minimization of the maximum peak to peak gain the general multiblock problem a comprehensive study of the general l sub 1 optimal multiblock problem and a new linear programming algorithm for computing suboptimal controllers are presented by formulating the interpolation conditions in a concise and natural way the general theory is developed in simpler terms and with a minimum number of assumptions in addition further insight is gained into the structure of the optimal solution and different classes of multiblock problems are distinguished this leads to a conceptually attractive iterative method for finding approximate solutions

multigrid accelerated tensor approximation of function related multidimensional arrays in this paper we describe and analyze a novel tensor approximation method for discretized multidimensional functions and operators in mathbb r d based on the idea of multigrid acceleration the approach stands on successive reiterations of the orthogonal tucker tensor approximation on a sequence of nested refined grids on the one hand it provides a good initial guess for the nonlinear iterations to find the approximating subspaces on finer grids on the other hand it allows us to transfer from the coarse to fine grids the important data structure information on the location of the so called most important fibers in directional unfolding matrices the method indicates linear complexity with respect to the size of data representing the input tensor in particular if the target tensor is given by using the rank r canonical model then our approximation method is proved to have linear scaling in the univariate grid size n and in the input rank r the method is tested by three dimensional 3d electronic structure calculations for the multigrid accelerated low tucker rank approximation of the all electron densities having strong nuclear cusps we obtain high resolution of their 3d convolution product with the newton potential the accuracy of order 10 6 in max norm is achieved on large n times n times n grids up to n 1 6 cdot10 4 with the time scale in several minutes

coupled fe be method for eigenvalue analysis of elastic structures submerged in an infinite fluid domain summary for thin elastic structures submerged in heavy fluid e g water a strong interaction between the structural domain and the fluid domain occurs and significantly alters the eigenfrequencies therefore the eigenanalysis of the fluid structure interaction system is necessary in this paper a coupled finite element and boundary element fe be method is developed for the numerical eigenanalysis of the fluid structure interaction problems the structure is modeled by the finite element method the compressibility of the fluid is taken into consideration and hence the helmholtz equation is employed as the governing equation and solved by the boundary element method bem the resulting nonlinear eigenvalue problem is converted into a small linear one by applying a contour integral method adequate modifications are suggested to improve the efficiency of the contour integral method and avoid missing the eigenvalues of interest the burton miller formulation is applied to tackle the fictitious eigenfrequency problem of the bem and the optimal choice of its coupling parameter is investigated for the coupled fe be method numerical examples are given and discussed to demonstrate the effectiveness and accuracy of the developed fe be method copyright 2016 john wiley sons ltd

minimization of the l sup infinity induced norm for sampled data systems it is shown that given any degree of accuracy there exists a standard discrete time l sup 1 problem that can be determined a priori whose solution yields a controller that is almost optimal in terms of the hybrid l sup infinity induced norm this is accomplished by first converting the hybrid system into an equivalent infinite dimensional discrete time system using the lifting technique in continuous time and then approximating the infinite dimensional parts of the system which model the intersample dynamics a thorough analysis of the approximation procedure is presented and it is shown that it is convergent at the rate of 1 n explicit bounds that are independent of the controller are obtained to characterize the approximation it is also shown that the geometry of the induced norm for the sampled data problem is different from that of the standard l sup 1 norm and hence there might not exist a linear isometry that maps the sampled data problem exactly to a standard discrete time problem

fr operator approach to the h sub 2 analysis and synthesis of sampled data systems recently a frequency domain operator called frequency response fr operator was defined and shown to represent the transfer characteristics of a stable sampled data system using this novel frequency domain notion and introducing its extended notion called hybrid fr operator we define an h sub 2 norm for sampled data systems in this paper then sampled data h sub 2 control problems are formulated and solved whereby the usefulness of these frequency domain notions is demonstrated both in the analysis and synthesis aspects of sampled data systems for the case of sampled data systems with hybrid i e both continuous time and discrete time input and output signals the h sub 2 norm defined by a hybrid fr operator turns out to be slightly different from that defined in previous studies the source of the discrepancy is also identified

computing the l2 induced norm of a compression operator abstract a new computationally viable approach is derived for computing the induced norm of a state space compression operator that is an integral operator defined on the finite length space l2 0 h determining this norm is a crucial component in the control analysis of both delay systems and sampled data systems the technique developed may have application to a wider variety of integral operators

regularity and numerical solution of eigenvalue problems with piecewise analytic data this paper discusses the regularity of the eigenfunctions of eigenvalue problems with piecewise analytic data and the approximation of the eigenvalues and eigenfunctions of such problems a detailed and systematic numerical study of these approximations is presented together with an analysis of the numerical results in light of the theoretical results the specific aim is to assess the reliability of the theoretical results which are of an asymptotic nature as a guide to practical computations which may take place in the pre asymptotic phase and to look for characteristic features of the numerical results that are not completely explained by known theoretical results

an integral method for solving nonlinear eigenvalue problems we propose a numerical method for computing all eigenvalues and the corresponding eigenvectors of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane the method uses complex integrals of the resolvent operator applied to at least k column vectors where k is the number of eigenvalues inside the contour the theorem of keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension k no initial approximations of eigenvalues and eigenvectors are needed the method is particularly suitable for moderately large eigenvalue problems where k is much smaller than the matrix dimension we also give an extension of the method to the case where k is larger than the matrix dimension the quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour

h 2 and hinf design of sampled data systems using lifting part ii properties of systems in the lifted domain an important class of sampled data systems becomes time invariant when the systems are lifted into an appropriate domain the purpose of the present paper is to study some basic properties of systems represented by state space equations in the lifted domain hidden modes invariant zeros and riccati equations are investigated and the connection between representations in continuous and lifted domains are clarified the material treated here is important for solving optimal control problems of generalized sampled data systems

a novel numerical method to determine the algebraic multiplicity of nonlinear eigenvalues we generalize the algebraic multiplicity of the eigenvalues of nonlinear eigenvalue problems neps to the rational form and give the extension of the argument principle in addition we propose a novel numerical method to determine the algebraic multiplicity of the eigenvalues of the neps in a given region by the contour integral method finally some numerical experiments are reported to illustrate the effectiveness of our method

kronecker product approximations for some function related matrices abstract a new approximation tool such as sums of kronecker products is recently found to provide a superb compression property on a series of numerical examples of quite a general nature the purpose of the paper is explanation of this phenomenon in the form of existence theorems for matrix approximations of low kronecker rank for some classes of function related matrices including important specimens from potential theory this lays the grounds for development of new approximation algorithms for example in the cases when a matrix is associated with a shift invariant function on the cartesian product of nonunform grids which is of great practical interest in the solution of integral equations on plates or screens

