a dc stable well balanced calderon preconditioned time domain electric field integral equation the marching on in time mot solution of the time domain electric field integral equation td efie has traditionally suffered from a number of problems including 1 instability 2 spurious static contributions plaguing the solution 3 low frequency breakdown and 4 dense discretization breakdown the first issue can be resolved by employing proper space time galerkin discretization schemes and accurate quadrature methods the second and the third issue have been resolved by the quasi helmholtz projected td efie qhp tdefie this contribution introduces a multiplicative preconditioner which can be applied to the qhp tdefie without further modifying the original scheme this preconditioner is based on calderon techniques and guarantees that the mot system can be solved efficiently using iterative methods not only for large time step sizes but also for dense spatial discretizations and for both simply and multiply connected geometries

analysis and regularization of the td efie low frequency breakdown low frequency breakdown phenomenon plaguing marching on in time solutions of time domain electric field integral equations is analyzed and cured a theoretical analysis of the breakdown identifies deficiencies in the treatments proposed to date a scheme for eliminating low frequency breakdown phenomena that leverages hierarchical regularization is presented it applies to arbitrary meshes and gives rise to linear systems that are better conditioned than those obtained with standard loop star tree bases the scheme s analytical properties and numerical results demonstrate its efficacy in accelerating the iterative solution of the time domain electric field integral equations

preconditioned mixed spectral element methods for elasticity and stokes problems this paper continues our study of efficient preconditioners for the indefinite systems obtained by discretizing the linear elasticity and stokes problems with mixed spectral methods in three dimensions our study is extended here in two new directions to include i more general multidomain mixed spectral element methods and ii a new class of block triangular preconditioners the stiffness matrices resulting from stokes and mixed elasticity problems have the structure of saddle point problems with a penalty term which is associated with the poisson ratio for elasticity problems or with stabilization techniques for stokes problems the main result of this paper shows that the convergence rate of the proposed preconditioned iterative methods is independent of the penalty parameter nu the number of spectral elements n and mildly dependent on the spectral degree n via the inf sup constant the number of iterations required by the block triangular preconditioner is much smaller than the one required by the block diagonal preconditioner on the other hand the block triangular preconditioner is no longer symmetric and has a small extra cost these results and several numerical experiments presented in the final section show that these algorithms are a practical and efficient strategy for the iterative solution of the indefinite systems arising from mixed spectral element discretizations of elliptic systems

moment information for probability distributions without solving the moment problem ii main mass tails and shape approximation how much information does a small number of moments carry about the unknown distribution function is it possible to explicitly obtain from these moments some useful information e g about the support the modality the general shape or the tails of a distribution without going into a detailed numerical solution of the moment problem in this previous and subsequent papers clear and easy to implement answers will be given to some questions of this type first the question of how to distinguish between the main mass interval and the tail regions in the case we know only a number of moments of the target distribution function will be addressed the answer to this question is based on a version of the chebyshev stieltjes markov inequality which provides us with upper and lower moment based bounds for the target distribution then exploiting existing asymptotic results in the main mass region an explicit moment based approximation of the target probability density function is provided although the latter cannot be considered in general as a satisfactory solution it can always serve as an initial approximation in any iterative scheme for the numerical solution of the moment problem numerical results illustrating all the theoretical statements are also presented

the discrete korn s type inequality in subspaces and iterative methods for thin elastic structures abstract when iterative methods are applied to thin elastic structures such as shells plates rods arches and beams they suffer from slow convergence with diminishing thickness to overcome this difficulty the authors have introduced the fundamental concept of the korn s type inequality in subspaces see proc r soc lond a 453 1997 2003 2016 here we present the discrete korn s type inequality in subspaces which when applied to discrete methods such as the finite element method enables the design of iterative algorithms for thin elastic structures with convergence rate independent of both the thickness and the discretisation parameters as a paradigm we consider the steepest descent method with the subspace correction preconditioning and present p version finite element results for a model shell problem which show that a simple modification of this method based on the discrete korn s type inequality in subspaces yields a radical improvement in the convergence

a lbm dlm fd method for 3d fluid structure interactions the previously developed lbm dlm fd method derived from the lattice boltzmann method and the distributed lagrange multiplier fictitious domain method is extended to deal with 3d fluid structure interactions in our current algorithm the fluid motion is solved by lbm the deformation of the solid body is solved by the finite element method and the lagrange multiplier is solved on the low order mesh three numerical examples are employed to validate the lbm dlm fd method and reveal the potential of the method to deal with the fluid elastic body interaction problems

