the convergence of spline collocation for strongly elliptic equations on curves most boundary element methods for two dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order the equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines the crucial assumption for the generalized boundary integral and integro differential operators is strong ellipticity our analysis is based on simple fourier expansions in particular we extend results by j saranen and w l wendland from constant to variable coefficient equations our results include the first convergence proof of midpoint collocation with piecewise constant functions i e the panel method for solving systems of cauchy singular integral equations

a wavelet algorithm for the boundary element solution of a geodetic boundary value problem abstract in this paper we consider a piecewise bilinear collocation method for the solution of a singular integral equation over a part of the surface of the earth this singular equation is the boundary integral equation corresponding to the oblique derivative boundary problem for laplace s equation we introduce special wavelet bases for the spaces of test and trial functions analogously to well known results on wavelet algorithms the stiffness matrices with respect to these bases can be reduced to sparse matrices such that the assembling of the matrices and the iterative solution of the matrix equations quicken though the theoretical results apply only to integral equations with smooth solutions over smooth manifolds we present numerical tests for a geometry as difficult as the surface of the earth

multiwavelets for second kind integral equations we consider a galerkin method for an elliptic pseudodifferential operator of order zero on a two dimensional manifold we use piecewise linear discontinuous trial functions on a triangular mesh and describe an orthonormal wavelet basis using this basis we can compress the stiffness matrix from n2 to o n log n nonzero entries and still obtain up to log n terms the same convergence rates as for the exact galerkin method

two level linear programming decentralized planning has long been recognized as an important decision making problem many approaches based on the concepts of large scale system decomposition have generally lacked the ability to model the type of truly independent subsystems which often exist in practice multilevel programming models partition control over decision variables among ordered levels within a hierarchical planning structure a planner at one level of the hierarchy may have his objective function and set of feasible decisions determined in part by other levels however his control instruments may allow him to influence the policies at other levels and thereby improve his own objective function this paper examines the special case of the two level linear programming problem geometric characterizations and algorithms are presented with some examples the goal is to demonstrate the tractability of such problems and motivate a wider interest in their study

efficient inversion of the galerkin matrix of general second order elliptic operators with nonsmooth coefficients this article deals with the efficient approximate inversion of finite element stiffness matrices of general second order elliptic operators with l coefficients it will be shown that the inverse stiffness matrix can be approximated by hierarchical matrices h matrices furthermore numerical results will demonstrate that it is possible to compute an approximate inverse with almost linear complexity

trigonometric hermite wavelet approximation for the integral equations of second kind with weakly singular kernel this paper is concerned with a trigonometric hermite wavelet galerkin method for the fredholm integral equations with weakly singular kernel the kernel function of this integral equation considered here includes two parts a weakly singular kernel part and a smooth kernel part the approximation estimates for the weakly singular kernel function and the smooth part based on the trigonometric hermite wavelet constructed by e quak trigonometric wavelets for hermite interpolation math comp 65 1996 683 722 are developed the use of trigonometric hermite interpolant wavelets for the discretization leads to a circulant block diagonal symmetrical system matrix it is shown that we only need to compute and store o n entries for the weakly singular kernel representation matrix with dimensions n 2 which can reduce the whole computational cost and storage expense the computational schemes of the resulting matrix elements are provided for the weakly singular kernel function furthermore the convergence analysis is developed for the trigonometric wavelet method in this paper

spline collocation for strongly elliptic equations on the torus we prove convergence and error estimates in sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus shaped domains in three or more dimensions for odd degree splines we prove convergence of nodal collocation for any strongly elliptic operator for even degree splines and midpoint collocation we find an additional condition for the convergence which is satisfied for the classical boundary integral operators our analysis is a generalization to higher dimensions of the corresponding analysis of arnold and wendland 4

approximation of integral operators by green quadrature and nested cross approximation we present a fast algorithm that constructs a data sparse approximation of matrices arising in the context of integral equation methods for elliptic partial differential equations the new algorithm uses green s representation formula in combination with quadrature to obtain a first approximation of the kernel function and then applies nested cross approximation to obtain a more efficient representation the resulting mathcal h 2 h2 matrix representation requires mathcal o n k o nk units of storage for an n times n n n matrix where k depends on the prescribed accuracy