localization theorems for nonlinear eigenvalue problems let t omega rightarrow mathbb c n times n be a matrix valued function that is analytic on some simply connected domain omega subset mathbb c a point lambda in omega is an eigenvalue if the matrix t lambda is singular in this article we describe new localization results for nonlinear eigenvalue problems that generalize gershgorin s theorem pseudospectral inclusion theorems and the bauer fike theorem we use our results to analyze three nonlinear eigenvalue problems an example from delay differential equations a problem due to hadeler and a quantum resonance computation

sampled data control of continuous time systems with an h optimality criterion abstract the synthesis of sampled data controllers for continuous time systems is studied in the context of h control theory the sampled data h optimal control problem consists of finding a sampled data controller which minimizes the l2 induced norm from the continuous time disturbance to the continuous time output the problem is here solved via a discrete system representation this is achieved by describing the closed loop system as a discrete system with finite state dimension and with a representation of the disturbance and the output signals of the discrete system as sequences of elements of certain infinite dimensional spaces via finite rank approximations of a compact integral operator associated with the system the sampled data h control problem can be solved to any degree of accuracy from the structure of the problem it follows that irrespective of the order of the approximation a feasible sampled data controller can be found which has the same order as the system

asymptotic and numerical techniques for resonances of thin photonic structures we consider the problem of calculating resonance frequencies and radiative losses of an optical resonator the optical resonator is in the form of a thin membrane with variable dielectric properties this work provides two very different approaches for doing such calculations the first is an asymptotic method which exploits the small thickness and high index of the membrane we derive a limiting resonance problem as the thickness goes to zero and for the case of a simple resonance find a first order correction the limiting problem and the correction are in one less space dimension which can make the approach very efficient convergence estimates are proved for the asymptotics the second approach based on the finite element method with a truncated perfectly matched layer is not restricted to thin structures we demonstrate the use of these methods in numerical calculations which further illustrate their differences the asymptotic method finds resonance by solving a dense but small nonlinear eige

grid based electronic structure calculations we present a fully grid based approach for solving hartree fock and all electron kohn sham equations based on low rank approximation of three dimensional electron orbitals due to the low rank structure the total complexity of the algorithm depends linearly with respect to the one dimensional grid size linear complexity allows for the usage of fine grids e g 81923 and thus cheap extrapolation procedure we test the proposed approach on closed shell atoms up to the argon several molecules and clusters of hydrogen atoms all tests show systematical convergence with the required accuracy

spectral analysis and singular value computations of the noncompact frequency response and compression operators in sampled data systems this paper is motivated by the problem of computing the frequency response gain of general sampled data systems with noncompact frequency response operators we first show that with the j unitary transformation the computation in the noncompact operator case can be reduced in principle to that in the compact operator case to which an existing efficient and reliable bisection method can be applied at the same time however we point out that there arise some critical problems in this reduction to the compact case which could be serious enough to invalidate the apparent success in the reduction through some spectral analysis of operators involving or related to the frequency response operators we eventually prove that these critical problems can be circumvented after all and we give an explicit result that shows how to compute the frequency response gain with a bisection method dealing only with finite dimensional matrices extending the arguments we also give a bisection method to compute the singular values of the frequency response operators and the associated compression operators

a new approach to the discretization of continuous time controllers a new method for discretizing continuous time controllers is derived using principles of controller approximation it focuses on the closed loop use of the discrete time controller the resulting approximation criterion is a measure for stability of the control system provides an upper bound on the sampling time for which stability can be guaranteed and because it is based on continuous time controller approximation it indicates the cost of discretization in terms of performance degradation the discrete time controller is obtained through minimization of this criterion which can be performed with standard software used in h sub infinity controller design

continuation of eigenvalues and invariant pairs for parameterized nonlinear eigenvalue problems invariant pairs have been proposed as a numerically robust means to represent and compute several eigenvalues along with the corresponding generalized eigenvectors for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter in this work we consider nonlinear eigenvalue problems that depend on an additional parameter and our interest is to track several eigenvalues as this parameter varies based on the concept of invariant pairs a theoretically sound and reliable numerical continuation procedure is developed particular attention is paid to the situation when the procedure approaches a singularity that is when eigenvalues included in the invariant pair collide with other eigenvalues for the real generic case it is proven that such a singularity only occurs when two eigenvalues collide on the real axis it is shown how this situation can be handled numerically by an appropriate expansion of the invariant pair the viability of our continuation procedure is illustrated by a numerical example

minimization of the l induced norm for sampled data systems the following problem is addressed given a continuous time plant with continuous time performance objectives expressed in terms of the l induced norm design a digital controller that delivers or optimizes this performance this problem differs from the standard discrete time methods in that it takes into consideration the inter sample behavior of the closed loop system the resulting closed loop system dynamics consist of both continuous time and discrete time dynamics and thus such systems are known as hybrid systems it is shown that given any degree of accuracy there exists a standard discrete time l 1 problem which can be determined a priori whose solution yields a controller that is almost optimal in terms of the hybrid l induced norm

l sub 1 analysis and design of sampled data systems the focus of this work is l sub 1 optimal control of sampled data systems a converging approximation procedure is derived to compute the l sub infinity induced norm of closed loop finite dimensional linear time invariant lti sampled data systems an approximation method is developed to synthesize l sub 1 optimal sampled data regulators finally an example is provided that illustrates the l sub 1 analysis and design techniques presented

l 1 optimal feedback controllers for mimo discrete time systems the problem considered in this paper is the design of a closed loop system consisting of a mimo discrete time plant and compensator in such a way that the system is internally stable and optimally tracks all persistent bounded inputs the solution consists of two parts first the calculation of the minimum value of the performance index a weighted transfer function which is done by solving a linear programming problem and second construction of the optimal transfer function by solving a set of linear equations it is shown that in general the discrete time problem will have a rational solution and thus appears to be of considerable practical significance

discrete perturbation estimates for eigenpairs of fredholm operator valued functions we present perturbation estimates for eigenvalue and eigenvector approximations for a class of fredholm operator valued functions our approach is based on perturbation estimates for the generalized resolvents and the exponential convergence of the contour integration by the trapezoidal rule we use discrete residual functions to estimate the resolvents a posteriori numerical experiments are also presented

