approximate tree kernels convolution kernels for trees provide simple means for learning with tree structured data the computation time of tree kernels is quadratic in the size of the trees since all pairs of nodes need to be compared thus large parse trees obtained from html documents or structured network data render convolution kernels inapplicable in this article we propose an effective approximation technique for parse tree kernels the approximate tree kernels atks limit kernel computation to a sparse subset of relevant subtrees and discard redundant structures such that training and testing of kernel based learning methods are significantly accelerated we devise linear programming approaches for identifying such subsets for supervised and unsupervised learning tasks respectively empirically the approximate tree kernels attain run time improvements up to three orders of magnitude while preserving the predictive accuracy of regular tree kernels for unsupervised tasks the approximate tree kernels even lead to more accurate predictions by identifying relevant dimensions in feature space

recent progress in unconstrained nonlinear optimization without derivatives we present an introduction to a new class of derivative free methods for unconstrained optimization we start by discussing the motivation for such methods and why they are in high demand by practitioners we then review the past developments in this field before introducing the features that characterize the newer algorithms in the context of a trust region framework we focus on techniques that ensure a suitable geometric quality of the considered models we then outline the class of algorithms based on these techniques as well as their respective merits we finally conclude the paper with a discussion of open questions and perspectives

on the convergence of the multidirectional search algorithm this paper presents the convergence analysis for the multidirectional search algorithm a direct search method for unconstrained minimization the analysis follows the classic lines of proofs of convergence for gradient related methods the novelty of the argument lies in the fact that explicit calculation of the gradient is unnecessary although it is assumed that the function is continuously differentiable over some subset of the domain the proof can be extended to treat most nonsmooth cases of interest the argument breaks down only at points where the derivative exists but is not continuous finally it is shown how a general convergence theory can be developed for an entire class of direct search methods which includes such methods as the factorial design algorithm and the pattern search algorithm that share a key feature of the multidirectional search algorithm

convergence of the nelder mead simplex method to a nonstationary point this paper analyzes the behavior of the nelder mead simplex method for a family of examples which cause the method to converge to a nonstationary point all the examples use continuous functions of two variables the family of functions contains strictly convex functions with up to three continuous derivatives in all the examples the method repeatedly applies the inside contraction step with the best vertex remaining fixed the simplices tend to a straight line which is orthogonal to the steepest descent direction it is shown that this behavior cannot occur for functions with more than three continuous derivatives the stability of the examples is analyzed

fortified descent simplicial search method a general approach we propose a new simplex based direct search method for unconstrained minimization of a real valued function f of n variables as in other methods of this kind the intent is to iteratively improve an n dimensional simplex through certain reflection expansion contraction steps the method has three novel features first a user chosen integer bar m k specifies the number of good vertices to be retained in constructing the initial trial simplices reflected then either expanded or contracted at iteration k second a trial simplex is accepted only when it satisfies the criteria of fortified descent which are stronger than the criterion of strict descent used in most direct search methods third the number of additional function evaluations needed to check a trial reflected expanded simplex for fortified descent can be controlled if one of the initial trial simplices satisfies the fortified descent criteria it is accepted as the new simplex otherwise the simplex is shrunk a fraction of the way toward a best vertex and the process is restarted etc until either a trial simplex is accepted or the simplex effectively has shrunk to a single point we prove several theoretical properties of the new method if f is continuously differentiable bounded below and uniformly continuous on its lower level set and we choose bar m k with the same value at all iterations k then every cluster point of the generated sequence of iterates is a stationary point the same conclusion holds if the function is continuously differentiable bounded below and we choose bar m k 1 at all iterations k

an extension of the discontinuous galerkin method for the singular poisson equation we present a numerical method for solving a singular poisson equation which solution contains jumps and kinks due to a singular right hand side equations of this type may arise e g within the pressure computation of incompressible multiphase flows the method is an extension to the well known discontinuous galerkin dg method being able to represent the jumps and kinks with subcell accuracy in the proposed method an ansatz function which already fulfills the jump condition is subtracted from the original problem thereby reducing it to a standard poisson equation without a jump invoking a technique that we refer to as patching the construction of the ansatz function can be limited to a very narrow domain around the jump position thus making the construction numerically cheap and easy under optimal conditions the method shows a convergence order of p 1 for dg polynomial degree p still in the worst case a convergence order of approximately 2 4 is preserved for dg polynomial degree of 2