hierarchical optimization a satisfactory solution abstract hierarchical optimization or multi level programming techniques are extensions of stackelberg games for solving decentralized planning problems with multiple decision makers in a hierarchical organization they become more important for contemporary decentralized organizations where each unit or department seeks its own interests traditional approaches include vertex enumeration algorithms and approaches based on kuhn tucker conditions or penalty functions these are not only technically inefficient but also lead to a paradox that the follower s decision power dominates the leader s in this study concepts of memberships of optimalities and degrees of decision powers are proposed to solve the above problems efficiently in the proposed hierarchy decision process the leader first sets memberships of optimalities of his her possible objective values and decisions as well as his her decision power and then asks the follower for his her optima calculated in isolation under given constraints the follower s decision with the corresponding levels of optimalities and decision powers are submitted to and modified by the leader with considerations of overall benefit for the organization and distribution of decision power until a best preferred solution is reached

galerkin analysis for schrodinger equation by wavelets we consider the perturbed schrodinger equation which is an elliptic operator with unbounded coefficients we use wavelets adapted to the schrodinger operator to deal with problems on the unbounded domain the wavelets are constructed from hermite functions which characterizes the space generated by the schrodinger operator we show that the galerkin matrix can be pre conditioned by a diagonal matrix so that its condition number is uniformly bounded moreover we introduce a periodic pseudo differential operator and show that its discrete galerkin matrix under periodic wavelet system is equal to the galerkin matrix for the equation with unbounded coefficients under the hermite system the convergence is proved in the l2 topology

a penalty function method for solving inverse optimal value problem in order to consider the inverse optimal value problem under more general conditions we transform the inverse optimal value problem into a corresponding nonlinear bilevel programming problem equivalently using the kuhn tucker optimality condition of the lower level problem we transform the nonlinear bilevel programming into a normal nonlinear programming the complementary and slackness condition of the lower level problem is appended to the upper level objective with a penalty then we give via an exact penalty method an existence theorem of solutions and propose an algorithm for the inverse optimal value problem also analysis the convergence of the proposed algorithm the numerical result shows that the algorithm can solve a wider class of inverse optimal value problem

patterns and localized structures in a hybrid non equilibrium ising model we study a hybrid non equilibrium pattern formation model combining short scale ising model with a continuous slow or long scale inhibitor the computation combines alternates monte carlo algorithm with updating of the inhibitor field it is very fast and allows us to study the influence of various factors such as scale ratios coupling strength bias temperature level of noise anisotropy etc on the pattern formation and behavior of emerging non equilibrium structures

estimation of distribution algorithm for a class of nonlinear bilevel programming problems in this paper a novel evolutionary algorithm called estimation of distribution algorithm eda is proposed for solving a special class of nonlinear bilevel programming problems blpps in which the lower level problem is a convex programming problem for each given upper level decision this special type of blpp is transformed into a equivalent single level constrained optimization problem using the karush kuhn er conditions of the lower level problem then we propose an eda based on the statistical information of the superior candidate solutions to solve the transformed problem we stress that the new population of individuals is sampled from the probabilistic distribution of those superior solutions thus one of the main advantages of eda over most other meta heuristics is its ability to adapt the operators to the structure of the problem although adaptation in eda is usually limited by the initial choice of the probabilistic model in addition two specific rules are established in the initialization procedure to make use of the hierarchical structure of blpps and to handle the constraints moreover without requiring the differentiability of the objective function or the convexity of the search space of the equivalent problem the proposed algorithm can address nonlinear blpps with non differentiable or non convex upper level objective function and upper level constraint functions finally the proposed algorithm has been applied to 16 benchmark problem in five of these problems all of the upper level variables and lower level variables are 10 dimensional the numerical results compared with those of other methods reveal the feasibility and effectiveness of the proposed algorithm

solving quadratic convex bilevel programming problems using a smoothing method abstract in this paper we present a smoothing sequential quadratic programming to compute a solution of a quadratic convex bilevel programming problem we use the karush kuhn tucker optimality conditions of the lower level problem to obtain a nonsmooth optimization problem known to be a mathematical program with equilibrium constraints the complementary conditions of the lower level problem are then appended to the upper level objective function with a classical penalty these complementarity conditions are not relaxed from the constraints and they are reformulated as a system of smooth equations by mean of semismooth equations using fisher burmeister functional then using a quadratic sequential programming method we solve a series of smooth regular problems that progressively approximate the nonsmooth problem some preliminary computational results are reported showing that our approach is efficient