a 2 5d boundary element formulation for modeling damped waves in arbitrary cross section waveguides and cavities a regularized 2 5d boundary element method bem is proposed to predict the dispersion properties of damped stress guided waves in waveguides and cavities of arbitrary cross section the wave attenuation induced by material damping is introduced using linear viscoelastic constitutive relations and described in a spatial manner by the imaginary component of the axial wavenumber the discretized dispersive wave equation results in a nonlinear eigenvalue problem which is solved obtaining complex axial wavenumbers for a fixed frequency using a contour integral algorithm due to the singular characteristics and the multivalued feature of the wave equation the requirement of holomorphicity inside the contour region over the complex wavenumber plane is fulfilled by the introduction of the sommerfeld branch cuts and by the choice of the permissible riemann sheets a post processing analysis is developed for the extraction of the energy velocity of propagative guided waves the reliability of the method is demonstrated by comparing the results obtained for a rail and a bar with square cross section with those obtained from a 2 5d finite element formulation also known in literature as semi analytical finite element safe method next to show the potential of the proposed numerical framework dispersion properties are predicted for surface waves propagating along cylindrical cavities of arbitrary cross section it is demonstrated that the attenuation of surface waves approaches asymptotically the attenuation of rayleigh waves

optimal rejection of persistent bounded disturbances in this paper we formulate the problem of optimal disturbance rejection in the case where the disturbance is generated as the output of a stable system in response to an input which is assumed to be of unit amplitude but is otherwise arbitrary the objective is to choose a controller that minimizes the maximum amplitude of the plant output in response to such a disturbance mathematically this corresponds to requiring uniformly good disturbance rejection over all time since the problem of optimal tracking is equivalent to that of optimal disturbance rejection if a feedback controller is used see 7 sect 5 6 the theory presented here can also be used to design optimal controllers that achieve uniformly good tracking over all time rather than a tracking error whose l 2 norm is small as is the case with the currently popular h infty theory the present theory is a natural counterpart to the existing theory of optimal disturbance rejection the so called h infty theory which is based on the assumption that the disturbance to be rejected is generated by a stable system whose input is square integrable and has unit energy it is shown that the problem studied here has quite different features from its predecessor complete solutions to the problem are given in several important cases including those where the plant is minimum phase or when it has only a single unstable zero in other cases procedures are given for obtaining bounds on the solution and for obtaining suboptimal controllers

microwave tomography using topology optimization techniques microwave tomography is a technique in which microwaves illuminate a specimen and measurements of the scattered electrical field are used to determine and depict the specimen s dielectric and conductive properties this article presents a new method to perform the reconstruction the reconstruction method is illustrated by assuming time harmonic scattering in two space dimensions in a setup tailored for medical applications we prove that the resulting constrained nonlinear least squares problem admits a solution the governing helmholtz equation is discretized by using the finite element method and the dielectric properties are allowed to attain different values at each element within a given region the reconstruction algorithm uses methodologies borrowed from topology optimization of linearly elastic structures numerical examples illustrate the reconstruction method in a parameter range typical for human tissue and for the challenging case where the size of the object is in the same order as the wavelength a reasonable estimate of the dielectric properties is obtained by using one observation per 20 unknowns when the permittivity is allowed to vary continuously within a given interval using a priori information that the permittivity attains only certain values results in a good estimate and a sharp image as opposed to topology optimization for structures there is no indication of mesh dependency and checkerboarding when forcing the permittivity to attain discrete values

computing the l 0 h induced norm of a compression operator via fast lifting abstract this paper studies computing the induced norm of a compression operator defined on the banach space l 0 h which is a difficult problem since it is an infinite rank operator two methods are provided for this problem each of which can compute an upper bound and a lower bound of the induced norm by using an idea of staircase or piecewise linear approximation staircase approximation and piecewise linear approximation are applied through fast lifting by which the interval 0 h is divided into m subintervals with equal width and the approximation errors in these methods are ensured to be reciprocally proportional to m or m 2 the effectiveness of the proposed methods is demonstrated through numerical examples

a projection method for generalized eigenvalue problems using numerical integration in this paper we propose a method for finding certain eigenvalues of a generalized eigenvalue problem that lie in a given domain of the complex plane the proposed method projects the matrix pencil onto a subspace associated with the eigenvalues that are located in the domain via numerical integration

solution of the helmholtz eigenvalue problem via the boundary element method the numerical solution of the helmholtz eigenvalue problem is considered the application of the boundary element method reduces it to that of a non linear eigenvalue problem through a polynomial approximation with respect to the wavenumber the non linear eigenvalue problem is reduced to a standard generalized eigenvalue problem the method is applied to the test problems of a three dimensional sphere with an axisymmetric boundary condition and a two dimensional square

a contour integral approach to the computation of invariant pairs abstract we study some aspects of the invariant pair problem for matrix polynomials as introduced by betcke and kressner 3 and by beyn and thummler 6 invariant pairs extend the notion of eigenvalue eigenvector pairs providing a counterpart of invariant subspaces for the nonlinear case we compute formulations for the condition numbers and the backward error for invariant pairs and solvents we then adapt the sakurai sugiura moment method 1 to the computation of invariant pairs including some classes of problems that have multiple eigenvalues numerical refinement via a variant of newton s method is also studied furthermore we investigate the relation between the matrix solvent problem and the triangularization of matrix polynomials

efficient and reliable hp fem estimates for quadratic eigenvalue problems and photonic crystal applications we present a posteriori analysis of higher order finite element approximations hp fem for quadratic fredholm valued operator functions residual estimates for approximations of the algebraic eigenspaces are derived and we reduce the analysis of the estimator to the analysis of an associated boundary value problem for the reasons of robustness we also consider approximations of the associated invariant pairs we show that our estimator inherits the efficiency and reliability properties of the underlying boundary value estimator as a model problem we consider spectral problems arising in analysis of photonic crystals in particular we present an example where a targeted family of eigenvalues cannot be guaranteed to be semisimple numerical experiments with hp fem show the predicted convergence rates the measured effectivities of the estimator compare favorably with the performance of the same estimator on the associated boundary value problem we also present a benchmark estimator based on the dual weighted residual dwr approach which is more expensive to compute but whose measured effectivities are close to one

on spurious solutions in finite element approximations of resonances in open systems in this paper we discuss problems arising when computing resonances with a finite element method in the pre asymptotic regime we detect for the one dimensional case spurious solutions in finite