controller design via nonsmooth multidirectional search we propose an algorithm which combines multidirectional search mds with nonsmooth optimization techniques to solve difficult problems in automatic control applications include static and fixed order output feedback controller design simultaneous stabilization h 2 h infty synthesis and much else we show how to combine direct search techniques with nonsmooth descent steps in order to obtain convergence certificates in the presence of nonsmoothness our technique is efficient when small and medium size controllers for plants with large state dimension are sought our numerical testing includes several benchmark examples for instance our algorithm needs 0 41 s to compute a static output feedback stabilizing controller for the boeing 767 flutter benchmark problem e e j davison ifac technical committee reports pergamon press oxford 1990 a system with 55 states the first static controller without performance specifications for this system was obtained in j burke a lewis and m overton siam j optim 15 2003 pp 751 779

on a convergent dsa preconditioned source iteration for a dgfem method for radiative transfer abstract we consider the numerical approximation of the radiative transfer equation using discontinuous angular and continuous spatial approximations for the even parts of the solution the even parity equations are solved using a diffusion synthetic accelerated source iteration we provide a convergence analysis for the infinite dimensional iteration as well as for its discretized counterpart the diffusion correction is computed by a subspace correction which leads to a convergence behavior that is robust with respect to the discretization the proven theoretical contraction rate deteriorates for scattering dominated problems we show numerically that the preconditioned iteration is in practice robust in the diffusion limit moreover computations for the lattice problem indicate that the presented discretization does not suffer from the ray effect the theoretical methodology is presented for plane parallel geometries with isotropic scattering but the approach and proofs generalize to multi dimensional problems and more general scattering operators verbatim

from electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex the electrostatic interpretation of the jacobi gauss quadrature points is exploited to obtain interpolation points suitable for approximation of smooth functions defined on a simplex moreover several new estimates based on extensive numerical studies for approximation along the line using jacobi gauss lobatto quadrature points as the nodal sets are presented the electrostatic analogy is extended to the two dimensional case with the emphasis being on nodal sets inside a triangle for which two very good matrices of nodal sets are presented the matrices are evaluated by computing the lebesgue constants and they share the property that the nodes along the edges of the simplex are the gauss lobatto quadrature points of the chebyshev and legendre polynomials respectively this makes the resulting nodal sets particularly well suited for integration with conventional spectral methods and supplies a new nodal basis for h p finite element methods

experimental complexity analysis of continuous constraint satisfaction problems continuous constraint satisfaction problems are at the core of many real world applications including planning scheduling control and diagnosis of physical systems such as car planes and factories yet in order to be effective in such resource constrained environments constraint based techniques must take into account the complexity of continuous constrained problems in this paper we study complexity phase transition phenomena of continuous constraint satisfaction problems csps first we analyze three continuous constraint satisfaction formulations based on discrete 3 sat problems which have a strong relation between structure and search cost then we propose a generic benchmarking model for comparing continuous csps and algorithms and present two example problems based on sine functions in our experiments these problems are solved using both local and global search methods besides comparing the complexities of different continuous csps and search algorithms we also connect these back to results from similar studies on sat problems in solving continuous 3 sat and sine based csps we find that the median search cost is characterized by simple parameters such as the constraint to variable ratio and constraint tightness and that discrete search algorithms such as gsat have continuous counter parts with similar behavior regarding local versus global search techniques for constraint solving our results show that local search methods are more efficient for weakly constrained problems whereas global search methods work better on highly constrained problems

direct search methods then and now we discuss direct search methods for unconstrained optimization we give a modern perspective on this classical family of derivative free algorithms focusing on the development of direct search methods during their golden age from 1960 to 1971 we discuss how direct search methods are characterized by the absence of the construction of a model of the objective we then consider a number of the classical direct search methods and discuss what research in the intervening years has uncovered about these algorithms in particular while the original direct search methods were consciously based on straightforward heuristics more recent analysis has shown that in most but not all cases these heuristics actually suffice to ensure global convergence of at least one subsequence of the sequence of iterates to a first order stationary point of the objective function