production planning problem with sequence dependent setups as a bilevel programming problem abstract each of n products is to be processed on two machines in order to satisfy known demands in each of t periods only one product can be processed on each machine at any given time each switch from one item to another requires sequence dependent setup time the objective is to minimize the total setup time and the sum of the costs of production storage and setup we consider the problem as a bilevel mixed 0 1 integer programming problem the objective of the leader is to assign the products to the machines in order to minimize the total sequence dependent setup time while the objective of the follower is to minimize the production storage and setup cost of the machine we develop a heuristics based on tabu search for solving the problem at the end some computational results are presented

general theory of instabilities for patterns with sharp interfaces in reaction diffusion systems an asymptotic method for finding instabilities of arbitrary d dimensional large amplitude patterns in a wide class of reaction diffusion systems is presented the complete stability analysis of two and three dimensional localized patterns is carried out it is shown that in the considered class of systems the criteria for different types of instabilities are universal the specific nonlinearities enter the criteria only via three numerical constants of order 1 the analysis performed explains the self organization scenarios observed in the recent experiments and numerical simulations of some concrete reaction diffusion systems textcopyright 1996 the american physical society

fuzzy goal programming approach to multilevel programming problems the purpose of this paper is to propose a procedure for solving multilevel programming problems in a large hierarchical decentralized organization through linear fuzzy goal programming approach here the tolerance membership functions for the fuzzily described objectives of all levels as well as the control vectors of the higher level decision makers are defined by determining individual optimal solution of each of the level decision makers since the objectives are potentially conflicting in nature a possible relaxation of the higher level decision is considered for avoiding decision deadlock then fuzzy goal programming approach is used for achieving highest degree of each of the membership goals by minimizing negative deviational variables sensitivity analysis with variation of tolerance values on decision vectors is performed to present how the solution is sensitive to the change of tolerance values the efficiency of our concept is ascertained by comparing results with other fuzzy programming approaches

a penalty function method for solving weak price control problem abstract the price control problem is an important linear bilevel programming problem in this paper we are concerned with a class of weak price control problems with non unique lower level solutions for such problems we study the existence of solution via a penalty method then we propose a simple algorithm for this problem and present the convergence of the algorithm the example illustrates that the method is feasible and efficient

an efficient point algorithm for a linear two stage optimization problem this paper presents an algorithm using sensitivity analysis to solve a linear two stage optimization problem the underlying theory rests on a set of first order optimality conditions that parallel the kuhn tucker conditions associated with a one dimensional parametric linear program the solution to the original problem is uncovered by systematically varying the parameter over the unit interval and solving the corresponding linear program finite convergence is established under nondegenerate assumptions the paper also discusses other solution techniques including branch and bound and vertex enumeration and gives an example highlighting their computational and storage requirements by these measures the algorithm presented here has an overall advantage finally a comparison is drawn between bicriteria and bilevel programming and underscored by way of an example

traveling spike autosolitons in the gray scott model abstract we developed singular perturbation techniques based on the strong separation of time and length scales to construct the solutions in the form of the traveling spike autosolitons self sustained solitary waves in the gray scott model of an autocatalytic reaction we found that when the inhibitor diffusion is sufficiently slow the ultrafast traveling spike autosolitons are realized in a wide range of parameters when the diffusion of the inhibitor is sufficiently fast the slower traveling spike autosolitons with the diffusion precursor are realized we asymptotically calculated the main parameters such as speed and amplitude of these autosolitons as well as the regions of their existence in the gray scott model we also showed that in certain parameter regions the traveling spike autosolitons coexist with static

on a new class of bilevel programming problems and its use for reformulating mixed integer problems abstract we extend some known results about the bilevel linear problem blp a hierarchical two stage optimization problem showing how it can be used to reformulate any mixed integer linear problem then we introduce some new concepts which might be useful to fasten almost all the known algorithms devised for blp as this kind of reformulation appears to be somewhat artificial we define a natural generalization of blp the bilevel linear quadratic problem bl qp and show that most of the exact and or approximate algorithms originally devised for the blp such as gsa or k th best can be extended to this new class of bilevel programming problems for bl qp more natural reformulations of mips are available leading to the use of known nonexact algorithms for blp as heuristic approaches to mips we report some contrasting results obtained in the network design problem case showing that although the direct application of our dual gsa algorithm is not of any practical use we obtain as a by product a good theoretical characterization of the optimal solutions set of the ndp along with a powerful scheme for constructing fast algorithms for the minimum cost flow problem with piecewise convex linear cost functions