time domain adaptive integral method for surface integral equations an efficient marching on in time mot scheme is presented for solving electric magnetic and combined field integral equations pertinent to the analysis of transient electromagnetic scattering from perfectly conducting surfaces residing in an unbounded homogenous medium the proposed scheme is the extension of the frequency domain adaptive integral pre corrected fast fourier transform fft method to the time domain fields on the scatterer that are produced by space time sources residing on its surface are computed 1 by locally projecting for each time step all sources onto a uniform auxiliary grid that encases the scatterer 2 by computing everywhere on this grid the transient fields produced by the resulting auxiliary sources via global multilevel blocked space time ffts 3 by locally interpolating these fields back onto the scatterer surface as this procedure is inaccurate when source and observer points reside close to each other and 4 near fields are computed classically albeit pre corrected for errors introduced through the use of global ffts the proposed scheme has a computational complexity and memory requirement of o n sub t n sub s log sup 2 n sub s and o n sub s sup 3 2 when applied to quasiplanar structures and of o n sub t n sub s sup 3 2 log sup 2 n sub s and o n sub s sup 2 when used to analyze scattering from general surfaces here n sub s and n sub t denote the number of spatial and temporal degrees of freedom of the surface current density these computational cost and memory requirements are contrasted to those of classical mot solvers which scale as o n sub t n sub s sup 2 and o n sub s sup 2 respectively a parallel implementation of the scheme on a distributed memory computer cluster that uses the message passing interface is described simulation results demonstrate the accuracy efficiency and the parallel performance of the implementation

on the application of maximum entropy to the moments problem the maximum entropy approach to the solution of classical inverse problem of moments in which one seeks to reconstruct a function p x where x 0 from the values of a finite set n 1 of its moments is studied it is shown that for n 4 such a function always exists while for n 2 and n 3 the acceptable values of the moments are singled out analytically the paper extends to the general case where the results were previously bounded to the case n 2

two level domain decomposition preconditioning for the p version finite element method in three dimensions this paper presents a flexible approach to domain decomposition preconditioning for problems expressed in terms of local stiffness matrices of subdomains evaluation of the preconditioner requires fully parallelizable local computations and the solution of a global auxiliary problem with few variables per subdomain convergence factors can be bounded using properties of the local stiffness matrices only the method is applied to the p version finite element method for three dimensional elasticity we treat each element as a subdomain and compute explicit convergence bounds

moment information and entropy evaluation for probability densities abstract how much information does the sequence of integer moments carry about the corresponding unknown absolutely continuous distribution we prove that a reliable evaluation of the corresponding shannon entropy can be done by exploiting some known theoretical results on the entropy convergence uniquely involving exact moments without solving the underlying moment problem all the procedure essentially rests on the solution of linear systems with nearly singular matrices and hence it requires both calculations in high precision and a pre conditioning technique numerical examples are provided to support the theoretical results

numerical inversion of the mellin transform on the real line for heavy tailed probability density functions a method for numerical inversion on the real line of the mellin transform without reduction of the problem to the inversion of laplace transform is described maximum entropy technique is invoked in choosing the analytical form of the approximant function entropy convergence and then l 1 norm convergence is proved a stability analysis in evaluating entropy and expected values is illustrated an upper bound of the error in the expected values computation is provided in terms of entropy

the problem of selecting the shape functions for a p type finite element abstract the paper addresses the question of the optimal selection of the shape functions for p type finite elements and discusses the effectivity of the conjugate gradient and multilevel iteration method for solving the corresponding linear system the selection of the shape functions is of major importance for the performance of the solver based on iterative methods neither the theory nor practice of the optimal selection of the shape functions is available yet we have seen that the condensation approach which has obvious advantages from the point of parallel computations is a very effective tool for keeping the condition number under the control and is especially advantageous for the conjugate gradient method

fast analysis of large finite arrays with a combined multiresolution sm aim approach we present a synthesis of the sparse matrix adaptive integral method sm aim and the multiresolution mr approach for the analysis of electrically large finite arrays with planar or 3 d radiating elements the two methods were separately introduced previously the use of the mr has the effect of a preconditioner and speeds up the convergence rate of the sm aim of almost two orders of magnitude with a total reduction of the numerical complexity with respect to the standard mom of almost three orders of magnitude

a multilevel matrix decomposition algorithm for analyzing scattering from large structures a multilevel algorithm is presented for analyzing scattering from electrically large surfaces the algorithm accelerates the iterative solution of integral equations that arise in computational electromagnetics the algorithm permits a fast matrix vector multiplication by decomposing the traditional method of moment matrix into a large number of blocks with each describing the interaction between distant scatterers the multiplication of each block by a trial solution vector is executed using a multilevel scheme that resembles a fast fourier transform fft and that only relies on well known algebraic techniques the computational complexity and the memory requirements of the proposed algorithm are o n log sup 2 n

a domain decomposition algorithm for elliptic problems in three dimensions most domain decomposition algorithms have been developed for problems in two dimensions one reason for this is the difficulty in devising a satisfactory easy to implement robust method of providing global communication of information for problems in three dimensions several methods that work well in two dimension do not perform satisfactorily in three dimensions a new iterative substructuring algorithm for three dimensions is proposed it is shown that the condition number of the resulting preconditioned problem is bounded independently of the number of subdomains and that the growth is quadratic in the logarithm of the number of degrees of freedom associated with a subdomain the condition number is also bounded independently of the jumps in the coefficients of the differential equation between subdomains the new algorithm also has more potential parallelism than the iterative substructuring methods previously proposed for problems in three dimensions

a marching on in time hierarchical scheme for the solution of the time domain electric field integral equation a spatially hierarchical current expansion is used to construct a marching on in time based time domain electric field integral equation solver the formulation gives rise to well conditioned linear systems and is immune to time domain low frequency breakdown phenomena numerical results that demonstrate the effectiveness of the proposed hierarchical approach in comparison with existing technologies are presented