asymptotic properties of some triangulations of the sphere in this paper we analyse a method for triangulating the sphere originally proposed by baumgardner and frederickson in 1985 the method is essentially a refinement procedure for arbitrary spherical triangles that fit into a hemisphere refinement is carried out by dividing each triangle into four by introducing the midpoints of the edges as new vertices and connecting them in the usual red way we show that this process can be described by a sequence of piecewise smooth mappings from a reference triangle onto the spherical triangle we then prove that the whole sequence of mappings is uniformly bi lipschitz and converges uniformly to a non smooth parameterization of the spherical triangle recovering the baumgardner and frederickson spherical barycentric coordinates we also prove that the sequence of triangulations is quasi uniform that is areas of triangles and lengths of the edges are roughly the same at each refinement level some numerical experiments confirm the theoretical results

on the convergence of grid based methods for unconstrained optimization the convergence of direct search methods for unconstrained minimization is examined in the case where the underlying method can be interpreted as a grid or pattern search over successively refined meshes an important aspect of the main convergence result is that translation rotation scaling and shearing of the successive grids are allowed

on the convergence of pattern search algorithms we introduce an abstract definition of pattern search methods for solving nonlinear unconstrained optimization problems our definition unifies an important collection of optimization methods that neither compute nor explicitly approximate derivatives we exploit our characterization of pattern search methods to establish a global convergence theory that does not enforce a notion of sufficient decrease our analysis is possible because the iterates of a pattern search method lie on a scaled translated integer lattice this allows us to relax the classical requirements on the acceptance of the step at the expense of stronger conditions on the form of the step and still guarantee global convergence

a simple based discontinuous galerkin solver for steady incompressible flows in this paper we present how the well known simple algorithm can be extended to solve the steady incompressible navier stokes equations discretized by the discontinuous galerkin method the convective part is discretized by the local lax friedrichs fluxes and the viscous part by the symmetric interior penalty method within the simple algorithm the equations are solved in an iterative process the discretized equations are linearized and an equation for the pressure is derived on the discrete level the equations obtained for each velocity component and the pressure are decoupled and therefore can be solved sequentially leading to an efficient solution procedure the extension of the proposed scheme to the unsteady case is straightforward where fully implicit time schemes can be used various test cases are carried out the poiseuille flow the channel flow with constant transpiration the kovasznay flow the flow into a corner and the backward facing step flow using a mixed order formulation i e order k for the velocity and order k 1 for the pressure the scheme is numerically stable for all test cases convergence rates of k 1 and k in the l 2 norm are observed for velocity and pressure respectively a study of the convergence behavior of the simple algorithm shows that no under relaxation for the pressure is needed which is in strong contrast to the application of the simple algorithm in the context of the finite volume method or the continuous finite element method we conclude that the proposed scheme is efficient to solve the steady incompressible navier stokes equations in the context of the discontinuous galerkin method comprising hp accuracy

pattern search methods for linearly constrained minimization we extend pattern search methods to linearly constrained minimization we develop a general class of feasible point pattern search algorithms and prove global convergence to a karush kuhn tucker point as in the case of unconstrained minimization pattern search methods for linearly constrained problems accomplish this without explicit recourse to the gradient or the directional derivative of the objective key to the analysis of the algorithms is the way in which the local search patterns conform to the geometry of the boundary of the feasible region

minimizing roundoff errors of prefix sums via dynamic construction of huffman trees the prefix sum operation which returns all prefix sums on a sequence of numbers plays an important role in many applications we study how to efficiently evaluate prefix sums on positive floating point numbers such that the worst case roundoff error of each sum is minimized a direct approach to this problem builds a huffman tree for each prefix subsequence from scratch requiring exactly quadratic time for every input x we can do better by taking advantage of the current huffman tree to build the next huffman tree using dynamic insertions and deletions on huffman trees consequently subquadratic time suffices for various input patterns we also provide experimental comparisons of all the algorithms studied in this paper on inputs that are randomly and uniformly generated

a switching method for nonlinear systems a new iterative method is proposed for the solution of nonlinear systems the method does not use explicit derivative informations and at each iteration automatically selects one of two distinct iterative schemes a direct search method and a damped approximate newton s method so the method is referred as switching method it is shown that the method is a global method with quadratic convergence numerical results show the very good practical performance of the method