the computation of resonances in open systems using a perfectly matched layer in this paper we consider the problem of computing resonances in open systems we first characterize resonances in terms of improper eigen functions of the helmholtz operator on an unbounded domain the perfectly matched layer pml technique has been successfully applied to the computation of scattering problems we shall see that the application of pml converts the resonance problem to a standard eigenvalue problem still on an infinite domain this new eigenvalue problem involves an operator which resembles the original helmholtz equation transformed by a complex shift in the coordinate system our goal will be to approximate the shifted operator first by replacing the infinite domain by a finite computational domain with a convenient boundary condition and second by applying finite elements on the computational domain we shall prove that the first of these steps leads to eigenvalue convergence to the desired resonance values which is free from spurious computational eigenvalues provided that the size of computational domain is sufficiently large the analysis of the second step is classical finally we illustrate the behavior of the method applied to numerical experiments in one and two spatial dimensions

perturbation theory of nonlinear non self adjoint eigenvalue problems simple eigenvalues abstract the study of the vibrational modes and stability of a given physical system is strongly tied to the efficient numerical evaluation of its eigenvalues the operators governing the eigenproblem are in general nonlinear in the eigenvalue and non self adjoint which makes the repeated solution of the eigenvalue problem necessary for example when the effect of several parameter values on the system needs to be assessed expensive this study reviews the adjoint based incremental procedure for calculating the coefficients of power series expansions of simple non degenerate eigenvalues and their eigenvectors these expansions approximate the eigenvalues to any desired order in a finite region an efficient numerical implementation of the theory is proposed and it is shown how high order power series approximations of the eigenvalues give very accurate results within the radius of convergence of the power series which is finite and generally not small furthermore the domain of convergence of the power series might be extended by considering pade expansions of the eigenvalues examples involving the stability of the orr sommerfeld equation the biharmonic equation for the vibrational modes of a membrane and the emission of sound from a rijke tube associated with thermoacoustic feedback are used to assess and validate the theory

algorithms for numerical analysis in high dimensions nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension the separated representation introduced previously allows many operations to be performed with scaling that is formally linear in the dimension in this paper we further develop this representation by i discussing the variety of mechanisms that allow it to be surprisingly efficient ii addressing the issue of conditioning iii presenting algorithms for solving linear systems within this framework and iv demonstrating methods for dealing with antisymmetric functions as arise in the multiparticle schrodinger equation in quantum mechanics numerical examples are given

the hyperbolic cross space approximation of electronic wavefunctions the electronic schrodinger equation describes the motion of electrons under coulomb interaction forces in the field of clamped nuclei and forms the basis of quantum chemistry the present article continues the author s recent work numer math 98 731 759 2004 on the regularity of its solutions the electronic wavefunctions it was shown in the mentioned article that these wavefunctions possess square integrable high order mixed weak derivatives as they are needed to justify and underpin sparse grid and hyperbolic cross space approximation techniques theoretically how fast the norms of these derivatives can increase with the number of electrons is studied in the present article provided that all quantities are properly related to a characteristic lengthscale of the considered atomar or molecular system the norms of the derivatives remain bounded independent of the number of electrons and the number the positions and the charges of the nuclei

rational approximation of l sub 1 optimal controllers for siso systems the l sub 1 optimal control problem with rational controllers for continuous time systems is considered in which it is shown that the optimal l sub 1 performance index with rational controllers is equal to that of irrational controllers a sequence of rational controllers that approximates the optimal index is constructed convergence properties of such a sequence are studied that the corresponding sequence of objective transfer functions is shown to converge in weak topology in bv r sub in the time domain and uniformly in a wider sense in the frequency domain

recursive integral method for transmission eigenvalues the transmission eigenvalue problem arose in the inverse scattering theory for inhomogeneous media and has important applications in a variety of inverse problems for target identification and nondestructive testing the problem is numerically challenging because of the nonself adjointness and complicated spectrum in this talk we introduce a novel recursive contour integral method using the spectrum projection the method can compute both real and complex eigenvalues and does not require any a priori information numerical examples show that the method is effective and robust

computing the common zeros of two bivariate functions via bezout resultants the common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor chebyshev basis from here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and bezout matrices with polynomial entries using techniques including domain subdivision bezoutian regularization and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree ge 1 000 1 000 we analyze the resultant method and its conditioning by noting that the bezout matrices are matrix polynomials two implementations are available one on the matlab central file exchange and another in the roots command in chebfun2 that is adapted to suit chebfun s methodology

a general framework for linear periodic systems with applications to h sup infinity sampled data control the authors present a framework for dealing with continuous time periodic systems the main tool is a lifting technique which provides a strong correspondence between continuous time periodic systems and certain types of discrete time time invariant systems with infinite dimensional input and output spaces despite the infinite dimensionality of the input and output spaces a lifted system has a finite dimensional state space if the original system does this fact permits rather constructive methods for analyzing these systems as a demonstration of the utility of this framework the authors use it to describe the continuous time i e intersample behavior of sampled data systems and to obtain a complete solution to the problem of parameterizing all controllers that constrain the l sup 2 induced norm of a sampled data system to within a certain bound

numerical solution of the hartree fock equation in multilevel tensor structured format in this paper we describe a novel method for a robust and accurate iterative solution of the self consistent hartree fock equation in mathbb r 3 based on the idea of tensor structured computation of the electron density and the nonlinear hartree and nonlocal exchange operators at all steps of the iterative process we apply the self consistent field scf iteration to the galerkin discretization in a set of low separation rank basis functions that are solely specified by the respective values on a three dimensional cartesian grid the approximation error is estimated by o h 3 where h o n 1 is the mesh size of an n times n times n tensor grid while the numerical complexity to compute the galerkin matrices scales linearly in n log n we propose the tensor truncated version of the scf iteration using the traditional direct inversion in the iterative subspace scheme enhanced by the multilevel acceleration with the grid dependent termination criteria at each discretization level this implies that the overall computational cost scales almost linearly in the univariate problem size n numerical illustrations are presented for the all electron case of h 2 o and the pseudopotential case of ch 4 and ch 3 oh molecules the proposed scheme is not restricted to a priori given rank 1 basis sets allowing analytically integrable convolution transform with the newton kernel that opens further perspectives for promotion of the tensor structured methods in computational quantum chemistry

