hidden conic quadratic representation of some nonconvex quadratic optimization problems the problem of minimizing a quadratic objective function subject to one or two quadratic constraints is known to have a hidden convexity property even when the quadratic forms are indefinite the equivalent convex problem is a semidefinite one and the equivalence is based on the celebrated s lemma in this paper we show that when the quadratic forms are simultaneously diagonalizable sd it is possible to derive an equivalent convex problem which is a conic quadratic cq one and as such is significantly more tractable than a semidefinite problem the sd condition holds for free for many problems arising in applications in particular when deriving robust counterparts of quadratic or conic quadratic constraints affected by implementation error the proof of the hidden cq property is constructive and does not rely on the s lemma this fact may be significant in discovering hidden convexity in some nonquadratic problems

the regularizing properties of anisotropic radial basis functions in the present work we consider the problem of interpolating scattered data using radial basis functions rbf in general it is well known that this leads to a discrete linear inverse problem that needs to be regularized in order to provide a meaningful solution the work focuses on a metric regularization approach based on a new class of rbf called anisotropic rbf the work provides theoretical justifications for the regularization approach and it considers a suitable proposal for the metric supporting it by numerical examples

multilevel sparse kernel based interpolation a multilevel kernel based interpolation method suitable for moderately high dimensional function interpolation problems is proposed the method termed multilevel sparse kernel based interpolation mlski for short uses both levelwise and directionwise multilevel decomposition of structured or mildly unstructured interpolation data sites in conjunction with the application of kernel based interpolants with different scaling in each direction the multilevel interpolation algorithm is based on a hierarchical decomposition of the data sites whereby at each level the detail is added to the interpolant by interpolating the resulting residual of the previous level on each level anisotropic radial basis functions are used for solving a number of small interpolation problems which are subsequently linearly combined to produce the interpolant mlski can be viewed as an extension of d boolean interpolation which is closely related to ideas in sparse grid and hyperbolic crosses literature to kernel ba

the trust region subproblem with non intersecting linear constraints this paper studies an extended trust region subproblem etrs in which the trust region intersects the unit ball with m m linear inequality constraints when m 0 m 1 m 0 m 1 or m 2 m 2 and the linear constraints are parallel it is known that the etrs optimal value equals the optimal value of a particular convex relaxation which is solvable in polynomial time however it is also known that when m ge 2 m 2 and at least two of the linear constraints intersect within the ball i e some feasible point of the etrs satisfies both linear constraints at equality then the same convex relaxation may admit a gap with etrs this paper shows that the convex relaxation has no gap for arbitrary m m as long as the linear constraints are non intersecting

regression based inverse distance weighting with applications to computer experiments inverse distance weighting idw is a simple method for multivariate interpolation but has poor prediction accuracy in this article we show that the prediction accuracy of idw can be substantially improved by integrating it with a linear regression model this new predictor is quite flexible computationally efficient and works well in problems having high dimensions and or large datasets we also develop a heuristic method for constructing confidence intervals for prediction this article has supplementary material online

on a subproblem of trust region algorithms for constrained optimization we study a subproblem that arises in some trust region algorithms for equality constrained optimization it is the minimization of a general quadratic function with two special quadratic constraints properties of such subproblems are given it is proved that the hessian of the lagrangian has at most one negative eigenvalue and an example is presented to show that the hessian may have a negative eigenvalue when one constraint is inactive at the solution

on the construction of polyharmonic b splines in this paper we present two constructions providing polyharmonic b splines with a quasi rotational invariant behavior their properties are discussed and it is shown that in this way we improve the decay when email protected

representing implicit surfaces satisfying lipschitz conditions by 4 dimensional point sets abstract for any given implicit surface satisfying two lipschitz conditions this work triangulates the surface using a set of 4 dimensional points with small hausdorff distances every 4 dimensional point is a 3 dimensional point with a weight compared to traditional triangulation approaches our method does not explicitly require the storage of connectivity information instead the connectivity information can be implicitly generated using a concept called the alpha shape with two calculated bounds the density of the 4 dimensional points adapts to the local curvature moreover the generated triangulation has been mathematically proven to be homeomorphic to the original implicit surface the hausdorff distance between the triangulation and the original surface is demonstrated to be small enough according to the experimental results