energy dependent mesh adaptivity of discontinuous isogeometric discrete ordinate methods with dual weighted residual error estimators in this paper a hanging node discontinuous galerkin isogeometric discretisation of the multigroup discrete ordinates sn equations is presented in which each energy group has its own mesh the equations are discretised using non uniform rational b splines nurbs which allows the coarsest mesh to exactly represent the geometry for a wide range of engineering problems of interest this would not be the case using straight sided finite elements information is transferred between meshes via the construction of a supermesh this is a non trivial task for two arbitrary meshes but is significantly simplified here by deriving every mesh from a common coarsest initial mesh in order to take full advantage of this flexible discretisation goal based error estimators are derived for the multigroup discrete ordinates equations with both fixed extraneous and fission sources and these estimators are used to drive an adaptive mesh refinement amr procedure the method is applied to a variety of test cases for both fixed and fission source problems the error estimators are found to be extremely accurate for linear nurbs discretisations with degraded performance for quadratic discretisations owing to a reduction in relative accuracy of the exact adjoint solution required to calculate the estimators nevertheless the method seems to produce optimal meshes in the amr process for both linear and quadratic discretisations and is 100 more accurate than uniform refinement for the same amount of computational effort for a 67 group deep penetration shielding problem

finding local optima of high dimensional functions using direct search methods this paper focuses on a subclass of box constrained non linear optimization problems we are particularly concerned with settings where gradient information is unreliable or too costly to calculate and the function evaluations themselves are very costly this encourages the use of derivative free optimization methods and especially a subclass of these referred to as direct search methods the thrust of our investigation is twofold first we implement and evaluate a number of traditional direct search methods according to the premise that they should be suitable as local optimizers when used in a metaheuristic framework second we introduce a new direct search method based on scatter search designed to remedy the lack of a good derivative free method for solving problems of high dimensions our new direct search method has convergence properties comparable to those of existing methods in addition to being able to solve larger problems more effectively

skew symmetric nitsche s formulation in isogeometric analysis dirichlet and symmetry conditions patch coupling and frictionless contact abstract a simple skew symmetric nitsche s formulation is introduced into the framework of isogeometric analysis iga to deal with various problems in small strain elasticity essential boundary conditions symmetry conditions for kirchhoff plates patch coupling in statics and in modal analysis as well as signorini contact conditions for linear boundary or interface conditions the skew symmetric formulation is parameter free for contact conditions it remains stable and accurate for a wide range of the stabilization parameter several numerical tests are performed to illustrate its accuracy stability and convergence performance we investigate particularly the effects introduced by nitsche s coupling including the convergence performance and condition numbers in statics as well as the extra outlier frequencies and corresponding eigenmodes in structural dynamics we present the hertz test the block test and a 3d self contact example showing that the skew symmetric nitsche s formulation is a suitable approach to simulate contact problems in iga

pattern search methods for user provided points application to molecular geometry problems this paper deals with the application of pattern search methods to the numerical solution of a class of molecular geometry problems with important applications in molecular physics and chemistry the goal is to find a configuration of a cluster or a molecule with minimum total energy the minimization problems in this class of molecular geometry problems have no constraints and the objective function is smooth the difficulties arise from the existence of several local minima and especially from the expensive function evaluation total energy and the possible nonavailability of first order derivatives we introduce a pattern search approach that attempts to exploit the physical nature of the problem by using energy lowering geometrical transformations and to take advantage of parallelism without the use of derivatives numerical results for a particular instance of this new class of pattern search methods are presented showing the promise of our approach the new pattern search methods can be used in any other context where there is a user provided scheme to generate points leading to a potential objective function decrease

performance of discontinuous galerkin methods for elliptic pdes in this paper we compare the performance of several discontinuous galerkin dg methods for elliptic partial differential equations pdes on a model problem theoretical estimates of the condition number of the stiffness matrix are given for dg methods whose bilinear form is symmetric and which are shown to be numerically sharp then the efficiency of the methods in the computation of both the potential and its gradient is tested on unstructured triangular meshes

h p finite element trace inequalities for the pyramid polynomial trace inequalities typically involve an unknown constant depending on the order n of the polynomial the dependence of these constants on n was made more explicit in warburton and hesthaven 2003 for the general d simplex and they were determined for quadrilaterals hexes and wedges in hillewaert 2013 in this note we derive explicit expressions for the constant in the trace inequality for two sets of basis functions on the pyramid

generalized pattern search methods for a class of nonsmooth optimization problems with structure we propose a generalized pattern search gps method to solve a class of nonsmooth minimization problems where the set of nondifferentiability is included in the union of known hyperplanes and therefore is highly structured both unconstrained and linearly constrained problems are considered at each iteration the set of poll directions is enforced to conform to the geometry of both the nondifferentiability set and the boundary of the feasible region near the current iterate this is the key issue to guarantee the convergence of certain subsequences of iterates to points which satisfy first order optimality conditions numerical experiments on some classical problems validate the method