h optimal sampled data control in continuous time systems we consider the design of h optimal discrete time digital controllers in continuous time systems an apparent difficulty especially in utilizing modern transform domain analysis in this context stems from the absence of an appropriate transfer function model for the hybrid time discrete and continuous closed loop system this difficulty is overcome through the introduction of an equivalent difference equation model for the continuous time system with distributed inputs and outputs equivalence being in the sense that the continuous and discrete time inputs and outputs are essentially identical using the interplay between the discrete and the continuous time models solutions of the well known purely continuous time and purely discrete time standard problems extend to solutions of the problem at hand they comprise riccati equation characterizations of feasible combinations of sampling rates and bounds on the closed loop induced input output norm and parameterization of compensators we consid

l induced norm analysis of sampled data systems via piecewise constant and linear approximations this paper deals with the l 1 analysis of linear sampled data systems by which we mean the computation of the l induced norm of linear sampled data systems two computation methods based on piecewise constant and piecewise linear approximations are provided through fast lifting by which the sampling interval 0 h is divided into m subintervals with an equal width even though the central part of the method with the former approximation essentially coincides with a conventional method via fast sample fast hold fsfh approximation after all we show that both methods successfully lead to the upper and lower bounds of the l induced norm whose gap converges to 0 at the rate of 1 m in the former approximation and 1 m 2 in the latter extended approximation such achievements are in sharp contrast with an existing result on the former i e fsfh approximation which only shows the convergence rate of the error in the resulting estimate of the l induced norm without providing any readily computable upper and lower bounds a numerical example is given to illustrate the effectiveness of these methods

nonlinear eigenvalue problems and contour integrals in this paper beyn s algorithm for solving nonlinear eigenvalue problems is given a new interpretation and a variant is designed in which the required information is extracted via the canonical polyadic decomposition of a hankel tensor a numerical example shows that the choice of the filter function is very important particularly with respect to where it is positioned in the complex plane

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

a low rank approach to the computation of path integrals we present a method for solving the reaction diffusion equation with general potential in free space it is based on the approximation of the feynman kac formula by a sequence of convolutions on sequentially diminishing grids for computation of the convolutions we propose a fast algorithm based on the low rank approximation of the hankel matrices the algorithm has complexity of o n r m log m n r 2 m flops and requires o m r floating point numbers in memory where n is the dimension of the integral r n and m is the mesh size in one dimension the presented technique can be generalized to the higher order diffusion processes

fast and accurate tensor approximation of a multivariate convolution with linear scaling in dimension in the present paper we present the tensor product approximation of a multidimensional convolution transform discretized via a collocation projection scheme on uniform or composite refined grids examples of convolving kernels are provided by the classical newton slater exponential and yukawa potentials 1 email protected e l x and e l x email protected with email protected r d for piecewise constant elements on the uniform grid of size n d we prove quadratic convergence o h 2 in the mesh parameter h 1 n and then justify the richardson extrapolation method on a sequence of grids that improves the order of approximation up to o h 3 a fast algorithm of complexity o dr 1r 2nlogn is described for tensor product convolution on uniform composite grids of size n d where r 1 r 2 are tensor ranks of convolving functions we also present the tensor product convolution scheme in the two level tucker canonical format and discuss the consequent rank reduction strategy finally we give numerical illustrations confirming a the approximation theory for convolution schemes of order o h 2 and o h 3 b linear logarithmic scaling of 1d discrete convolution on composite grids c linear logarithmic scaling in n of our tensor product convolution method on an nxnxn grid in the range email protected 16384

a new approach to the solution of the l sub 1 control problem the scaled q method we explore an approach for solving multiple input multiple output mimo l sub 1 optimal control problems this approach which we refer to as the scaled q approach is introduced to alleviate many of the difficulties facing the numerical solution of optimal l sub 1 control problems in particular the computations of multivariable zeros and their directions are no longer required the scaled q method also avoids the pole zero cancellation difficulties that existing methods based on zero interpolation face when attempting to recover the optimal controller from an optimal closed loop map because the scaled q approach is based on solving a regularized auxiliary problem for which the solution is always guaranteed to exist it can be used no matter where the system zeros are including the stability boundary upper and lower bounds that converge to the optimal cost are provided and all solutions are based on finite dimensional linear programming for which efficient software exists

h 2 and h infty design of sampled data systems using lifting part i general framework and solutions this paper presents a complete solution to the h2 and h infty problems for sampled data systems as opposed to previous works in the area it is assumed here that all or some of the sampling function the discrete time controller and the hold function are available for design the solution is obtained by transforming the problem to discrete time using the well known lifting technique it is then shown that the desired components of the sampled data controller can be peeled off from the inherently infinite dimensional description in the lifted domain the procedure for doing this last step is central to the approach in this paper both new and revised solutions are presented in this paper the solution to the h2 problems is completely new the solution to the h infty problems is presented in a unifying framework and is more transparent than the previous existing solutions in the literature transparency pays in the form of clearer results in particular a separation structure is established between the design of sub optimal hold and sampler

tensor decomposition in electronic structure calculations on 3d cartesian grids in this paper we investigate a novel approach based on the combination of tucker type and canonical tensor decomposition techniques for the efficient numerical approximation of functions and operators in electronic structure calculations in particular we study applicability of tensor approximations for the numerical solution of hartree fock and kohn sham equations on 3d cartesian grids we show that the orthogonal tucker type tensor approximation of electron density and hartree potential of simple molecules leads to low tensor rank representations this enables an efficient tensor product convolution scheme for the computation of the hartree potential using a collocation type approximation via piecewise constant basis functions on a uniform nxnxn grid combined with the richardson extrapolation our approach exhibits o h 3 convergence in the grid size h o n 1 moreover this requires o 3rn r 3 storage where r denotes the tucker rank of the electron density with r o logn almost uniformly in n for example calculations of the coulomb matrix and the hartree fock energy for the ch 4 molecule with a pseudopotential on the c atom achieved accuracies of the order of 10 6 hartree with a grid size n of several hundreds since the tensor product convolution in 3d is performed via 1d convolution transforms our scheme markedly outperforms the 3d fft in both the computing time and storage requirements

numerical operator calculus in higher dimensions when an algorithm in dimension one is extended to dimension d in nearly every case its computational cost is taken to the power d this fundamental difficulty is the single greatest impediment to solving many important problems and has been dubbed the curse of dimensionality for numerical analysis in dimension d we propose to use a representation for vectors and matrices that generalizes separation of variables while allowing controlled accuracy basic linear algebra operations can be performed in this representation using one dimensional operations thus bypassing the exponential scaling with respect to the dimension although not all operators and algorithms may be compatible with this representation we believe that many of the most important ones are we prove that the multiparticle schrodinger operator as well as the inverse laplacian can be represented very efficiently in this form we give numerical evidence to support the conjecture that eigenfunctions inherit this property by computing the ground state eigenfunction for a simplified schrodinger operator with 30 particles we conjecture and provide numerical evidence that functions of operators inherit this property in which case numerical operator calculus in higher dimensions becomes feasible