indefinite trust region subproblems and nonsymmetric eigenvalue perturbations this paper extends the theory of trust region subproblems in two ways i it allows indefinite inner products in the quadratic constraint and ii it uses a two sided upper and lower bound quadratic constraint characterizations of optimality are presented that have no gap between necessity and sufficiency conditions for the existence of solutions are given in terms of the definiteness of a matrix pencil a simple dual program is introduced that involves the maximization of a strictly concave function on an interval this dual program simplifies the theory and algorithms for trust region subproblems it also illustrates that the trust region subproblems are implicit convex programming problems and thus explains why they are so tractable the duality theory also provides connections to eigenvalue perturbation theory trust region subproblems with zero linear term in the objective function correspond to eigenvalue problems and adding a linear term in the objective function is seen to correspond to a p

some equivalent results with yakubovich s s lemma this paper presents an elementary proof of nonstrict finsler s theorem and discusses five closely related theorems yakubovich s s lemma yuan s alternative theorem nonstrict finsler s theorem strict finsler s theorem and finsler calabi s theorem this paper discusses how one is related with another at the end of this article we present some extensions of these results to an infinite dimensional vector space

trace minimization and definiteness of symmetric pencils abstract a symmetric matrix pencil a b of order n is called positive definite if there is a such that the matrix a b is positive definite we consider the case with b nonsingular and show that the definiteness is closely related to the existence of min tr x t ax under the condition x t bx j 1 where j 1 is a given diagonal matrix of order n and j 2 1 i we also prove an analog of the cauchy interlacing theorem for some such pencils

radial basis functions for the multivariate interpolation of large scattered data sets an efficient method for the multivariate interpolation of very large scattered data sets is presented it is based on the local use of radial basis functions and represents a further improvement of the well known shepard s method although the latter is simple and well suited for multivariate interpolation it does not share the good reproduction quality of other methods widely used for bivariate interpolation on the other band radial basis functions which have proven to be highly useful for multivariate scattered data interpolation have a severe drawback they are unable to interpolate large sets in an efficient and numerically stable way and maintain a good level of reproduction quality at the same time both problems have been circumvented using radial basis functions to evaluate the nodal function of the modified shepard s method this approach exploits the flexibility of the method and improves its reproduction quality the proposed algorithm has been implemented and numerical results confirm its efficiency

polyharmonic splines an approximation method for noisy scattered data of extra large size the aim of this paper is to provide a fast method with a good quality of reproduction to recover functions from very large and irregularly scattered samples of noisy data which may present outliers to the given sample of size n we associate a uniform grid and around each grid point we condense the local information given by the noisy data by a suitable estimator the recovering is then performed by a stable interpolation based on isotropic polyharmonic b splines due to the good approximation rate we need only m n degrees of freedom to recover the phenomenon faithfully

a locally corrected nystrom method for em scattering with a fourier mode expansion computation of corrected weights in the locally corrected nystrom method is a time consuming procedure since the method exhibits high order convergence properties the expense of computing the corrected weights is worthwhile when many computations will be performed on a fixed geometry such situations occur when solutions are computed over a range of frequencies and when a series of mode solutions in a series expansion are needed here an explanation is given as to how corrected weights can be computed once and then reused over a range of frequencies and mode numbers for any fixed body of revolution bor geometry the corrected weights need not be recomputed but only adjusted by multiplication

connectivity of quadratic hypersurfaces and its applications in optimization part i general theory in this paper we establish a basic theory on the connectivity of quadratic hypersurfaces in addition we examine how both the connectivity and the hidden convexity of quadratic functions are related to each other and prove several connectivity theorems on quadratic functions that combine and generalize dines s and brickman s results on the hidden convexity of quadratic mappings in real field many classical results in quadratic optimization are reproved by the unified approach and more properties involving the matrix pencils are given