a direct search algorithm for optimization with noisy function evaluations we consider the unconstrained optimization of a function when each function evaluation is subject to a random noise we assume that there is some control over the variance of the noise term in the sense that additional computational effort will reduce the amount of noise this situation may occur when function evaluations involve simulation or the approximate solution of a numerical problem it also occurs in an experimental setting when averaging repeated observations at the same point can lead to a better estimate of the underlying function value we describe a new direct search algorithm for this type of problem we prove convergence of the new algorithm when the noise is controlled so that the standard deviation of the noise approaches zero faster than the step size we also report some numerical results on the performance of the new algorithm

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

clustering tree structured data on manifold tree structured data usually contain both topological and geometrical information and are necessarily considered on manifold instead of euclidean space for appropriate data parameterization and analysis in this study we propose a novel tree structured data parameterization called topology attribute matrix t a matrix so the data clustering task can be conducted on matrix manifold we incorporate the structure constraints embedded in data into the non negative matrix factorization method to determine meta trees from the t a matrix and the signature vector of each single tree can then be extracted by meta tree decomposition the meta tree space turns out to be a cone space in which we explore the distance metric and implement the clustering algorithm based on the concepts like frechet mean finally the t a matrix based clustering tambac framework is evaluated and compared using both simulated data and real retinal images to illustrate its efficiency and accuracy

optimization by direct search new perspectives on some classical and modern methods direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis nonetheless users remained loyal to these methods most of which were easy to program some of which were reliable in the past fifteen years these methods have seen a revival due in part to the appearance of mathematical analysis as well as to interest in parallel and distributed com puting this review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results the underlying principles allow general ization to handle bound constraints and linear constraints we also discuss extensions to problems with nonlinear constraints

direct search methods on parallel machines direct search methods are methods designed to solve unconstrained minimization problems of the form mathop min limits x in mathbb r n f x where f n these methods are distinguished by the fact that they neither use nor require explicit derivative information the search for a local minimizer is driven solely by function information popular methods in this class include the factorial design algorithm of box 1 the pattern search algorithm of hooke and jeeves 4 and the simplex method of neider and mead 5

two level additive schwarz methods for a discontinuous galerkin approximation of second order elliptic problems we present some two level nonoverlapping and overlapping additive schwarz methods for a discontinuous galerkin method for solving second order elliptic problems it is shown that the condition numbers of the preconditioned systems are of the order o frac h h for the nonoverlapping schwarz methods and of the order o frac h delta for the overlapping schwarz methods where h and h stand for the fine mesh size and the coarse mesh size respectively and delta denotes the size of the overlaps between subdomains numerical experiments are provided to gauge the efficiency of the methods and to validate the theory

detection and remediation of stagnation in the nelder mead algorithm using a sufficient decrease condition the nelder mead algorithm can stagnate and converge to a nonoptimal point even for very simple problems in this note we propose a test for sufficient decrease which if passed for all iterations will guarantee convergence of the nelder mead iteration to a stationary point if the objective function is smooth and the diameters of the nelder mead simplices converge to zero failure of this condition is an indicator of potential stagnation as a remedy we propose a new step which we call an oriented restart that reinitializes the simplex to a smaller one with orthogonal edges whose orientation is determined by an approximate descent direction from the current best point we also give results that apply when the objective function is a low amplitude perturbation of a smooth function we illustrate our results with some numerical examples

a discontinuous galerkin method for the mono energetic fokker planck equation based on a spherical interior penalty formulation we present a new discretization of the mono energetic fokker planck equation we build on previous work kophazi and lathouwers 2015 where we devised an angular discretization for the boltzmann equation allowing for both heterogeneous and anisotropic angular refinement the angular discretization is based on a discontinuous finite element method on the unit sphere here we extend the methodology to include the effect of the fokker planck scatter operator describing small angle particle scatter we describe the construction of an interior penalty method on the sphere surface results are provided for a variety of test cases ranging from purely angular to fully three dimensional the results show that the scheme can resolve highly forward peaked flux distributions with forward peaked scatter