the numerical solution of the helmholtz equation for wave propagation problems in underwater acoustics abstract the helmholtz equation k2n2 u 0 with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems a numerical algorithm has been developed and a computer code implemented that can effectively solve this equation in the intermediate frequency range the equation is discretized by using the finite element method thus allowing for the modeling of complicated geometries including interfaces and complicated boundary conditions a global radiation boundary condition is imposed at the far field boundary that is exact for an arbitrary number of propagating modes the resulting large nonselfadjoint system of linear equations with indefinite symmetric part is solved by using the preconditioned conjugate gradient method applied to the normal equations a new preconditioner based on the multigrid method is developed this preconditioner is vectorizable and is extremely effective over a wide range of frequencies provided the number of grid levels is reduced for large frequencies a heuristic argument is given that indicates the superior convergence properties of this preconditioner the relevant limit to analyze convergence is for k increasing and a fixed prescribed accuracy level the efficiency and robustness of the numerical algorithm are confirmed for large acoustic models including interfaces with strong velocity contrasts

tensor structured factorized calculation of two electron integrals in a general basis in this paper the problem of efficient grid based computation of the two electron integrals tei in a general basis is considered we introduce the novel multiple tensor factorizations of the tei unfolding matrix which decrease the computational demands for the evaluation of tei in several aspects using the reduced higher order svd the redundancy free product basis set is constructed that diminishes dramatically the initial number o n b 2 of three dimensional 3 d convolutions defined over cross products of n b basis functions to o n b scaling the tensor structured numerical integration with the 3 d newton convolving kernel is performed in one dimensional 1 d complexity thus enabling high resolution over fine 3 d cartesian grids furthermore using the quantized approximation of long vectors ensures the logarithmic storage complexity in the grid size finally we present and analyze two approaches to compute the cholesky decomposition of the tei matrix based on two types of precomput

modified fast sample fast hold approximation and γ independent h discretisation for general sampled data systems by fast lifting this article is concerned with the fast lifting approach to h analysis and design of sampled data systems and extends our preceding study on modified fast sample fast hold fsfh approximation in which the direct feedthrough matrix d 11 from the disturbance w to the controlled output z was assumed to be zero more precisely this article removes this assumption and shows that a independent h discretisation is still possible in a nontrivial fashion by applying what we call quasi finite rank approximation of an infinite rank operator and then the loop shifting technique as in the case of d 11 0 the modified fsfh approach retains the feature that both the upper and lower bounds of the h norm or the frequency response gain can be computed where the gap between the upper and lower bounds can be bounded with the approximation parameter n and is independent of the discrete time controller this feature is significant in applying the new method especially to control system design and this study in

on manifolds of tensors of fixed tt rank recently the format of tt tensors hackbusch and kuhn in j fourier anal appl 15 706 722 2009 oseledets in siam j sci comput 2009 submitted oseledets and tyrtyshnikov in siam j sci comput 31 5 2009 oseledets and tyrtyshnikov in linear algebra appl 2009 submitted has turned out to be a promising new format for the approximation of solutions of high dimensional problems in this paper we prove some new results for the tt representation of a tensor u in mathbb r n 1 times cdots times n d and for the manifold of tensors of tt rank underline r as a first result we prove that the tt or compression ranks r i of a tensor u are unique and equal to the respective separation ranks of u if the components of the tt decomposition are required to fulfil a certain maximal rank condition we then show that the set mathbb t of tt tensors of fixed rank underline r locally forms an embedded manifold in mathbb r n 1 times cdots times n d therefore preserving the essential theoretical properties of the tucker format but often showing an improved scaling behaviour extending a similar approach for matrices conte and lubich in m2an 44 759 2010 we introduce certain gauge conditions to obtain a unique representation of the tangent space mathcal t u mathbb t of mathbb t and deduce a local parametrization of the tt manifold the parametrisation of mathcal t u mathbb t is often crucial for an algorithmic treatment of high dimensional time dependent pdes and minimisation problems lubich in from quantum to classical molecular dynamics reduced methods and numerical analysis 2008 we conclude with remarks on those applications and present some numerical examples

computing a alpha log a and related matrix functions by contour integrals new methods are proposed for the numerical evaluation of f mathbf a or f mathbf a b where f mathbf a is a function such as mathbf a 1 2 or log mathbf a with singularities in infty 0 and mathbf a is a matrix with eigenvalues on or near 0 infty the methods are based on combining contour integrals evaluated by the periodic trapezoid rule with conformal maps involving jacobi elliptic functions the convergence is geometric so that the computation of f mathbf a b is typically reduced to one or two dozen linear system solves which can be carried out in parallel

krylov type methods for tensor computations i several krylov type procedures are introduced that generalize matrix krylov methods for tensor computations they are denoted minimal krylov recursion maximal krylov recursion and contracted tensor product krylov recursion it is proved that for a given tensor a with multilinear rank p q r the minimal krylov recursion extracts the correct subspaces associated to the tensor in p q r number of tensor vector vector multiplications an optimized minimal krylov procedure is described that for a given multilinear rank of an approximation produces a better approximation than the standard minimal recursion we further generalize the matrix krylov decomposition to a tensor krylov decomposition the tensor krylov methods are intended for the computation of low multilinear rank approximations of large and sparse tensors but they are also useful for certain dense and structured tensors for computing their higher order singular value decompositions or obtaining starting points for the best low rank computations of tensors a set of numerical experiments using real world and synthetic data sets illustrate some of the properties of the tensor krylov methods

exploiting efficient representations in large scale tensor decompositions decomposing tensors into simple terms is often an essential step toward discovering and understanding underlying processes or toward compressing data however storing the tensor and computing its

an accurate and efficient acoustic eigensolver based on a fast multipole bem and a contour integral method an accurate numerical solver is developed in this paper for eigenproblems governed by the helmholtz equation and formulated through the boundary element method a contour integral method is used to convert the nonlinear eigenproblem into an ordinary eigenproblem so that eigenvalues can be extracted accurately by solving a set of standard boundary element systems of equations in order to accelerate the solution procedure the parameters affecting the accuracy and efficiency of the method are studied and two contour paths are compared moreover a wideband fast multipole method is implemented with a block idr s solver to reduce the overall solution cost of the boundary element systems of equations with multiple right hand sides the burton miller formulation is employed to identify the fictitious eigenfrequencies of the interior acoustic problems with multiply connected domains the actual effect of the burton miller formulation on tackling the fictitious eigenfrequency problem is investigated and the optimal choice of the coupling parameter as i k is confirmed through exterior sphere examples furthermore the numerical eigenvalues obtained by the developed method are compared with the results obtained by the finite element method to show the accuracy and efficiency of the developed method