a note on proximity of distributions in terms of coinciding moments a bound on the absolute difference between two distributions based on entropy is introduced such a bound can be made as precise as desired according to the number of common moments assumed the approximating distribution is obtained by maximizing the entropy constrained by moments

a numerical study on neumann neumann and feti methods for hp approximations on geometrically refined boundary layer meshes in two dimensions abstract we present extensive numerical tests showing the performance and robustness of certain balancing neumann neumann and one level feti methods for the solution of algebraic linear systems arising from hp finite element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in two dimensions the numerical results confirm the theoretical results derived in a toselli x vasseur technical report 02 15 seminar fur angewandte mathematik eth zurich september 2002 the condition numbers are independent of the aspect ratio of the mesh and of potentially large jumps of the coefficients in addition they only grow polylogarithmically with the polynomial degree as in the case of p approximations on shape regular meshes

multiple step random walks distribution functions from toeplitz matrices abstract toeplitz matrices are used to calculate the distribution function for a random walk on a lattice in continuous time where multiple hops covering an arbitrary number of lattice sites are allowed the method employed is similar to that used to calculate the complete end to end distance distribution for polymers

a multiresolution method of moments for triangular meshes this paper presents the construction use and properties of a multiresolution wavelet basis for the method of moments mom analysis of metal antennas scatterers and microwave circuits discretized by triangular meshes several application examples show fast convergence of iterative solvers and accurate solutions with highly sparse mom matrices the proposed basis is organized in hierarchical levels and keeps the different scales of the problem directly into the basis functions representation the current is divided into a solenoidal and a quasi irrotational part which allows mapping these two vector parts onto fully scalar quantities where the wavelets are defined as a byproduct this paper also presents a way to construct hierarchical sets of rao wilton glisson rwg functions on a family of meshes obtained by subsequent refinement i e with the rwg of coarser meshes expressed as linear combinations of those of finer meshes

loop star decomposition of basis functions in the discretization of the efie a general and readily applicable scheme is presented for the determination of the basis functions that allow the decomposition of the surface current into a solenoidal part and a nonsolenoidal remainder the proposed approach brings into correspondence these two parts with two scalar functions and generates the known loop and star basis functions the completeness of the loop star basis is discussed employing the presented scheme the issue of the irrotational property of the nonsolenoidal functions is addressed

lyapunov exponents and the natural invariant density determination of chaotic maps an iterative maximum entropy ansatz we apply the maximum entropy principle to construct the natural invariant density and the lyapunov exponent of one dimensional chaotic maps using a novel function reconstruction technique that is based on the solution of the hausdorff moment problem via maximizing shannon entropy we estimate the invariant density and the lyapunov exponent of nonlinear maps in one dimension from a knowledge of finite number of moments the accuracy and the stability of the algorithm are illustrated by comparing our results to a number of nonlinear maps for which the exact analytical results are available furthermore we also consider a very complex example for which no exact analytical result for the invariant density is available a comparison of our results to those available in the literature is also discussed

an iterative solver for p version finite elements in three dimensions abstract a new iterative method for systems arising from p version finite elements is presented and computational experience reported for conforming solid serendipity elements of order up to p 8 and real world structures with 5000 elements and over 1 million degrees of freedom the method uses a domain decomposition technique with each element taken to be a subdomain and a coarse space consisting of p 1 elements to overcome the i o bottleneck only access to a fast matrix vector multiplication instead of stiffness data is needed during iterations in terms of wall clock time the method outperforms on ibm rs 6000 an earlier method running on cray xmp

marching on in time solution of the time domain magnetic field integral equation using a predictor corrector scheme an explicit marching on in time mot scheme for solving the time domain magnetic field integral equation td mfie is presented the proposed mot td mfie solver uses rao wilton glisson basis functions for spatial discretization and a pe ce m type linear multistep method for time marching unlike previous explicit mot td mfie solvers the time step size can be chosen as large as that of the implicit mot td mfie solvers without adversely affecting accuracy or stability an algebraic stability analysis demonstrates the stability of the proposed explicit solver its accuracy and efficiency are established via numerical examples

on the convergence of moment problems we study the problem of estimating a nonnegative density given a finite number of moments such problems arise in numerous practical applications as the number of moments increases the estimates will always converge weak as measures but need not converge weakly in l this is related to the existence of functions on a compact metric space which are not essentially riemann integrable in some suitable sense we characterize the type of weak convergence we can expect in terms of riemann integrability and in some cases give error bounds when the estimates are chosen to minimize an objective function with weakly compact level sets such as the bolzmann shannon entropy they will converge weakly in l when an lp norm 1 p is used as the objective the estimates actually converge in norm these results provide theoretical support to the growing popularity of such methods in practice

a polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensions iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems a p version finite element method based on continuous piecewise q p functions is considered for second order elliptic problems in three dimensions this special method can also be viewed as a conforming spectral element method an iterative method is designed for which the condition number of the relevant operator grows only in proportion to 1 log p 2 this bound is independent of jumps in the coefficient of the elliptic problem across the interfaces between the subregions numerical results are also reported which support the theory