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

interactive fuzzy programming for multi level 0 1 programming problems through genetic algorithms abstract in this paper we propose interactive fuzzy programming for multi level 0 1 programming problems through genetic algorithms our method is supposed to apply to hierarchical decision problems in which decision making at each level is sequential from upper to lower level and decision makers are essentially cooperative after determining the fuzzy goals of the decision makers at all levels a satisfactory solution is derived efficiently by updating the satisfactory degrees of the decision makers at the upper level with considerations of overall satisfactory balance among all levels an illustrative numerical example for three level 0 1 programming problems is provided to demonstrate the feasibility of the proposed method

computational difficulties of bilevel linear programming we show using small examples that two algorithms previously published for the bilevel linear programming problem blp may fail to find the optimal solution and thus must be considered to be heuristics a proof is given that solving blp problems is np hard which makes it unlikely that there is a good exact algorithm

interactive fuzzy programming for multilevel linear programming problems this paper presents interactive fuzzy programming for multilevel linear programming problems in fuzzy programming for multilevel linear programming problems recently developed by lai et al since the fuzzy goals are determined for both an objective function and decision variables at the upper level undesirable solutions are produced when these fuzzy goals are inconsistent in order to overcome such problems after eliminating the fuzzy goals for decision variables interactive fuzzy programming for multilevel linear programming problems is presented in our interactive method after determining the fuzzy goals of the decision makers at all levels a satisfactory solution is derived efficiently by updating the satisfactory degrees of decision makers at the upper level with considerations of overall satisfactory balance among all levels illustrative numerical examples for two level and for three level linear programming problems are provided to demonstrate the feasibility of the proposed method

on the fast matrix multiplication in the boundary element method by panel clustering the boundary element method bem leads to a system of linear equations with a full matrix while fem yields sparse matrices this fact seems to require much computational work for the definition of the matrix for the solution of the system and in particular for the matrix vector multiplication which always occurs as an elementary in this paper a method for the approximate matrix vector multiplication is described which requires much less arithmetical work in addition the storage requirements are strongly reduced

computation of the optimal tolls on the traffic network abstract the present paper is devoted to the computation of optimal tolls on a traffic network that is described as fuzzy bilevel optimization problem as a fuzzy bilevel optimization problem we consider bilinear optimization problem with crisp upper level and fuzzy lower level an effective algorithm for computation optimal tolls for the upper level decision maker is developed under assumption that the lower level decision maker chooses the optimal solution as well the algorithm is based on the membership function approach this algorithm provides us with a global optimal solution of the fuzzy bilevel optimization problem

wavelet methods for pdes some recent developments abstract this paper is concerned with recent developments of wavelet schemes for the numerical treatment of operator equations with special emphasis on two issues adaptive solution concepts and nontrivial domain geometries after describing a general multiresolution framework the key features of wavelet bases are highlighted namely locality norm equivalences and cancellation properties assuming first that wavelet bases with these properties are available on the relevant problem domains the relevance of these features for a wide class of stationary problems is explained in subsequent sections the main issues are preconditioning and the efficient adaptive application of wavelet representations of the involved operators we indicate then how these ingredients combined with concepts from nonlinear or best n term approximation culminate in an adaptive wavelet scheme for elliptic selfadjoint problems covering boundary value problems as well as boundary integral equations these schemes can be shown to exhibit convergence rates that are in a certain sense asymptotically optimal we conclude this section with some brief remarks on data structures and implementation interrelations with regularity in a certain scale of besov spaces and strategies of extending such schemes to unsymmetric or indefinite problems we address then the adaptive evaluation of nonlinear functionals of wavelet expansions as a central task arising in connection with nonlinear problems wavelet bases on nontrivial domains are discussed next the main issues are the development of fourier free construction principles and criteria for the validity of norm equivalences finally we indicate possible combinations of wavelet concepts with conventional discretizations such as finite element or finite volume schemes in connection with convection dominated and hyperbolic problems

composite wavelet bases for operator equations this paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit n cube these bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains this covers also elliptic boundary value problems although this study is primarily motivated by our previous analysis of wavelet methods for pseudo differential equations with special emphasis on boundary integral equations in this case it is natural to model the boundary surface as a union of parametric images of the unit cube it will be shown how to construct wavelet bases on the surface which are composed of wavelet bases defined on each surface patch here the relevant properties are the validity of norm equivalences in certain ranges of sobolev scales as well as appropriate moment conditions