numerical solution of high dimensional stationary fokker planck equations via tensor decomposition and chebyshev spectral differentiation abstract this paper focuses on the curse of dimensionality in the numerical solution of the stationary fokker planck equation for systems with state independent excitation a tensor decomposition approach is combined with chebyshev spectral differentiation to drastically reduce the number of degrees of freedom required to maintain accuracy as dimensionality increases following the enforcement of the normality condition via a penalty method the discretized system is solved using alternating least squares algorithm numerical results for a variety of systems including separable non separable systems linear nonlinear systems and systems with without closed form stationary solutions up to ten dimensional state space are presented to illustrate the effectiveness of the proposed method

wedderburn rank reduction and krylov subspace method for tensor approximation part 1 tucker case new algorithms are proposed for the tucker approximation of a 3 tensor accessed only through a tensor by vector by vector multiplication subroutine in the matrix case the krylov methods are methods of choice to approximate the dominant column and row subspaces of a sparse or structured matrix given through a matrix by vector operation using the wedderburn rank reduction formula we propose a matrix approximation algorithm that computes the krylov subspaces and can be generalized to 3 tensors the numerical experiments show that on quantum chemistry data the proposed tensor methods outperform the minimal krylov recursion of savas and elden

fast multidimensional convolution in low rank tensor formats via cross approximation we propose a new cross conv algorithm for approximate computation of convolution in different low rank tensor formats tensor train tucker hierarchical tucker it has better complexity with respect to the tensor rank than previous approaches the new algorithm has a high potential impact in different applications the key idea is based on applying cross approximation in the frequency domain where convolution becomes a simple elementwise product we illustrate efficiency of our algorithm by computing the three dimensional newton potential and by presenting preliminary results for solution of the hartree fock equation on tensor product grids

sparse finite elements for elliptic problems with stochastic loading we formulate elliptic boundary value problems with stochastic loading in a bounded domain d d we show well posedness of the problem in stochastic sobolev spaces and we derive a deterministic elliptic pde in d d for the spatial correlation of the random solution we show well posedness and regularity results for this pde in a scale of weighted sobolev spaces with mixed highest order derivatives discretization with sparse tensor products of any hierarchic finite element fe spaces in d yields optimal asymptotic rates of convergence for the spatial correlation even in the presence of singularities or for spatially completely uncorrelated data multilevel preconditioning in d d allows iterative solution of the discrete equation for the correlation kernel in essentially the same complexity as the solution of the mean field equation

induced norms for sampled data systems abstract in this paper we consider a general linear interconnection of a continuous time plant and a discrete time controller via sample and hold devices when the closed loop sampled data feedback system is internally stable bounded inputs produce bounded outputs we present some explicit formulae for the induced norm of the closed loop system with l i e peak value and 1 i e integral absolute norms on the input and output signals

spectral approximation of quadratic operator polynomials arising in photonic band structure calculations galerkin spectral approximation theory for non self adjoint quadratic operator polynomials with periodic coefficients is considered the main applications are complex band structure calculations in metallic photonic crystals periodic waveguides and metamaterials we show that the spectrum of the considered operator polynomials consists of isolated eigenvalues of finite multiplicity with a nonzero imaginary part the spectral problem is equivalent to a non compact block operator matrix and norm convergence is shown for a block operator matrix having the same generalized eigenvectors as the original operator convergence rates of finite element discretizations are considered and numerical experiments with the p p version and the h h version of the finite element method confirm the theoretical convergence rates

a function space approach to sampled data control systems and tracking problems this paper presents a new framework for hybrid sampled data control systems instead of considering the state only at sampling instants this paper introduces a function piece during the sampling period as the state and gives an infinite dimensional model with such a state space this gives the advantage that sampled data systems with built in intersample behavior can be regarded as linear time invariant discrete time systems as a result the approach makes it possible to introduce such time invariant concepts as transfer functions poles and zeros to the sampled data systems even with the presence of the intersample behavior in particular tracking problems can be studied in this setting in a simple and unified way and ripples are completely characterized as a mismatch between the intersample reference signal and transmission zero directions this leads to the internal model principle for sampled data systems

l_ 1 discretization for sampled data controller synthesis via piecewise linear approximation this paper develops a new discretization method with piecewise linear approximation for the l 1 optimal controller synthesis problem of sampled data systems which is the problem of minimizing the l infty induced norm of sampled data systems we apply fast lifting on the top of the lifting technique by which the sampling interval 0 h is divided into m subintervals with an equal width the signals on each subinterval are then approximated by linear functions by introducing two types of linearizing operators for input and output which leads to piecewise linear approximation of sampled data systems by using the arguments of preadjoint operators we provide an important inequality that forms a theoretical basis for tackling the l 1 optimal controller synthesis problem of sampled data systems more efficiently than the conventional method more precisely a mathematical basis for the piecewise linear approximation method associated with the convergence rate is shown through this inequality and this suggests that the piecewise linear approximation method may drastically outperform the conventional method in the l 1 optimal controller synthesis problem of sampled data systems we then provide a discretization procedure of sampled data systems by which the l 1 optimal controller synthesis problem is converted to the discrete time l 1 optimal controller synthesis problem finally effectiveness of the proposed method is demonstrated through a numerical example

numerical algorithms based on analytic function values at roots of unity let f z be an analytic or meromorphic function in the closed unit disk sampled at the n th roots of unity based on these data how can we approximately evaluate f z or f m z at a point z in the disk how can we calculate the zeros or poles of f in the disk these questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations partial differential equations and large scale linear algebra we analyze some of the possibilities and emphasize the distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of cauchy integrals we then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension finally we highlight the power of rational in comparison with polynomial approximations for some of these problems

