regularity estimates for convex multifunctions the main result of the paper contains an exact formula for the rate of regularity of a set valued mapping with a convex graph as a consequence we find an exact expression for the rate of regularity of set valued mappings associated with so called constraint systems it turns out that the rate is equal to the upper bound of robinson type estimates over all norms in the graph space of the homogenized mapping majorizing the norm of the underlying space we further introduce a concept of a perfectly regular mapping and find some criteria for perfect regularity

constructing nested bases approximations from the entries of non local operators in this article a method for constructing nested bases approximations to large scale fully populated discretizations of integral operators is introduced the scheme uses only few of the matrix entries for approximating the whole matrix in this sense it is similar to the adaptive cross approximation method however its computational complexity is improved

on multivariate lagrange interpolation lagrange interpolation by polynomials in several variables is studied through a finite difference approach we establish an interpolation formula analogous to that of newton and a remainder formula both of them in terms of finite differences we prove that the finite difference admits an integral representation involving simplex spline functions in particular this provides a remainder formula for lagrange interpolation of degree n of a function f which is a sum of integrals of certain n 1 st directional derivatives of f multiplied by simplex spline functions we also provide two algorithms for the computation of lagrange interpolants which use only addition scalar multiplication and point evaluation of polynomials

a theory of pseudoskeleton approximations abstract let an m n matrix a be approximated by a rank r matrix with an accuracy e we prove that it is possible to choose r columns and r rows of a forming a so called pseudoskeleton component which approximates a with o e r m n accuracy in the sense of the 2 norm on the way to this estimate we study the interconnection between the volume i e the determinant in the absolute value and the minimal singular value r of r r submatrices of an n r matrix with orthogonal columns we propose a lower bound better than one given by chandrasekaran and ipsen and by hong and pan for the maximum of r over all these submatrices and formulate a hypothesis on a tighter bound

strong conical hull intersection property bounded linear regularity jameson s property g and error bounds in convex optimization the strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in euclidean space these fundamental notions occur in various branches of convex optimization constrained approximation convex feasibility problems linear inequalities for instance it is shown that the standard constraint qualification from convex analysis implies bounded linear regularity which in turn yields the strong conical hull intersection property jameson s duality for two cones which relates bounded linear regularity to property g is re derived and refined for polyhedral cones a statement dual to hoffman s error bound result is obtained a sharpening of a result on error bounds for convex inequalities by auslender and crouzeix is presented finally for two subspaces property g is quantified by the angle between the subspaces

hybrid cross approximation of integral operators the efficient treatment of dense matrices arising e g from the finite element discretisation of integral operators requires special compression techniques in this article we use the matrix representation that approximates the dense stiffness matrix in admissible blocks corresponding to subdomains where the underlying kernel function is smooth by low rank matrices the low rank matrices are assembled by a new hybrid algorithm hca that has the same proven convergence as standard interpolation but also the same efficiency as the heuristic adaptive cross approximation aca

data sparse approximation of non local operators by h2 matrices many of today s most efficient numerical methods are based on multilevel decompositions the multigrid algorithm is based on a hierarchy of grids wavelet techniques use a hierarchy of basis functions while fast panel clustering and multipole methods employ a hierarchy of clusters the high efficiency of these methods is due to the fact that the hierarchies are nested i e that the information present on a coarser level is also present on finer levels thus allowing efficient recursive algorithms h2 matrices employ nested local expansion systems in order to approximate matrices in optimal or for some problem classes at least optimal up to logarithmic factors order of complexity this paper presents a criterion for the approximability of general matrices in the h2 matrix format and an algorithm for finding good nested expansion systems and constructing the approximation efficiently

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

fast directional multilevel algorithms for oscillatory kernels this paper introduces a new directional multilevel algorithm for solving n body or n point problems with highly oscillatory kernels these systems often result from the boundary integral formulations of scattering problems and are difficult due to the oscillatory nature of the kernel and the non uniformity of the particle distribution we address the problem by first proving that the interaction between a ball of radius r and a well separated region has an approximate low rank representation as long as the well separated region belongs to a cone with a spanning angle of o 1 r and is at a distance which is at least o r 2 away from from the ball we then propose an efficient and accurate procedure which utilizes random sampling to generate such a separated low rank representation based on the resulting representations our new algorithm organizes the high frequency far field computation by a multidirectional and multiscale strategy to achieve maximum efficiency the algorithm performs well on a large group of highly oscillatory kernels our algorithm is proved to have o n log n computational complexity for any given accuracy when the points are sampled from a two dimensional surface we also provide numerical results to demonstrate these properties

tt cross approximation for multidimensional arrays abstract as is well known a rank r matrix can be recovered from a cross of r linearly independent columns and rows and an arbitrary matrix can be interpolated on the cross entries other entries by this cross or pseudo skeleton approximation are given with errors depending on the closeness of the matrix to a rank r matrix and as well on the choice of cross in this paper we extend this construction to d dimensional arrays tensors and suggest a new interpolation formula in which a d dimensional array is interpolated on the entries of some tt cross tensor train cross the total number of entries and the complexity of our interpolation algorithm depend on d linearly so the approach does not suffer from the curse of dimensionality we also propose a tt cross method for computation of d dimensional integrals and apply it to some examples with dimensionality in the range from d 100 up to d 4000 and the relative accuracy of order 10 10 in all constructions we capitalize on the new tensor decomposition in the form of tensor trains tt decomposition