weighting method for bi level linear fractional programming problems in this paper we consider the solution of a bi level linear fractional programming problem bllfpp by weighting method a non dominated solution set is obtained by this method in this article decision makers dms provide their preference bounds to the decision variables that is the upper and lower bounds to the decision variables they control we convert the hierarchical system into scalar optimization problem sop by finding proper weights using the analytic hierarchy process ahp so that objective functions of both levels can be combined into one objective function here the relative weights represent the relative importance of the objective functions

a preconditioned minres method for the coupling of mixed fem and bem for some nonlinear problems we provide an efficient solution procedure for the linearized galerkin schemes arising from the combined use of mixed finite elements mixed fem and boundary elements bem to solve a class of nonlinear problems as a model we consider a nonlinear linear transmission problem appearing in electromagnetism and steady heat conduction since the corresponding continuous and discrete variational formulations become nonlinear twofold saddle point problems also called dual dual formulations we propose to apply newton s method to the galerkin schemes thus yielding linear systems with the same dual dual structure hence we follow previous works on this kind of operator equation and derive a preconditioned minimum residual minres method that guarantees a bounded number of iterations independent of the mesh size to solve these systems

on an isoperimetric problem with a competing nonlocal term ii the general case this paper is the continuation of a previous paper h knupfer and c b muratov comm pure appl math 66 2013 1129 1162 we investigate the classical isoperimetric problem modified by an addition of a nonlocal repulsive term generated by a kernel given by an inverse power of the distance in this work we treat the case of a general space dimension we obtain basic existence results for minimizers with sufficiently small masses for certain ranges of the exponent in the kernel we also obtain nonexistence results for sufficiently large masses as well as a characterization of minimizers as balls for sufficiently small masses and low spatial dimensionality the physically important special case of three space dimensions and coulombic repulsion is included in all the results mentioned above in particular our work yields a negative answer to the question if stable atomic nuclei at arbitrarily high atomic numbers can exist in the framework of the classical liquid drop model of nuclear matter in all cases the minimal energy scales linearly with mass for large masses even if the infimum of energy cannot be attained 2014 wiley periodicals inc

wavelet methods for fast resolution of elliptic problems this paper shows that the use of wavelets to discretize an elliptic problem with dirichlet or neumann boundary conditions has two advantages an explicit diagonal preconditioning makes the condition number of the corresponding matrix become bounded by a constant and the order of approximation is locally of spectral type in contrast with classical methods using a conjugate gradient method one thus obtains fast numerical algorithms of resolution a comparison is also drawn between wavelet and classical methods

fuzzy multiple level programming the use of fuzzy set concept to overcome the computational difficulties of multi level optimization for large systems appears to be a powerful approach and can solve practical large problems in a very simplified manner furthermore unlike the duo ploy or multi ploy approaches the proposed interactive concept provides flexibility which is essential due to the different styles of management in different organizations the approach is formulated based on exploring the fuzziness or indefiniteness which is almost always present in a decentralized large hierarchy organization the advantage of this approach is that not only the problem is much more simplified for solution purposes the original problem is also much more realistically represented

a computational study for bilevel quadratic programs using semidefinite relaxations abstract in this paper we deal with bilevel quadratic programming problems with binary decision variables in the leader problem and convex quadratic programs in the follower problem for this purpose we transform the bilevel problems into equivalent quadratic single level formulations by replacing the follower problem with the equivalent karush kuhn tucker kkt conditions then we use the single level formulations to obtain mixed integer linear programming milp models and semidefinite programming sdp relaxations thus we compute optimal solutions and upper bounds using linear programming lp and sdp relaxations our numerical results indicate that the sdp relaxations are considerably tighter than the lp ones consequently the sdp relaxations allow finding tight feasible solutions for the problem especially when the number of variables in the leader problem is larger than in the follower problem moreover they are solved at a significantly lower computational cost for large scale instances

kkt transformation approach for multi objective multi level linear programming problems abstract the earlier karush kuhn tucker kkt transformation method has been applied to multi level decentralized programming problems ml d pps when the decision variable set was divided into subsets where each decision maker dm of the system controlled only a particular subset but had no control over any decision variables of some other subset in this paper we give the mathematical formulation and corresponding development of ml d pps by kkt transformation when dms have absolute control over certain decision variables but some variables may be shared and hence controlled by two or more dms