an interior penalty stabilised incompressible discontinuous galerkin fourier solver for implicit large eddy simulations abstract we present an implicit large eddy simulation iles h p high order 2 unstructured discontinuous galerkin fourier solver with sliding meshes the solver extends the laminar version of ferrer and willden 2012 34 to enable the simulation of turbulent flows at moderately high reynolds numbers in the incompressible regime this solver allows accurate flow solutions of the laminar and turbulent 3d incompressible navier stokes equations on moving and static regions coupled through a high order sliding interface the spatial discretisation is provided by the symmetric interior penalty discontinuous galerkin ip dg method in the x y plane coupled with a purely spectral method that uses fourier series and allows efficient computation of spanwise periodic three dimensional flows since high order methods e g discontinuous galerkin and fourier are unable to provide enough numerical dissipation to enable under resolved high reynolds computations i e as necessary in the iles approach we adapt the laminar version of the solver to increase controllably the dissipation and enhance the stability in under resolved simulations the novel stabilisation relies on increasing the penalty parameter included in the dg interior penalty ip formulation the latter penalty term is included when discretising the linear viscous terms in the incompressible navier stokes equations these viscous penalty fluxes substitute the stabilising effect of non linear fluxes which has been the main trend in implicit les discontinuous galerkin approaches the ip dg penalty term provides energy dissipation which is controlled by the numerical jumps at element interfaces e g large in under resolved regions such as to stabilise under resolved high reynolds number flows this dissipative term has minimal impact in well resolved regions and its implicit treatment does not restrict the use of large time steps thus providing an efficient stabilization mechanism for iles the ip dg stabilisation is complemented with a spectral vanishing viscosity svv method in the z direction to enhance stability in the continuous fourier space the coupling between the numerical viscosity in the dg plane and the svv damping provides an efficient approach to stabilise high order methods at moderately high reynolds numbers we validate the formulation for three turbulent flow cases a circular cylinder at re 3900 a static and pitch oscillating naca 0012 airfoil at re 10000 and finally a rotating vertical axis turbine at re 40000 with reynolds based on the circular diameter airfoil chord and turbine diameter respectively all our results compare favourably with published direct numerical simulations large eddy simulations or experimental data we conclude that the dg fourier high order solver with ip svv stabilisation proves to be a valuable tool to predict turbulent flows and associated statistics for both static and rotating machinery

a space angle dgfem approach for the boltzmann radiation transport equation with local angular refinement in this paper a new method for the discretization of the radiation transport equation is presented based on a discontinuous galerkin method in space and angle that allows for local refinement in angle where any spatial element can support its own angular discretization to cope with the discontinuous spatial nature of the solution a generalized riemann procedure is required to distinguish between incoming and outgoing contributions of the numerical fluxes a new consistent framework is introduced that is based on the solution of a generalized eigenvalue problem the resulting numerical fluxes for the various possible cases where neighboring elements have an equal higher or lower level of refinement in angle are derived based on tensor algebra and the resulting expressions have a very clear physical interpretation the choice of discontinuous trial functions not only has the advantage of easing local refinement it also facilitates the use of efficient sweep based solvers due to decoupling of unknowns on a large scale thereby approaching the efficiency of discrete ordinates methods with local angular resolution the approach is illustrated by a series of numerical experiments results show high orders of convergence for the scalar flux on angular refinement the generalized riemann upwinding procedure leads to stable and consistent solutions further the sweep based solver performs well when used as a preconditioner for a krylov method

a derivative free multidimensional optimal search method using lucas number we consider a unimodal continuous function of n variables over an n dimensional rectangular parallelopiped and explain a search method to find the domain of uncertainty which contains the point that maximizes the function our search method does not require differentiability assumption the function is projected along each dimension an algorithm is developed to eliminate the subset of the domain at each step which does not contain the solution lucas numbers are used in the elimination process

an angular multigrid preconditioner for the radiation transport equation with fokker planck scattering in a previous paper hennink and lathouwers 2017 we developed a finite element discretization for the boltzmann transport equation with forward peaked scatter modeled by the fokker planck approximation the discretization was based on the discontinuous galerkin method in both space and angle it was expected and found that the regular source iteration algorithm for the boltzmann equation is not effective in solving the discretized system and becomes excessively expensive for problems with many angular degrees of freedom the purpose of this paper is to develop a multigrid scheme as preconditioner for the above mentioned discretization the method exploits the nested nature of the meshes and the natural prolongation restriction between meshes by galerkin projection a set of test problems ranging from pure spherical diffusion to the complete boltzmann transport problem in 3d are presented to illustrate that the method is very effective resulting in iteration counts nearly independent of problem size even for highly non isotropically refined angular meshes