stieltjes moment problem via fractional moments stieltjes moment problem is considered to recover a probability density function from the knowledge of its infinite sequence of ordinary moments the approximate density is obtained through maximum entropy technique under the constraint of few fractional moments the latter are numerically obtained from the infinite sequence of ordinary moments and are chosen in such a way as to convey the maximum information content carried by the ordinary moments as a consequence a model with few parameters is obtained and intrinsic numerical instability is avoided it is proved that the approximate density is useful for calculating expected values and some other interesting probabilistic quantities

hierarchical bases for nonhierarchic 3 d triangular meshes we describe a novel basis of hierarchical multiscale functions that are linear combinations of standard rao wilton glisson rwg functions when the basis is used for discretizing the electric field integral equation efie for pec objects it gives rise to a linear system immune from low frequency breakdown and well conditioned for dense meshes the proposed scheme can be applied to any mesh with triangular facets and therefore it can be used as if it were an algebraic preconditioner the properties of the new system are confirmed by numerical results that show fast convergence rates of iterative solvers significantly better than those for the loop tree basis as a byproduct of the basis generation a generalization of the rwg functions to nonsimplex cells is introduced

parallel algebraic multigrids for structural mechanics this paper presents the results of a comparison of three parallel algebraic multigrid amg preconditioners for structural mechanics applications in particular we are interested in investigating both the scalability and robustness of the preconditioners numerical results are given for a range of structural mechanics problems with various degrees of difficulty and they show that algebraic multigrid methods can be applied successfully even to difficult problems in structural mechanics

the p version of the finite element method for three dimensional curved thin walled structures in this paper we present an implementation of a three dimensional p version for structural problems of solids with almost arbitrarily curved surfaces applying the blending function method complex structures can often be modelled by a few p elements being the basis for a higher order approximation numerical examples will demonstrate that the p version with anisotropic ansatz spaces allows to predict the structural behaviour of three dimensional plates and shells with approximately the same amount of degrees of freedom as in the two dimensional case yet significantly more accurate due to the three dimensional model furthermore it is advantageous to compute complex structures exclusively with three dimensional discretizations as no special elements are needed to model the transition from dimensionally reduced formulations like plates or shells to fully three dimensional solid elements using the p version with anisotropic ansatz spaces the whole structure can be efficiently discretized with solid elements even if the aspect ratio of the elements becomes very large copyright 2001 john wiley sons ltd

on the application of maximum entropy to the moments problem the use of the maximum entropy condition to obtain a solution to the moments problem in which one seeks to reconstruct a function from the values of a finite set of its moments is of interest in a variety of physical applications unfortunately the standard analysis which is available and on which these applications are based takes as its starting point precise values of the moments whereas the physical applications involve imprecise data in this paper it will be shown how to solve the moments problem using the maximum entropy condition starting from imprecise data values with errors specified by a 2 function

on the proximity of distributions in terms of coinciding fractional moments a bound on the absolute difference between two distributions based on entropy is introduced such a bound can be made as precise as desired according to the number of common fractional moments assumed the approximating distribution is obtained by maximizing the entropy constrained by fractional moments

spectral element method for three dimensional elliptic problems with smooth interfaces abstract in this paper we propose a least squares spectral element method for three dimensional elliptic interface problems the differentiability estimates and the main stability theorem using non conforming spectral element functions are proven the proposed method is free from any kind of first order reformulation a suitable preconditioner is constructed with help of the regularity estimate and proposed stability estimates which is used to control the condition number we show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them we obtain the error estimates which show the exponential accuracy of the method numerical results are obtained for both straight and curved interfaces to show the efficiency of the proposed method

loop star basis functions and a robust preconditioner for efie scattering problems an electric field integral equation efie formulation using the loop star basis functions has been developed for modeling plane wave scattering from perfect conducting objects a stability analysis at the dc limit shows that the use of the rao wilton glisson rwg basis functions results in a singular matrix operator however the use of the loop star basis functions results in a well conditioned matrix moreover a preconditioner constructed from a two step process based on near interactions and an incomplete factorization with a heuristic drop strategy has been proposed in conjunction with the conjugate gradient method to solve the resulting matrix equation the approach is shown to be effective for resolving both the low frequency instability and the bad conditioning of the efie method the computational complexity of the proposed approach is shown to be o n sup 2

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

the truncated stieltjes moment problem solved by using kernel density functions in this work the problem of the approximate numerical determination of a semi infinite supported continuous probability density function pdf from a finite number of its moments is addressed the target space is carefully defined and an approximation theorem is proved establishing that the set of all convex superpositions of appropriate kernel density functions kdfs is dense in this space a solution algorithm is provided based on the established approximate representation of the target pdf and the exploitation of some theoretical results concerning moment sequence asymptotics the solution algorithm also permits us to recover the tail behavior of the target pdf and incorporate this information in our solution a parsimonious formulation of the proposed solution procedure based on a novel sequentially adaptive scheme is developed enabling a very efficient moment data inversion the whole methodology is fully illustrated by numerical examples

overlapping schwarz methods for unstructured spectral elements a parallel and scalable domain decomposition method for unstructured and hybrid spectral element discretizations of elliptic problems is introduced and studied the spectral elements are affine images of the reference triangle or square in two dimensions and of the reference tetrahedron pyramid prism or cube in three dimensions the method is based on overlapping schwarz techniques applied to the schur complement of the discrete system and is implemented as a preconditioner for a krylov space method numerical results in two and three dimensions show that the iteration counts of our method are bounded by a constant independent of the spectral degree and the number of subdomains the resulting elliptic solver can be used in navier stokes simulations using the spectral element code nektar