interpolation of besov spaces we investigate besov spaces and their connection with dyadic spline approximation in lp q 0 p oo our main results are the determination of the interpolation spaces between a pair of besov spaces an atomic decomposition for functions in a besov space the characterization of the class of functions which have certain prescribed degree of approximation by dyadic splines

a neural network approach for solving nonlinear bilevel programming problem a neural network model is presented for solving nonlinear bilevel programming problem which is a np hard problem the proposed neural network is proved to be lyapunov stable and capable of generating approximal optimal solution to the nonlinear bilevel programming problem the asymptotic properties of the neural network are analyzed and the condition for asymptotic stability solution feasibility and solution optimality are derived the transient behavior of the neural network is simulated and the validity of the network is verified with numerical examples

a neurodynamic optimization approach to bilevel quadratic programming this paper presents a neurodynamic optimization approach to bilevel quadratic programming bqp based on the karush kuhn tucker kkt theorem the bqp problem is reduced to a one level mathematical program subject to complementarity constraints mpcc it is proved that the global solution of the mpcc is the minimal one of the optimal solutions to multiple convex optimization subproblems a recurrent neural network is developed for solving these convex optimization subproblems from any initial state the state of the proposed neural network is convergent to an equilibrium point of the neural network which is just the optimal solution of the convex optimization subproblem compared with existing recurrent neural networks for bqp the proposed neural network is guaranteed for delivering the exact optimal solutions to any convex bqp problems moreover it is proved that the proposed neural network for bilevel linear programming is convergent to an equilibrium point in finite time finally three numerical examples are elaborated to substantiate the efficacy of the proposed approach

the polynomial hierarchy and a simple model for competitive analysis the multi level linear programs of candler norton and townsley are a simple class of sequenced move games in which players are restricted in their moves only by common linear constraints and each seeks to optimize a fixed linear criterion function in his her own continuous variables and those of other players all data of the game and earlier moves are known to a player when he she is to move the one player case is just linear programming

wavelet galerkin methods for second kind integral equations we use vector valued multiwavelets on compact sets to develop a galerkin method for systems of integral equations of the second kind we propose a compression strategy for the coefficient matrix of the linear system obtained from this method and show that the compressed scheme preserves almost optimal convergence rate of the original scheme and yields a sparse matrix with a bounded condition number

minimality via second variation for a nonlocal isoperimetric problem we discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts we show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and in fact satisfy a quantitative isoperimetric inequality with respect to sets that are l 1 close the link with local minimizers for the diffuse interface ohta kawasaki energy is also discussed as a byproduct of the quantitative estimate we get new results concerning periodic local minimizers of the area functional and a proof via second variation of the sharp quantitative isoperimetric inequality in the standard euclidean case as a further application we address the global and local minimality of certain lamellar configurations

on the complexity of quantified linear systems in this paper we explore the computational complexity of the conjunctive fragment of the first order theory of linear arithmetic quantified propositional formulas of linear inequalities with k 1 quantifier alternations are log space complete in s k p or p k p depending on the initial quantifier we show that when we restrict ourselves to quantified conjunctions of linear inequalities i e quantified linear systems the complexity classes collapse to polynomial time in other words the presence of universal quantifiers does not alter the complexity of the linear programming problem which is known to be in p our result reinforces the importance of sentence formats from the perspective of computational complexity

a neural network for solving a convex quadratic bilevel programming problem a neural network is proposed for solving a convex quadratic bilevel programming problem based on lyapunov and lasalle theories we prove strictly an important theoretical result that for an arbitrary initial point the trajectory of the proposed network does converge to the equilibrium which corresponds to the optimal solution of a convex quadratic bilevel programming problem numerical simulation results show that the proposed neural network is feasible and efficient for a convex quadratic bilevel programming problem

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

quadrature methods for 2d and 3d problems in this paper we give an overview on well known stability and convergence results for simple quadrature methods based on low order composite quadrature rules and applied to the numerical solution of integral equations over smooth manifolds first we explain the methods for the case of second kind equations then we discuss what is known for the analysis of pseudodifferential equations we explain why these simple methods are not recommended for integral equations over domains with dimension higher than one finally for the solution of a two dimensional singular integral equation we prove a new result on a quadrature method based on product rules