new sequential and parallel derivative free algorithms for unconstrained minimization this paper presents sequential and parallel derivative free algorithms for finding a local minimum of smooth and nonsmooth functions of practical interest it is proved that under mild assumptions a sufficient decrease condition holds for a nonsmooth function based on this property the algorithms explore a set of search directions and move to a point with a sufficiently lower functional value if the function is strictly differentiable at its limit points a sub sequence of points generated by the algorithm converges to a first order stationary point nabla f x 0 if the function is convex around its limit points convergence of a subsequence to a point with nonnegative directional derivatives on a set of search directions is ensured preliminary numerical results on sequential algorithms show that they compare favorably with the recently introduced pattern search methods

on the constants in hp finite element trace inverse inequalities we derive inverse trace inequalities for hp finite elements utilizing orthogonal polynomials we show how to derive explicit expressions for the constants when considering triangular and tetrahedral elements we also discuss how to generalize this technique to the general d simplex

short note an explicit expression for the penalty parameter of the interior penalty method we derive an explicit expression for the penalty parameter of the interior penalty method for elliptic problems the expression yields a coercive bilinear form and is valid for general meshes comprising of geometrically nonconforming simplical elements

estimating the condition number of the frechet derivative of a matrix function the frechet derivative l f of a matrix function f mathbb c n times n to mathbb c n times n is used in a variety of applications and several algorithms are available for computing it we define a condition number for the frechet derivative and derive upper and lower bounds for it that differ by at most a factor 2 for a wide class of functions we derive an algorithm for estimating the 1 norm condition number that requires o n 3 flops given o n 3 flops algorithms for evaluating f and l f in practice it produces estimates correct to within a factor 6n numerical experiments show the new algorithm to be much more reliable than a previous heuristic estimate of conditioning

on relevant dimensions in kernel feature spaces we show that the relevant information of a supervised learning problem is contained up to negligible error in a finite number of leading kernel pca components if the kernel matches the underlying learning problem in the sense that it can asymptotically represent the function to be learned and is sufficiently smooth thus kernels do not only transform data sets such that good generalization can be achieved using only linear discriminant functions but this transformation is also performed in a manner which makes economical use of feature space dimensions in the best case kernels provide efficient implicit representations of the data for supervised learning problems practically we propose an algorithm which enables us to recover the number of leading kernel pca components relevant for good classification our algorithm can therefore be applied 1 to analyze the interplay of data set and kernel in a geometric fashion 2 to aid in model selection and 3 to denoise in feature space in order to yield better classification results

an introduction to kernel based learning algorithms this paper provides an introduction to support vector machines kernel fisher discriminant analysis and kernel principal component analysis as examples for successful kernel based learning methods we first give a short background about vapnik chervonenkis theory and kernel feature spaces and then proceed to kernel based learning in supervised and unsupervised scenarios including practical and algorithmic considerations we illustrate the usefulness of kernel algorithms by discussing applications such as optical character recognition and dna analysis

sparse adaptive finite elements for radiative transfer the linear radiative transfer equation a partial differential equation for the radiation intensity u x s with independent variables email protected email protected r n in the physical domain d of dimension n 2 3 and angular variable email protected s 2 email protected r 3 y 1 is solved in the n 2 dimensional computational domain dxs 2 we propose an adaptive multilevel galerkin finite element method fem for its numerical solution our approach is based on a a stabilized variational formulation of the transport operator b on so called sparse tensor products of two hierarchic families of finite element spaces in h 1 d and in l 2 s 2 respectively and c on wavelet thresholding techniques to adapt the discretization to the underlying problem an a priori error analysis shows under strong regularity assumptions on the solution that the sparse tensor product method is clearly superior to a discrete ordinates method as it converges with essentially optimal asymptotic rates while its complexity grows essentially only as that for a linear transport problem in r n numerical experiments for n 2 on a set of example problems agree with the convergence and complexity analysis of the method and show that introducing adaptivity can improve performance in terms of accuracy vs number of degrees even further