numerical inversion of laplace transform on the real line from expected values a numerical method for inversion of laplace transform f s of probability densities given for real s 0 only is proposed fractional calculus allows us to have expected values from f s or f s then the maximum entropy technique provides the analytical form of the approximant fractional moments are mainly investigated entropy and cross entropy convergence are proved some numerical tests are illustrated

convergence of best entropy estimates given a finite number of moments of an unknown density on a finite measure space the best entropy estimate that nonnegative density x with the given moments which minimizes the boltzmann shannon entropy i x x log x is considered a direct proof is given that i has the kadec property in l if yn converges weakly to 37 and i yn converges to i 37 then yn converges to 37 in norm as a corollary it is obtained that as the number of given moments increases the best entropy estimates converge in li norm to the best entropy estimate of the limiting problem which is simply in the determined case furthermore for classical moment problems on intervals with strictly positive and sufficiently smooth error bounds and uniform convergence are actually obtained

the problem of membrane locking in finite element analysis of cylindrical shells we analyse the problem of membrane locking in h p finite element models of a thin hemicylindrical shell roof loaded by a smoothly varying normal pressure distribution we show that in the standard finite element method locking occurs especially at low values ofp and when the finite element grid is not aligned with the axis of the cylinder a general strategy of avoiding locking by using modified bilinear forms is introduced and a special implementation of this strategy on aligned rectangular grids is considered

entropy estimate of probability densities having assigned moments stieltjes case we consider an absolutely continuous positive random variable having assigned the first algebraic moments in terms of given moments is provided an upper bound for entropy for all the distributions admitting moment generating function

multiresolution analysis of printed antennas and circuits a dual isoscalar approach a new approach is presented for the integral equation analysis of multilayer printed circuits antennas and arrays it is based on the introduction of new multiresolution vector functions with properties similar to those of the scalar wavelets and flexible enough to accommodate for general shapes the resulting method of moments mom impedance matrix shows a good conditioning that allows its safe sparsification and the efficient use of iterative methods for the solution of the linear system the results of the analysis of various real life structures show the advantage of the method

polynomial programming lp relaxations also converge we consider the global minimization of a multivariate polynomial on a compact semialgebraic set using a result of krivine j analyse math 12 1964 pp 307 326 c r acad sci paris 258 1964 pp 3417 3418 it is shown that lp relaxations based on products of the original constraints in the spirit of the reformulation linearization technique procedure of sherali and adams converge to the global optimum this extends to compact semialgebraic sets previous results known for polytopes and 0 1 programs the asymptotic behavior of the lp relaxations is also analyzed and it is shown that in important cases finite convergence cannot occur

entropy estimate of a probability density from its mellin transform entropy estimate of heavy tailed probability distribution from its mellin transform is considered such an estimate is provided uniquely in terms of expected values obtained from mellin transform some particular cases are analytically investigated as a low number of expected values are available

maximum entropy and lacunary stieltjes moment problem the lacunary stieltjes moment problem in the framework of the maximum entropy approach is considered a proof for the existence conditions of the solution is provided

efficient preconditioning for the p version finite element method in two dimensions parallel preconditioners are formulated and analyzed for systems of equations arising from the p version finite element method applied to second order self adjoint elliptic boundary value problems in two dimensions by using new theoretical results for polynomial spaces it is proved that the condition number of the preconditioned system grows as log 2 p where p is the degree of the polynomial space numerical results are presented showing that the condition number indeed grows very slowly with p

distributionally robust optimization under moment uncertainty with application to data driven problems stochastic programming can effectively describe many decision making problems in uncertain environments unfortunately such programs are often computationally demanding to solve in addition their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters in this paper we propose a model that describes uncertainty in both the distribution form discrete gaussian exponential etc and moments mean and covariance matrix we demonstrate that for a wide range of cost functions the associated distributionally robust or min max stochastic program can be solved efficiently furthermore by deriving a new confidence region for the mean and the covariance matrix of a random vector we provide probabilistic arguments for using our model in problems that rely heavily on historical data these arguments are confirmed in a practical example of portfolio selection where our framework leads to better performing policies on the true distribution underlying the daily returns of financial assets

convergent sdp relaxations in polynomial optimization with sparsity we consider a polynomial programming problem p on a compact basic semialgebraic set k subset r n described by m polynomial inequalities g j x geq0 and with criterion f in r x we propose a hierarchy of semidefinite relaxations in the spirit of those of waki e siam j optim 17 2006 pp 218 242 in particular the sdp relaxation of order r has the following two features a the number of variables is o kappa 2r where kappa max kappa 1 kappa 2 with kappa 1 resp kappa 2 being the maximum number of variables appearing in the monomials of f resp appearing in a single constraint g j x geq0 b the largest size of the linear matrix inequalities lmis is o kappa r this is to compare with the respective number of variables o n 2r and lmi size o n r in the original sdp relaxations defined in j b lasserre siam j optim 11 2001 pp 796 817 therefore great computational savings are expected in case of sparsity in the data g j f i e when k is small a frequent case in practical applications of interest the novelty with respect to h waki s kim m kojima and m maramatsu siam j optim 17 2006 pp 218 242 is that we prove convergence to the global optimum of p when the sparsity pattern satisfies a condition often encountered in large size problems of practical applications and known as the running intersection property in graph theory in such cases and as a by product we also obtain a new representation result for polynomials positive on a compact basic semialgebraic set a sparse version of putinar s positivstellensatz m putinar indiana univ math j 42 1993 pp 969 984