sequential experimental design based generalised anova over the last decade surrogate modelling technique has gained wide popularity in the field of uncertainty quantification optimization model exploration and sensitivity analysis this approach relies on experimental design to generate training points and regression interpolation for generating the surrogate in this work it is argued that conventional experimental design may render a surrogate model inefficient in order to address this issue this paper presents a novel distribution adaptive sequential experimental design da sed the proposed da sed has been coupled with a variant of generalised analysis of variance g anova developed by representing the component function using the generalised polynomial chaos expansion moreover generalised analytical expressions for calculating the first two statistical moments of the response which are utilized in predicting the probability of failure have also been developed the proposed approach has been utilized in predicting probability of failure of three structural mechanics problems it is observed that the proposed approach yields accurate and computationally efficient estimate of the failure probability

closed form modal green s functions for accelerated computation of bodies of revolution to solve problems of bodies of revolution bor using the method of moments the modal green s function is used which is defined as the radiation field of a circular loop antenna with sinusoidal current distribution it is a group of oscillating integrals that forms the bottleneck for further accelerating the computation of bor problems in this paper we propose closed form expressions for near axis far distance modal green s functions which can be used to accelerate the computation of bor problems especially when the object is slim a criterion is presented based on a rigorous error analysis to guarantee its range of applications both the scattering and radiation examples are presented to demonstrate the validity and efficiency of the proposed expressions

a note on quadratic forms we extend an interesting theorem of yuan 12 for two quadratic forms to three matrices let c 1 c 2 c 3 be three symmetric matrices in n n if max x t c 1 x x t c 2 x x t c 3 x 0 for all x n it is proved that there exist t i 0 i 1 2 3 such that i 1 3 t i 1 and i 1 3 t i c i has at most one negative eigenvalue

multistep scattered data interpolation using compactly supported radial basis functions a hierarchical scheme is presented for smoothly interpolating scattered data with radial basis functions of compact support a nested sequence of subsets of the data is computed efficiently using successive delaunay triangulations the scale of the basis function at each level is determined from the current density of the points using information from the triangulation the method is rotationally invariant and has good reproduction properties moreover the solution can be calculated and evaluated in acceptable computing time

scattered and track data interpolation using an efficient strip searching procedure abstract a new local algorithm for bivariate interpolation of large sets of scattered and track data is presented the method which changes partially depending on the kind of data is based on the partition of the interpolation domain in a suitable number of parallel strips and starting from these on the construction for any data point of a square neighbourhood containing a convenient number of data points then the well known modified shepard s formula for surface interpolation is applied with some effective improvements the proposed algorithm is very fast owing to the optimal nearest neighbour searching and achieves good accuracy computational cost and storage requirements are analyzed moreover the efficiency and reliability of the algorithm are shown by several numerical tests also performed by renka s algorithm for a comparison

a survey of the s lemma in this survey we review the many faces of the s lemma a result about the correctness of the s procedure the basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization convex geometry and linear algebra as well these were all active research areas but as there was little interaction between researchers in these different areas their results remained mainly isolated here we give a unified analysis of the theory by providing three different proofs for the s lemma and revealing hidden connections with various areas of mathematics we prove some new duality results and present applications from control theory error estimation and computational geometry

shape preserving surface reconstruction using locally anisotropic radial basis function interpolants in this paper we deal with the problem of reconstructing surfaces from unorganized sets of points while capturing the significant geometry details of the modelled surface such as edges flat regions and corners this is obtained by exploiting the good approximation capabilities of the radial basis functions rbf the local nature of the method proposed in 1 and introducing information on shape features and data anisotropies detected from the given surface points the result is a shape preserving reconstruction given by a weighted combination of locally aniso tropic interpolants for each local interpolant the anisotropy is obtained by replacing the euclidean norm with a suitable metric which takes into account the local distribution of the points thus hyperellipsoid basis functions named anisotropic rbfs are defined results from the application of the method to the reconstruction of object surfaces in 3 are presented confirming the effectiveness of the approach

