditions of applicability of the hill determinant method to the potentials v r v1r v2r2 and v x x2 lambda x2 1 gx2 are investigated it is shown that the procedure leads to the actual eigenvalues in the former case provided the ansatz is properly chosen but it completely fails to yield the right answer in the latter one

exact solutions for anharmonic oscillators completes the author s investigation see phys lett a vol 81 p 17 8 1981 and presents exact solutions and eigenvalues for anharmonic oscillators the solutions being given in the form of definite integrals the conditions for the validity of these results are investigated

perturbation theory around two dimensional critical systems through holomorphic decomposition the authors consider perturbations by relevant interactions of two dimensional conformally invariant rational field theories the meromorphic structure with respect to the scaling dimensions of the perturbing interactions of the correlation functions is made explicit through a successive application of stoke s theorem the resulting decomposition of the amplitudes into holomorphic and antiholomorphic factors yields a representation of the meromorphic structure in terms of the basic data of the rational field theory which are scaling dimensions operator product coefficients and braid matrices they exemplify the general deduction by a concrete calculation to the third order in the coupling constant of the perturbing interaction

on the use of consistent approximations in boundary element based shape optimization in the presence of uncertainty this work addresses certain aspects of shape optimization for linearly elastic systems in the presence of uncertainty the goal is to formulate and test a set of computationally tractable methods for incorporating uncertainty in the system the boundary element method is employed in solving the underlying elasticity equations the resulting minimax problem is studied in the context of the so called consistent approximations which allow for adaptive control of the discretization error during the iterative optimization process

polyhedral risk measures in stochastic programming we consider stochastic programs with risk measures in the objective and study stability properties as well as decomposition structures thereby we place emphasis on dynamic models i e multistage stochastic programs with multiperiod risk measures in this context we define the class of polyhedral risk measures such that stochastic programs with risk measures taken from this class have favorable properties polyhedral risk measures are defined as optimal values of certain linear stochastic programs where the arguments of the risk measure appear on the right hand side of the dynamic constraints dual representations for polyhedral risk measures are derived and used to deduce criteria for convexity and coherence as examples of polyhedral risk measures we propose multiperiod extensions of the conditional value at risk

risk averse shape optimization risk averse optimization has attracted much attention in finite dimensional stochastic programming in this paper we propose a risk averse approach in the infinite dimensional context of shape optimization we consider elastic materials under stochastic loading as measures of risk awareness we investigate the expected excess and the excess probability the developed numerical algorithm is based on a regularized gradient flow acting on an implicit description of the shapes based on level sets we incorporate topological derivatives to allow for topological changes in the shape optimization procedure numerical results in two dimensions demonstrate the impact of the risk averse modeling on the optimal shapes and on the cost distribution over the set of scenarios

energy levels of a three dimensional anharmonic oscillator with sextic perturbation renormalized series version of inner product and renormalized series techniques with hill determinant approach are used to calculate the energy eigenvalues for a three dimensional oscillator for several sets of parameters our techniques were modified to treat higher power of perturbation for two eigenstates e0 0 1 and e1 1 0 with odd parity

incorporating topological derivatives into level set methods the aim of this paper is to investigate the use of topological derivatives in combination with the level set method for shape reconstruction and optimization problems we propose a new approach generalizing the standard speed method which is obtained by using a source term in the level set equation that depends on the topological derivative of the objective functional the resulting approach can be interpreted as a generalized fixed point iteration for the optimality system with respect to topological and shape variations moreover we apply the new approach for a simple model problem in shape reconstruction where the topological derivative can be computed without additional effort finally we present numerical tests related to this model problem which demonstrate that the new method based on shape and topological derivative successfully reconstructs obstacles in situations where the standard level set approach fails

comments on a paper concerning the analytic continued fraction technique for bound states the author shows that the treatment of bound and confined states for a class of confinement potentials by application of the analytic continued fraction method as presented in a recent paper dalta et al ibid vol 13 p 3161 1980 is erroneous and may lead to wrong results for the eigenvalues of the relevant schrodinger equation

entropy of conformal perturbation defects we consider perturbation defects obtained by perturbing a 2d conformal field theory cft by a relevant operator on a half plane if the perturbed bulk theory flows to an infrared fixed point described by another cft the defect flows to a conformal defect between the ultraviolet and infrared fixed point cfts for short bulk renormalization group flows connecting two fixed points which are close in theory space we find a universal perturbative formula for the boundary entropy of the corresponding conformal perturbation defect we compare the value of the boundary entropy that our formula gives for the flows between nearby virasoro minimal models with the boundary entropy of the defect constructed by gaiotto 2012 j high energy phys jhep12 2012 103 and find a match at the first two orders in the expansion