real moments from moments and vice versa positive and negative real moments are calculated uniquely in terms of given moments when the underlying stieltjes moment problem is determinate as well under weak hypothese moments are calculated from real moments the approximate densities which allow us to calculate real moments and moments reciprocally are borrowed from maximum entropy or minimun directed divergence techniques

multivariate polynomial equations with multiple zeros solved by matrix eigenproblems the eigenproblem method calculates the solutions of systems of polynomial equations f 1 x 1 ldots x s 0 ldots f m x 1 ldots x s 0 it consists in fixing a suitable polynomial f and in considering the matrix a f corresponding to the mapping p mapsto f cdot p where the equivalence classes are modulo the ideal generated by f 1 ldots f m the eigenspaces contain vectors from which all solutions of the system can be read off this access was investigated in 1 and 16 mainly for the case that is nonderogatory in the present paper we study the case where f 1 ldots f m have multiple zeros in common we establish a kind of jordan decomposition of a f reflecting the multiplicity structure and describe the conditions under which a f is nonderogatory the algorithmic analysis of the eigenproblem in the general case is indicated

application of loop flower basis functions in the time domain electric field integral equation the loop flower basis functions are applied to get a low frequency stable time domain electric field integral equation td efie the quasi helmholtz decomposition is performed by the loop flower basis functions which are all defined based on mesh nodes to get a well posed equation a temporally differentiated form of the td efie is tested with the flower function and the undifferentiated form is tested with the loop function for a robust mot solver the averaging scheme is also adopted to alleviate the high frequency instabilities numerical verifications of perfect electric conductor pec transient scattering problems are provided to demonstrate the accuracy and stability of the proposed technique

maximum entropy summation of divergent perturbation series in this paper the principle of maximum entropy is used to predict the sum of a divergent perturbation series from the first few expansion coefficients the perturbation expansion for the ground state energy e g of the octic oscillator defined by h p2 2 x2 2 gx8 is a series of the form e g 1 2 1 n 1 angn this series is terribly divergent because for large n the perturbation coefficients an grow like 3n this growth is so rapid that the solution to the moment problem is not unique and ordinary pade summation of the divergent series fails a completely different kind of procedure based on the principle of maximum entropy for reconstructing the function e g from its perturbation coefficients is presented very good numerical results are obtained

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

a dc stable and large time step well balanced td efie based on quasi helmholtz projectors the marching on in time mot solution of the time domain electric field integral equation td efie has traditionally suffered from a number of issues including the emergence of spurious static currents dc instability and ill conditioning at large time steps low frequencies in this contribution a space time galerkin discretization of the td efie is proposed which separates the loop and star components of both the equation and the unknown judiciously integrating or differentiating these components with respect to time leads to an equation which is free from dc instability by choosing the correct temporal basis and testing functions for each of the components a stable mot system is obtained furthermore the scaling of these basis and testing functions ensure that the system remains well conditioned for large time steps the loop star decomposition is performed using quasi helmholtz projectors to avoid the explicit transformation to the unstable bases of loops and stars or trees and to avoid the search for global loops which is a computationally expensive operation

maximum entropy in the finite stieltjes and hamburger moment problem a necessary and sufficient condition for the existence of the maximum entropy me function defined in an infinite or semiinfinite interval is provided the conclusions reached show that except in a few particular cases the necessary and sufficient conditions for the existence of maximum entropy function are identical to the conditions for the solution of the moment problem when the first m 1 moments are assigned even if the conclusions reached are very similar to the hausdorff case the specificity of the hamburger and stieltjes cases demands a different handling a sufficient condition for the entropy convergence of the resulting sequence of maximum entropy estimators to the entropy of the recovering function is also provided

adaptive solution strategy for solving large systems of p type finite element equations a solution strategy is proposed and implemented for taking advantage of the hierarchical structure of linear equation sets arising from the p type finite element method using a hierarchical basis function set the algorithm dynamically branches to either direct or iterative solution methods in the iterative solution branch the substructure of the finite element equation set is used to generate a lower order preconditioner for a preconditioned conjugate gradient pcg method the convergence rate of the pcg algorithm is monitored to improve the heuristics used in the choice of the preconditioner the robustness and efficiency of the method are demonstrated on a variety of three dimensional examples utilizing both hexahedral and tetrahedral mesh discretizations this strategy has been implemented in a p version finite element code which has been used in an industrial environment for over two years to solve mechanical design problems