numerical solution of the helmholtz equation in 2d and 3d using a high order nystrom discretization we show how to solve time harmonic scattering problems by means of a high order nystrom discretization of the boundary integral equations of wave scattering in 2d and 3d the novel aspect of our new method is its use of local corrections to the discretized kernel in the vicinity of the kernel singularity enhanced by local corrections the new algorithm has the simplicity and speed advantages of the traditional nystrom method but also enjoys the advantages of high order convergence for controlling solution error we explain the practical details of implementing a scattering code based on a high order nystrom discretization and demonstrate by numerical example that a scattering code based on this algorithm can achieve high order convergence to the correct answer we also demonstrate its performance advantages over a high order galerkin code

locally corrected nystrom method for em scattering by bodies of revolution the nystrom method allows for high order solutions to integral equations modifying the nystrom method with local corrections for singular integrands allows the method to be applied to the inherently singular kernels of frequency domain electromagnetic integral equations here the efficiency and high order convergence properties of the nystrom method are brought together with the body of revolution bor geometry to create an efficient approach to solving this important class of problems in this paper we summarize the locally corrected nystrom method and the magnetic field integral equation mfie for a bor scatterer results are presented that demonstrate high order convergence and excellent agreement with reference data

locally corrected multidimensional quadrature rules for singular functions accurate numerical integration of singular functions usually requires either adaptivity or product integration both interfere with fast summation techniques and thus hamper large scale computations this paper presents a method for computing highly accurate quadrature formulas for singular functions which combine well with fast summation methods given the singularity and the n nodes we first construct weights which integrate smooth functions with order k accuracy then we locally correct a small number of weights near the singularity to achieve order k accuracy on singular functions as well the method is highly efficient and runs in o nk 2d n log 2 n time and o k 2d n space we derive precise error bounds and time estimates and confirm them with numerical results which demonstrate the accuracy and efficiency of the method in large scale computations as part of our implementation we also construct a new adaptive multidimensional product gauss quadrature routine with an effective error

potpourri of conjectures and open questions in nonlinear analysis and optimization we present a collection of fourteen conjectures and open problems in the fields of nonlinear analysis and optimization these problems can be classified into three groups problems of pure mathematical interest problems motivated by scientific computing and applications and problems whose solutions are known but for which we would like to know better proofs for each problem we provide a succinct presentation a list of appropriate references and a view of the state of the art of the subject

an adaptive method for computing invariant manifolds in non autonomous three dimensional dynamical systems abstract we present a computational method for determining the geometry of a class of three dimensional invariant manifolds in non autonomous aperiodically time dependent dynamical systems the presented approach can be also applied to analyse the geometry of 3d invariant manifolds in three dimensional time dependent fluid flows the invariance property of such manifolds requires that at any fixed time they are given by surfaces in r 3 we focus on a class of manifolds whose instantaneous geometry is given by orientable surfaces embedded in r 3 the presented technique can be employed in particular to compute codimension one invariant stable and unstable manifolds of hyperbolic trajectories in 3d non autonomous dynamical systems which are crucial in the lagrangian transport analysis the same approach can also be used to determine evolution of an orientable material surface in a fluid flow these developments represent the first step towards a non trivial 3d extension of the so called lobe dynamics a geometric invariant manifold based framework which has been very successful in the analysis of lagrangian transport in unsteady two dimensional fluid flows in the developed algorithm the instantaneous geometry of an invariant manifold is represented by an adaptively evolving triangular mesh with piecewise c 2 interpolating functions the method employs an automatic mesh refinement which is coupled with adaptive vertex redistribution a variant of the advancing front technique is used for remeshing whenever necessary such an approach allows for computationally efficient determination of highly convoluted evolving geometry of codimension one invariant manifolds in unsteady three dimensional flows we show that the developed method is capable of providing detailed information on the evolving lagrangian flow structure in three dimensions over long periods of time which is crucial for a meaningful 3d transport analysis

joint ranges of hermitian matrices and simultaneous diagonalization abstract let a a 1 a k be a k tuple of heritian operators on an n diensional inner product space x with unit sphere s u x u u 1 with 1 k as the corresponding quadratic forms i e j x x a j x for all x x j 1 k we study ranges of over various subsets t of x the interplay between geometric properties of t and algebraic properties of a is our main concern in particular we characterize those a for which s is identical with the convex hull of the joint spectrum of a defined by a c k a j u j u for some u s the closely related problem of diagonalizing several hermitian operators simultaneously is also studied