on the applicability of the hill determinant and the analytic continued fraction method to anharmonic oscillators the authors finish a line of argument started in an earlier paper and show that the application of the hill determinant and the associated analytic continued fraction method for the calculation of eigenvalues of anharmonic oscillators is of dubious validity and may thus lead to erroneous results in this context some theorems concerning the eigenvalues of the lambda x4 and the doubly anharmonic oscillators are proved

on the applicability of the hill determinant method the authors show that contrarily to what flessas and anagnostatos claim 1982 the so called hill determinant method is perfectly suited to the numerical resolution of schrodinger s equation

hill determinants and 1 n theory a computationally efficient form of the hill determinant method is devised giving both e and x2 values and is used to test the accuracy of that method in the light of recent criticisms l 1 and n 1 asymptotic theories are used to check the numerical results

on the topological derivative in shape optimization in this paper the topological derivative for an arbitrary shape functional is defined examples are provided for elliptic equations and the elasticity system in the plane the topological derivative can be used for solving shape optimization problems in structural mechanics

structural optimization using sensitivity analysis and a level set method in the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level set method for front propagation we implement this method in two and three space dimensions for a model of linear or nonlinear elasticity we consider various objective functions with weight and perimeter constraints the shape derivative is computed by an adjoint method the cost of our numerical algorithm is moderate since the shape is captured on a fixed eulerian mesh although this method is not specifically designed for topology optimization it can easily handle topology changes however the resulting optimal shape is strongly dependent on the initial guess

a level set method for topology optimization of heat conduction problem under multiple load cases in this paper we present a numerical approach of topology optimization under multiple load cases for heat conduction problem this framework is based on the theories of topological derivative and shape derivative for elliptic system we employ level set model to implicitly represent geometric boundary of thermal conductive material introducing topological derivative will generate new topology in the design domain which suppresses the dependence of initial topology guess to some extent the shape optimization is obtained by combining shape derivative with level set method the functional of quadratic temperature gradient is taken as the objective function in our analysis which is subjected to the state equation of steady heat conduction and volume constrain the shape of material domain is treated as the design variable and the final result is achieved by updating level set function gradually we develop an effective numerical technique to implement the optimal design with multiple load cases for heat conduction problem numerical examples demonstrate that our proposed approach is effective and robust for topology optimization of heat conduction problem

robust optimization methodology and applications robust optimization ro is a modeling methodology combined with computational tools to process optimization problems in which the data are uncertain and is only known to belong to some uncertainty set the paper surveys the main results of ro as applied to uncertain linear conic quadratic and semidefinite programming for these cases computationally tractable robust counterparts of uncertain problems are explicitly obtained or good approximations of these counterparts are proposed making ro a useful tool for real world applications we discuss some of these applications specifically antenna design truss topology design and stability analysis synthesis in uncertain dynamic systems we also describe a case study of 90 lps from the netlib collection the study reveals that the feasibility properties of the usual solutions of real world lps can be severely affected by small perturbations of the data and that the ro methodology can be successfully used to overcome this phenomenon

dual stochastic dominance and related mean risk models we consider the problem of constructing mean risk models which are consistent with the second degree stochastic dominance relation by exploiting duality relations of convex analysis we develop the quantile model of stochastic dominance for general distributions this allows us to show that several models using quantiles and tail characteristics of the distribution are in harmony with the stochastic dominance relation we also provide stochastic linear programming formulations of these models

risk aversion via excess probabilities in stochastic programs with mixed integer recourse we consider linear two stage stochastic programs with mixed integer recourse instead of basing the selection of an optimal first stage solution on expected costs alone we include into the objective a risk term reflecting the probability that a preselected cost threshold is exceeded after we have put the resulting mean risk model into perspective with stochastic dominance we study further structural properties of the model and derive some basic stability results in the algorithmic part of the paper we propose a scenario decomposition method and report initial computational experience

elementary bound states for the power law potentials the author shows how the exact polynomial exp polynomial bound state solutions to the radial schrodinger equation may be constructed for any potential of the form v r sigma delta in d gdelta rdelta where d is a finite set of arbitrary rational numbers and some of the couplings gdelta are not independent

shape optimization under uncertainty a stochastic programming perspective we present an algorithm for shape optimization under stochastic loading and representative numerical results our strategy builds upon a combination of techniques from two stage stochastic programming and level set based shape optimization in particular usage of linear elasticity and quadratic objective functions permits us to obtain a computational cost which scales linearly in the number of linearly independent applied forces which often is much smaller than the number of different realizations of the stochastic forces numerical computations are performed using a level set method with composite finite elements both in two and in three spatial dimensions

conformal perturbation theory beyond the leading order higher order conformal perturbation theory is studied for theories with and without boundaries we identify systematically the universal quantities in the beta function equations and we give explicit formulae for the universal coefficients at next to leading order in terms of integrated correlation functions as an example we analyse the radius dependence of the conformal dimension of some boundary operators for the case of a single neumann brane on a circle and for an intersecting brane configuration on a torus reproducing in both cases the expected geometrical answer