estimating the support of a high dimensional distribution suppose you are given some data set drawn from an underlying probability distribution p and you want to estimate a simple subset s of input space such that the probability that a test point drawn from p lies outside of s equals some a priori specified value between 0 and 1 we propose a method to approach this problem by trying to estimate a function f that is positive on s and negative on the complement the functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data it is regularized by controlling the length of the weight vector in an associated feature space the expansion coefficients are found by solving a quadratic programming problem which we do by carrying out sequential optimization over pairs of input patterns we also provide a theoretical analysis of the statistical performance of our algorithm the algorithm is a natural extension of the support vector algorithm to the case of unlabeled data

estimation of penalty parameters for symmetric interior penalty galerkin methods this paper presents computable lower bounds of the penalty parameters for stable and convergent symmetric interior penalty galerkin methods in particular we derive the explicit dependence of the coercivity constants with respect to the polynomial degree and the angles of the mesh elements numerical examples in all dimensions and for different polynomial degrees are presented we show the numerical effects of loss of coercivity

a globally convergent augmented lagrangian pattern search algorithm for optimization with general constraints and simple bounds we give a pattern search method for nonlinearly constrained optimization that is an adaption of a bound constrained augmented lagrangian method first proposed by conn gould and toint siam j numer anal 28 1991 pp 545 572 in the pattern search adaptation we solve the bound constrained subproblem approximately using a pattern search method the stopping criterion proposed by conn gould and toint for the solution of the subproblem requires explicit knowledge of derivatives such information is presumed absent in pattern search methods however we show how we can replace this with a stopping criterion based on the pattern size in a way that preserves the convergence properties of the original algorithm in this way we proceed by successive inexact bound constrained minimization without knowing exactly how inexact the minimization is as far as we know this is the first provably convergent direct search method for general nonlinear programming

a two mesh adaptive mesh refinement technique for sn neutral particle transport using a higher order dgfem we present a two mesh adaptive mesh refinement technique for the s n neutral particle transport equation in 2d a high order up to order 4 discontinuous galerkin dg finite element discretization is employed with standard upwinding a suitable diffusion synthetic acceleration dsa scheme is derived to precondition the transport solves the dsa scheme also based on a dg discretization is derived directly from the discretized high order dg s n transport equation results are presented to demonstrate the superiority of amr for dsa accelerated s n transport solves

minimising the error in eigenvalue calculations involving the boltzmann transport equation using goal based adaptivity on unstructured meshes this article presents a method for goal based anisotropic adaptive methods for the finite element method applied to the boltzmann transport equation the neutron multiplication factor k e f f is used as the goal of the adaptive procedure the anisotropic adaptive algorithm requires error measures for k e f f with directional dependence general error estimators are derived for any given functional of the flux and applied to k e f f to acquire the driving force for the adaptive procedure the error estimators require the solution of an appropriately formed dual equation forward and dual error indicators are calculated by weighting the hessian of each solution with the dual and forward residual respectively the hessian is used as an approximation of the interpolation error in the solution which gives rise to the directional dependence the two indicators are combined to form a single error metric that is used to adapt the finite element mesh the residual is approximated using a novel technique arising from the sub grid scale finite element discretisation two adaptive routes are demonstrated i a single mesh is used to solve all energy groups and ii a different mesh is used to solve each energy group the second method aims to capture the benefit from representing the flux from each energy group on a specifically optimised mesh the k e f f goal based adaptive method was applied to three examples which illustrate the superior accuracy in criticality problems that can be obtained

linear time approximation algorithms for computing numerical summation with provably small errors given a multiset x x1 xn of real numbers the floating point set summation problem asks for sn x1 xn let e n denote the minimum worst case error over all possible orderings of evaluating sn we prove that if x has both positive and negative numbers it is np hard to compute sn with the worst case error equal to e n we then give the first known polynomial time approximation algorithm that has a provably small error for arbitrary x our algorithm incurs a worst case error at most 2 lceil log n 1 rceil 1 e n all logarithms log in this paper are base 2 after x is sorted it runs in o n time for the case where x is either all positive or all negative we give another approximation algorithm with a worst case error at most lceil log log n rceil e n even for unsorted x this algorithm runs in o n time previously the best linear time approximation algorithm had a worst case error at most lceil log n rceil e n while e n was known to be attainable in o n log n time using huffman coding

mixed hp dgfem for incompressible flows we consider several mixed discontinuous galerkin approximations of the stokes problem and propose an abstract framework for their analysis using this framework we derive a priori error estimates for hp approximations on tensor product meshes we also prove a new stability estimate for the discrete divergence bilinear form