data driven inverse optimization with imperfect information in data driven inverse optimization an observer aims to learn the preferences of an agent who solves a parametric optimization problem depending on an exogenous signal thus the observer seeks the agent s objective function that best explains a historical sequence of signals and corresponding optimal actions we formalize this inverse optimization problem as a distributionally robust program minimizing the worst case risk that the estimated decision i e the decision implied by a particular candidate objective differs from the agent s actual response to a random signal we show that our framework offers attractive out of sample performance guarantees for different prediction errors and that the emerging inverse optimization problems can be reformulated as or approximated by tractable convex programs when the prediction error is measured in the space of objective values a main strength of the proposed approach is that it naturally generalizes to situations where the observer has imperfect information e g when the agent s true objective function is not contained in the space of candidate objectives when the agent suffers from bounded rationality or implementation errors or when the observed signal response pairs are corrupted by measurement noise

a penalty function method based on kuhn tucker condition for solving linear bilevel programming abstract using the kuhn tucker optimality condition of the lower level problem we transform the linear bilevel programming problem into a corresponding single level programming the complementary and slackness condition of the lower level problem is appended to the upper level objective with a penalty then we decompose the linear bilevel programming into a series of linear programming problems and get the optimal solution of the linear bilevel programming using linear programming method

on two level optimization decentralized planning has long been recognized as an important decision making problem many approaches based on the concepts of large scale system decomposition have generally lacked the ability to model the type of truly independent subsystems which have often existed in practice similarly tractable game theoretic approaches have not allowed for a continuous solution space in which one player s decision restricts the feasible choices of another this correspondence offers a formulation to help model this type of decision making hierarchy characterizations and algorithmic approaches for a class of two level problems are discussed

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

interactive fuzzy programming for two level linear and linear fractional production and assignment problems a case study abstract in this paper we deal with actual problems on production and work force assignment in a housing material manufacturer and a subcontract firm we formulate two kinds of two level programming problems one is a profit maximization problem of both the housing material manufacturer and the subcontract firm and the other is a profitability maximization problem of them applying the interactive fuzzy programming for two level linear and linear fractional programming problems we derive satisfactory solutions to the problems after comparing the two problems we discuss the results of the applications and examine actual planning of the production and the work force assignment of the two firms to be implemented

an approximate programming method based on the simplex method for bilevel programming problem many real problems can be modeled to the problems with a hierarchical structure and bilevel programming is a useful tool to solve the hierarchical optimization problems so the bilevel programming is widely applied and numerous methods have been proposed to solve this programming in this paper we propose an approximate programming algorithm to solve bilevel nonlinear programming problem finally the example illustrates the feasibility of the proposed algorithm

the semidiscrete filtered backprojection algorithm is optimal for tomographic inversion the filtered backprojection algorithm is probably the most often used reconstruction algorithm in two dimensional computerized tomography for a semidiscrete version in the parallel scanning geometry we prove optimal l2 convergence rates for density distributions in sobolev spaces additionally we show l2 convergence without rates when the density distribution is only in l2 the key to success is a new representation of the filtered backprojection which enables us to apply techniques from approximation theory our analysis provides further a modification of the shepp logan reconstruction filter with an improved convergence behavior numerical experiments in the fully discrete setting reproduce the theoretical predictions

capacity planning with competitive decision makers trilevel milp formulation degeneracy and solution approaches abstract capacity planning addresses the decision problem of an industrial producer investing on infrastructure to satisfy future demand with the highest profit traditional models neglect the rational behavior of some external decision makers by assuming either static competition or captive markets we propose a mathematical programing formulation with three levels of decision makers to capture the dynamics of duopolistic markets the trilevel model is transformed into a bilevel optimization problem with mixed integer variables in both levels by replacing the third level linear program with its optimality conditions we introduce new definitions required for the analysis of degeneracy in multilevel models and develop two novel algorithms to solve these challenging problems each algorithm is shown to converge to a different type of degenerate solution the computational experiments for capacity expansion in industrial gas markets show that no algorithm is strictly superior in terms of performance

stability of the static spike autosolitons in the gray scott model we performed an asymptotic linear stability analysis of the static spike autosolitons ass self sustained solitary inhomogeneous states in the gray scott model of an autocatalytic chemical reaction we found that in one dimension these ass destabilize with respect to pulsations or the onset of traveling motion when the inhibitor is slow enough in higher dimensions the one dimensional static spike ass are always unstable with respect to corrugation and wriggling the higher dimensional radially symmetric static spike ass may destabilize with respect to the radially nonsymmetric fluctuations leading to their splitting when the inhibitor is fast or with respect to pulsations when the inhibitor is slow