a new credibilistic clustering algorithm abstract this paper focuses on credibilistic clustering approach a data clustering method partitions unlabeled data sets into clusters and labels them for various goals such as computer vision and pattern recognition there are different models for objective function based fuzzy clustering such as fuzzy c means fcm possibilistic c mean pcm and their combinations credibilistic clustering is a new approach in this field in this paper a new credibilistic clustering model is introduced in which credibility measure is applied instead of possibility measure in possibilistic clustering also in objective function the separation of clusters is considered in addition to the compactness within clusters the steps of clustering are designed based on this approach finally the main issues about model are discussed and the results of computational experiments are presented to show the efficiency of the proposed model

regularity modulus of arbitrarily perturbed linear inequality systems we aim to quantify the stability of systems of possibly infinitely many linear inequalities under arbitrary perturbations of the data our focus is on the aubin property also called pseudo lipschitz of the solution set mapping or equivalently on the metric regularity of its inverse mapping the main goal is to determine the regularity modulus of the latter mapping exclusively in terms of the system s data in our context both the right and the left hand side of the system are subject to possible perturbations this fact entails notable differences with respect to previous developments in the framework of linear systems with perturbations of the right hand side in these previous studies the feasible set mapping is sublinear which is not our current case and the well known radius theorem constitutes a useful tool for determining the modulus in our current setting we do not have an explicit expression for the radius of metric regularity and we have to tackle the modulus directly as an application we approach under appropriate assumptions the regularity modulus for a semi infinite system associated with the lagrangian dual of an ordinary nonlinear programming problem

construction of data sparse mathcal h 2 matrices by hierarchical compression hierarchical matrices mathcal h matrices provide an elegant approach to handling large densely populated matrices the matrix is split into a hierarchy of blocks and each block is approximated by a low rank matrix in factorized form it has been demonstrated that this representation can be used to treat integral and partial differential equations solve matrix equations from the field of control theory and evaluate matrix functions efficiently mathcal h 2 matrices use a refined representation that employs a multilevel structure in order to reduce the storage requirements of hierarchical matrices it has been shown that mathcal h 2 matrices can significantly reduce storage requirements for large problems in particular when combined with modern error control schemes until now all algorithms for constructing an efficient approximation of a general matrix by an mathcal h 2 matrix required a representation of the entire original matrix to be kept in storage therefore the storage requirements of mathcal h 2 matrix algorithms could be far larger than those of the final approximation this paper presents a new approach that allows us to construct an mathcal h 2 matrix without storing the entire original matrix the central idea is to approximate submatrices and combine them by an efficient new algorithm to form approximations of larger matrices until the entire matrix has been treated using this new approach many mathcal h matrix algorithms can be refitted easily to compute results in the more efficient representation possible applications include efficient matrix arithmetics for the construction of preconditioners the approximation of matrix functions or solutions of matrix equations or efficient compression schemes based on the popular cross approximation algorithms

approximation of boundary element matrices this article considers the problem of approximating a general asymptotically smooth function in two variables typically arising in integral formulations of boundary value problems by a sum of products of two functions in one variable from these results an iterative algorithm for the low rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed this algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance

linear regularity equirregularity and intersection mappings for convex semi infinite inequality systems in this paper we introduce the concepts of linear regularity and equirregularity for an arbitary family of set valued mappings between extended metric spaces the concept of linear regularity is inspired in the same property for a family of sets then we analyze the relationship between the metric regularity moduli of the mappings in the family and the modulus of the associated intersection mapping we are particularly concerned with the solution set of a system of infinitely many convex inequalities our framework allows for right hand side perturbations as well as for linear perturbations of the left hand side of all the inequalities

adaptive variable rank approximation of general dense matrices in order to handle large dense matrices arising in the context of integral equations efficiently panel clustering approaches such as the popular multipole expansion method have proven to be very useful these techniques split the matrix into blocks approximate the kernel function on each block by a degenerate expansion and discretize this expansion in order to find an efficient low rank approximation of each block typical expansion schemes use the same expansion order in each block and since the matrix approximation error has to be kept consistent with the discretization error this allows us to handle n times n matrices by algorithms with a complexity of mathcal o n log alpha n for alpha geq 1 recently variable order expansion schemes have been introduced which choose different ranks for different blocks and have been demonstrated to reach a complexity of mathcal o n while keeping the matrix approximation error and the discretization error consistent this paper introduces an algorithm which can construct variable rank approximations for general matrices without the need of an in depth analysis of the underlying operators the matrix is fed into the algorithm and the algorithm approximates it up to an arbitrary precision

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

black box approximation of tensors in hierarchical tucker format we derive and analyse a scheme for the approximation of order d tensors a rn1 nd in the hierarchical h tucker format a dimension multilevel variant of the tucker format and strongly related to the tt tensor train format for a fixed rank parameter k the storage complexity of a tensor in h tucker format is odk3 k i 1dni and we present a heuristic algorithm that finds an approximation to a tensor in the h tucker format in odk4 log d k2 i 1dni by inspection of only odk3 log d k2 i 1dni entries under mild assumptions tensors in the h tucker format are reconstructed for general tensors we derive error bounds that are based on the approximability of matrices matricizations of the tensor by few outer products of its rows and columns the construction parallelizes with respect to the order d and we also propose an adaptive approach that aims at finding the rank parameter for a given target accuracy e automatically

multilevel matrix multiplication and fast solution of integral equations abstract a fast multigrid approach is described for the task of calculating k x y u y dy for each x r d discretizing by an equidistant grid with n points and meshsize h and approximating the integrations to o h2s accuracy it is shown that the complexity of this calculation can be reduced from o n2 to o sn provided the kernel k is sufficiently smooth for potential type kernels the complexity is reduced to o sn log n corresponding integral equations can be solved to a similar accuracy in basically the same amount of work using a special kind of distributed relaxation in a multigrid algorithm one and two dimensional numerical tests and theoretical derivations of optimal strategies are reported the method is applicable to the task of multiplying by any matrix with appropriate smoothness properties including most types of many body interactions