a joint marginal approach to parametric polynomial optimization given a compact parameter set mathbf y subset mathbb r p we consider polynomial optimization problems mathbf p mathbf y on mathbb r n whose description depends on the parameter mathbf y in mathbf y we assume that one can compute all moments of some probability measure varphi on mathbf y absolutely continuous with respect to the lebesgue measure e g mathbf y is a box or a simplex and varphi is uniformly distributed we then provide a hierarchy of semidefinite relaxations whose associated sequence of optimal solutions converges to the moment vector of a probability measure that encodes all information about all global optimal solutions mathbf x mathbf y of mathbf p mathbf y as mathbf y in mathbf y in particular one may approximate as closely as desired any polynomial functional of the optimal solutions like e g their varphi mean in addition using this knowledge on moments the measurable function mathbf y mapsto x k mathbf y of the k th coordinate of optimal solutions can be estimated e g by maximum entropy methods also for a boolean variable x k one may approximate as closely as desired its persistency varphi mathbf y x k mathbf y 1 i e the probability that in an optimal solution mathbf x mathbf y the coordinate x k mathbf y takes the value 1 last but not least from an optimal solution of the dual semidefinite relaxations one provides a sequence of polynomial resp piecewise polynomial lower approximations with l 1 varphi resp varphi almost uniform convergence to the optimal value function

the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

approximation schemes for infinite linear programs this paper presents approximation schemes for an infinite linear program in particular it is shown that under suitable assumptions the program s optimum value can be approximated by the values of finite dimensional linear programs and that in addition every accumulation point of a sequence of optimal solutions for the approximating programs is an optimal solution for the original problem

a comparison of preconditioned iterative techniques using rapid operator application against direct solution methods the p version of the finite element method which uses high order hierarchic basis functions within an element allows for ease of adaptive computation and high accuracy with relatively few degrees of freedom however the solution time and storage requirements for large 3 d problems can be very high when direct methods are used to factor global matrices iterative techniques with rapid operator application have minimal storage requirements because neither element nor global matrices are formed but may not converge rapidly enough we study an approach that attempts to combine the two methods by using direct factorization of the matrix corresponding to low level polynomials as a preconditioner to an iterative method at a higher polynomial level this resulting solution scheme converges more rapidly than an unpreconditioned scheme but has much lower storage requirements than direct methods we illustrate the effectiveness of this scheme on some large three dimensional problems in structural analysis

maximum entropy in the hamburger moments problem the maximum entropy approach to the solution of the hamburger inverse problem of moments in which one seeks to recover a positive density function p x where x from the values of a finite n 1 of its moments is considered the obtained results show that unexpected upper bounds for the moments do not exist in the general hamburger finite moment problem unlike in the symmetric case previously considered some physical examples illustrating the use of partial information to determine the approximate function are presented

an aggregation multilevel method using smooth error vectors many algebraic multilevel methods for solving linear systems assume that the slowly converging algebraically smooth error is locally constant this assumption is often not true and can lead to poor performance of the method other multilevel methods require a description of the algebraically smooth error via knowledge of the near nullspace of the operator but this information may not always be available this paper presents an aggregation multilevel method for problems where the near nullspace of the operator is not known the method uses samples of low energy error vectors to construct its interpolation operator the basis vectors for an aggregate are computed via a singular value decomposition of the sample vectors locally over that aggregate compared to many other methods that automatically adjust to the near nullspace this method does not require that the element stiffness matrices are available

success or failure of a firm under different financing policies a dynamic stochastic model abstract the value of a firm evolves in time according to a standard ito s stochastic differential equation and under different types of financial support in particular both the cases of legal and illegal financing are considered the ruin success probabilities for the firm are evaluated in closed form via special functions in the various settings some functionals of the process which are relevant from the viewpoint of the financiers via kac s lemma are shown to obey a pde which is studied both with analytic tools second order odes of hypergeometric type and with numerical ones the case of insufficient payments is analyzed via stochastic calculus and a new model for loan renegotiation is offered

optimization of polynomials on compact semialgebraic sets a basic closed semialgebraic subset s of r n is defined by simultaneous polynomial inequalities g 1 ge 0 dotsc g m ge 0 we give a short introduction to lasserre s method for minimizing a polynomial f on a compact set s of this kind it consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs we give a new short proof for the convergence of the optimal values of these relaxations to the infimum f ast of f on s which is constructive and elementary in the case where f possesses a unique minimizer x ast we prove that every sequence of nearly optimal solutions of the successive relaxations gives rise to a sequence of points in r n converging to x ast

maximum entropy in the generalized moment problem the generalized moment problem in the framework of the maximum entropy approach is considered a proof for the existence conditions of the solution is provided

time domain single source integral equations for analyzing scattering from homogeneous penetrable objects single source time domain electric and magnetic field integral equations for analyzing scattering from homogeneous penetrable objects are presented their temporal discretization is effected by using shifted piecewise polynomial temporal basis functions and a collocation testing procedure thus allowing for a marching on in time mot solution scheme unlike dual source formulations single source equations involve space time domain operator products for which spatial discretization techniques developed for standalone operators do not apply here the spatial discretization of the single source time domain integral equations is achieved by using the high order divergence conforming basis functions developed by graglia et al alongside the high order divergence and quasi curl conforming dqcc basis functions of valdes et al the combination of these two sets allows for a well conditioned mapping from div to curl conforming function spaces that fully respects the space mapping properties of the space time operators involved numerical results corroborate the fact that the proposed procedure guarantees accuracy and stability of the mot scheme