a construction of interpolating wavelets on invariant sets we introduce the concept of a refinable set relative to a family of contractive mappings on a metric space and demonstrate how such sets are useful to recursively construct interpolants which have a multiscale structure the notion of a refinable set parallels that of a refinable function which is the basis of wavelet construction the interpolation points we recursively generate from a refinable set by a set theoretic multiresolution are analogous to multiresolution for functions used in wavelet construction we then use this recursive structure for the points to construct multiscale interpolants several concrete examples of refinable sets which can be used for generating interpolatory wavelets are included

wavelet galerkin method for fourth order linear and nonlinear differential equations abstract in this paper we propose a wavelet galerkin method for fourth order linear and nonlinear differential equations using compactly supported daubechies wavelets 2 term connection coefficients have been effectively used for a computationally economical evaluation of higher order derivatives the orthogonality and compact support properties of basis functions lead to highly sparse linear systems the quasilinearization strategy is effectively employed in dealing with wavelet coefficients of nonlinear problems the stability and the convergence analysis in the form of error analysis have been carried out an efficient compression algorithm is proposed to reduce the computational cost of the method finally the method is tested on several examples and found to be in good agreement with exact solution

multilevel preconditioning this paper is concerned with multilevel techniques for preconditioning linear systems arising from galerkin methods for elliptic boundary value problems a general estimate is derived which is based on the characterization of besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions the result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers in particular our analysis indicates how to realize even uniformly bounded condition numbers for example the general results are used to show that the bramble pasciak xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform furthermore they are applied to a general multivariate setting of refinable shift invariant spaces in particular covering those induced by various types of wavelets

a bilevel bottleneck programming problem this paper studies a bilevel programming problem with linear constraints and in which the objective functions at both levels are concave bottleneck functions which are to be minimized the problem is a non convex programming problem it is shown that an optimal solution to the problem is attainable at an extreme point of the underlying region the outer level objective function values are ranked in increasing order until a value is reached one of the solutions corresponding to which is feasible for the problem this solution is then the required global optimal solution

pessimistic bilevel optimization we study a variant of the pessimistic bilevel optimization problem which comprises constraints that must be satisfied for any optimal solution of a subordinate lower level optimization problem we present conditions that guarantee the existence of optimal solutions in such a problem and we characterize the computational complexity of various subclasses of the problem we then focus on problem instances that may lack convexity but that satisfy a certain independence property we develop convergent approximations for these instances and we derive an iterative solution scheme that is reminiscent of the discretization techniques used in semi infinite programming we also present a computational study that illustrates the numerical behavior of our algorithm on standard benchmark instances

prewavelet analysis of the heat equation summary we consider the heat equation in a smooth domain of r 2 with dirichlet and neumann boundary conditions it is solved by using its integral formulation with double layer potentials where the unknown the jump of the solution through the boundary belongs to an anisotropic sobolev space we approximate by the galerkin method and use a prewavelet basis on t 0 t which characterizes the anisotropic space the use of prewavelets allows to compress the stiffness matrix from o n 2 to o n log n when n is the size of the matrix and the condition number of the compressed matrix is uniformly bounded as the initial one in the prewavelet basis finally we show that the compressed scheme converges as fast as the galerkin one even for the dirichlet problem which does not admit a coercive variational formulation

a fast trigonometric collocation method for some elliptic pseudodifferential equations abstract in this paper we propose and analyze a fast trigonometric collocation method for a class of periodic elliptic pseudodifferential equations whose pseudodifferential operators can always be represented as the sum of a principal part and a smoothing operator we show that the whole matrix representation for the principal part in our discrete linear system can be generated by only computing o n entries rather than computing all entries of the matrix where 2 n or 2 n 1 is the size of the matrix the dense matrix for the smoothing operator can be compressed into a sparse matrix with only o n log n nonzero entries we also prove that our proposed method preserves the optimal convergent order the same as without compression some numerical experiments for solving three cases of boundary integral equations are presented to demonstrate its approximate accuracy and computational efficiency verifying the theoretical estimates

solving a bilevel linear program when the inner decision maker controls few variables abstract this paper presents an algorithm that always finds an optimum of a bilevel linear program blp problem if one exists moreover if the inner decision maker does not control too many variables the algorithm is efficient in particular if the number of variables controlled by the inner decision maker can be considered to be constant then the algorithm is a polynomial time algorithm the algorithm is also well suited to parallel implementation