recovering a probability density function from its mellin transform the recovering of a probability density function f x from its mellin transform m s is considered the approximate f m x is chosen resorting to maximum entropy technique constrained by the first derivatives of m s evaluated at s 1 so the basic properties of a probability density are saved existence conditions of the approximate f m x entropy convergence and then l 1 norm convergence are proved some numerical examples are reported resorting to the mellin transform is an alternative to laplace one as the recovered probability distribution is heavy tailed or equivalently its probability density function has abscissa convergence laplace transform equal to 0

preconditioned conjugate residual methods for mixed spectral discretizations of elasticity and stokes problems abstract mixed spectral methods applied to elasticity and stokes problems in three dimensions produce large symmetric indefinite systems of equations these systems are sparse and more ill conditioned than those produced by h version finite element methods these stiffness matrices have the structure of saddle point problems with a penalty term in this paper preconditioned iterative methods of conjugate residual type are introduced for these systems the convergence rates of the resulting algorithms are proven to be independent of the penalty parameter and mildly dependent on the spectral degree n several numerical experiments on three dimensional model problems confirm the results obtained

the korn s type inequality in subspaces and thin elastic structures we present the concept of the korn s type inequality in subspaces generalized to embrace thin elastic structures of various kinds such as shells plates rods arches and beams of arbitrary geometries which are viewed and analysed as three dimensional solids we show that based on this inequality we can eliminate the deterioration of the convergence from which iterative algorithms are known to suffer when applied to thin elastic structures as a paradigm we consider a class of iterative algorithms and show that a simple modification based on the korn s type inequality in subspaces yields a radical improvement of their convergence

explicit solution of the time domain volume integral equation using a stable predictor corrector scheme an explicit marching on in time mot scheme for solving the time domain volume integral equation is presented the proposed method achieves its stability by employing at each time step a corrector scheme which updates corrects fields computed by the explicit predictor scheme the proposed method is computationally more efficient when compared to the existing filtering techniques used for the stabilization of explicit mot schemes numerical results presented in this paper demonstrate that the proposed method maintains its stability even when applied to the analysis of electromagnetic wave interactions with electrically large structures meshed using approximately half a million discretization elements

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

least squares spectral element preconditioners for fourth order elliptic problems abstract in this paper we propose preconditioners for the system of linear equations that arise from a discretization of fourth order elliptic problems in two and three dimensions d 2 3 using spectral element methods these preconditioners are constructed using separation of variables and can be diagonalized and hence easy to invert for second order elliptic problems this technique has proven to be successful and performs better than other preconditioners in the framework of least squares methods we show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them numerical results for the condition number reflects the effectiveness of the preconditioners

parallel domain decomposition solver for adaptive hp finite element methods in this paper the development and implementation of highly parallelizable domain decomposition solvers for adaptive hp finite element methods is discussed two level orthogonalization is used to obtain a reduced system which is preconditioned by a coarse grid operator the condition number of the preconditioned system for poisson problems in two space dimensions is proved to be bounded by c 1 hp h 2 1 log p 2 and cp 1 hp h 2 1 log p 2 for different choices of coarse grid operators where h is the subdomain size p is the maximum spectral order h is the size of the smallest element in the subdomain and c is a constant independent of the mesh parameters the work here extends the work of bramble et al math comp 47 1986 pp 103 134 on the h version and babuska et al siam j numer anal 29 1991 pp 624 661 on the p version of the finite element method a preliminary version of this solver was first announced by oden patra and feng in domain decomposition solver for adaptive hp finite elements vii conference on domain decomposition state college pa october 1993 numerical experiments show fast convergence of the solver and good control of the condition number on a variety of discretizations

iterative methods for p version finite elements preconditioning thin solids abstract we present new preconditioning strategy for thin 3d p version elements used to model plate and shell structures as well as the microstructure of composite materials the strategy is incorporated in an adaptive preconditioner suitable for large real world problems the method is shown to perform well for several large real world problems including a model of the skin of an aircraft with over 1 6 million degrees of freedom and a highly accurate model of a 28 ply laminated fiber plate

exponentially accurate nonconforming least squares spectral element method for elliptic problems on unbounded domain abstract non conforming approximation methods are becoming increasingly popular because of the potential to apply to multi material and multi model analysis for both bounded and unbounded domains in this paper we present a least square approximation based method to solve the one or two dimensional elliptic problems on an unbounded domain the method gives exponential accuracy and shows superior performance when compared to other numerical methods differentiability estimates and the main stability estimate theorem using a non conforming spectral element method are also discussed the exponential convergence rate of the proposed method is also shown through rigorous error estimate and specific numerical examples

discrete probability distributions in the generalized moment problem the discrete maximum entropy probability distribution which can take on a finite number of values and whose first generalized moments are assigned is considered it is proved that the necessary and sufficient conditions for the existence of a maximum entropy solution are identical to the general ones for the finite moment problem the entropy decreasing by adding one more moment is studied instability of the distribution to be recovered is proved when an increasing number of moments is used

entropy convergence in stieltjes and hamburger moment problem in the classical stieltjes and hamburger moment problem the sequence of maximum entropy approximants whose first m moments are equal to given ones is considered it is proved that whenever an infinite moment problem is determined then maximum entropy approximants converge in entropy to the function characterized by given moments entropy convergence is proved by using exclusively existence and uniqueness conditions