hp finite elements for elliptic eigenvalue problems error estimates which are explicit with respect to lambda h and p convergence rates for finite element discretizations of elliptic eigenvalue problems in the literature usually are of the following form if the mesh width h is fine enough then the eigenvalues resp eigenfunctions converge at some well defined rate in this paper we will determine the maximal mesh width h 0 more precisely the minimal dimension of a finite element space so that the asymptotic convergence estimates hold for h leq h 0 this mesh width will depend on the size and spacing of the exact eigenvalues the spatial dimension and the local polynomial degree of the finite element space for example in the one dimensional case the condition lambda 3 4 h 0 lesssim1 is sufficient for piecewise linear finite elements to compute an eigenvalue lambda with optimal convergence rates as h 0 geq h rightarrow0 it will turn out that the condition for eigenfunctions is slightly more restrictive furthermore we will analyze the dependence of the ratio of the errors of the galerkin approximation and of the best approximation of an eigenfunction on lambda and h in this paper the error estimates for the eigenvalue function are limited to the selfadjoint case however the regularity theory and approximation property cover also the nonselfadjoint case and hence pave the way towards the error analysis of nonselfadjoint eigenvalue function problems

approximation of the electron density of aluminium clusters in tensor product format the tensor structured methods developed recently for the accurate calculation of the hartree and the non local exchange operators have been applied successfully to the ab initio numerical solution of the hartree fock equation for some molecules in the present work we show that the rank structured representation can be gainfully applied to the accurate approximation of the electron density of large aluminium clusters we consider the tucker type decomposition of the electron density of certain aluminium clusters originating from finite element calculations in the framework of the orbital free density functional theory numerical investigations of the tucker approximation of the corresponding electron density reveal the exponential decay of the approximation error with respect to the tucker rank the resulting low rank tensor representation reduces dramatically the storage needs and the computational complexity of the consequent tensor operations on the electron density as main result the rank of the tucker approximation for the accurate representation of the electron density is small and only weakly dependent on the system size for the systems studied here this shows good promise for resolving the electronic structure of materials using tensor structured techniques

i 1 optimal control of multivariable systems with output norm constraints in this paper we consider the l1 optimal control problem for general rational plants it is shown that for plants with no poles or zeros on the unit circle an optimal compensator exists and that the resulting closed loop transfer function is polynomial whenever there are at least as many controls as regulated outputs and at least as many measurements as exogeneous inputs exactly or approximately optimal rational compensators can be obtained by solving a sequence of finite linear programs for the coefficients of a polynomial closed loop transfer function no assumptions on plant poles or zeros are required to obtain at least approximately optimal compensators it is shown that constrained problems in which a set of outputs is regulated subject to l norm constraints on another set of outputs can be solved using a slight modification of the same algorithm

further results on the l 1 analysis of sampled data systems via kernel approximation approach abstractthis paper gives two methods for the l1 analysis of sampled data systems by which we mean computing the l induced norm of sampled data systems this is achieved by developing what we call the kernel approximation approach in the setting of sampled data systems we first consider the lifting treatment of sampled data systems and give an operator theoretic representation of their input output relation we further apply the fast lifting technique by which the sampling interval 0 h is divided into m subintervals with an equal width and provide methods for computing the l induced norm in contrast to a similar approach developed earlier called the input approximation approach we use an idea of kernel approximation in which the kernel function of an input operator and the hold function of an output operator are approximated by piecewise constant or piecewise linear functions furthermore it is shown that the approximation errors in the piecewise constant approximation or piecewise linear appro

spectral instability for some schrodinger operators we show that the condition numbers of isolated eigenvalues of typical non self adjoint differential operators such as the harmonic oscillator may be extremely large we describe a stable procedure for computing the condition numbers for schrodinger operators in one dimension and apply it to the complex resonances of a typical operator with a dilation analytic potential

low rank tensor methods in galerkin based isogeometric analysis abstract the global patch wise geometry map which describes the computational domain is a new feature in isogeometric analysis this map has a global tensor structure inherited from the parametric spline geometry representation the use of this global structure in the discretization of partial differential equations may be regarded as a drawback at first glance as opposed to the purely local nature of high order classical finite elements in this work we demonstrate that it is possible to exploit the regularity of this structure and to identify the great potential for the efficient implementation of isogeometric discretizations first we formulate tensor product b spline bases as well as the corresponding mass and stiffness matrices as tensors in order to reveal their intrinsic structure second we derive an algorithm for the separation of variables in the integrands arising in the discretization this is possible by means of low rank approximation of the integral kernels we arrive at a compact separated representation of the integrals the separated form implies an expression of galerkin matrices as kronecker products of matrix factors with small dimensions this representation is very appealing due to the reduction in both memory consumption and computation times our benchmarks performed using the c library g smo demonstrate that the use of tensor methods in isogeometric analysis possesses significant advantages

the h 2 problem for sampled data systems m for sampled data systems in this paper we pose and give a complete solution to an analog of the h2 problem for sampled data system we motivate and develop a natural generalization of the h2 cost to periodic systems which is then applied to the continuous time closed loop mapping in a sampled data control system it is shown that the cost criterion developed is actually a norm in an space of hilbert schmidt operator valued functions we give state space solutions to the optimal and suboptimal controllers synthesis problems in this new norm by establishing an equivalent standard h2 problem

tensor train decomposition a simple nonrecursive form of the tensor decomposition in d dimensions is presented it does not inherently suffer from the curse of dimensionality it has asymptotically the same number of parameters as the canonical decomposition but it is stable and its computation is based on low rank approximation of auxiliary unfolding matrices the new form gives a clear and convenient way to implement all basic operations efficiently a fast rounding procedure is presented as well as basic linear algebra operations examples showing the benefits of the decomposition are given and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19 dimensional operator

quasi newton methods on grassmannians and multilinear approximations of tensors in this paper we proposed quasi newton and limited memory quasi newton methods for objective functions defined on grassmannians or a product of grassmannians specifically we defined bfgs and limited memory bfgs updates in local and global coordinates on grassmannians or a product of these we proved that when local coordinates are used our bfgs updates on grassmannians share the same optimality property as the usual bfgs updates on euclidean spaces when applied to the best multilinear rank approximation problem for general and symmetric tensors our approach yields fast robust and accurate algorithms that exploit the special grassmannian structure of the respective problems and which work on tensors of large dimensions and arbitrarily high order extensive numerical experiments are included to substantiate our claims

designing rational filter functions for solving eigenvalue problems by contour integration abstract solving nonlinear eigenvalue problems by contour integration requires an effective discretization for the corresponding contour integrals in this paper it is shown that good rational filter functions can be computed using nonlinear least squares optimization techniques as opposed to designing those functions based on a thorough understanding of complex analysis the conditions that such an effective filter function should satisfy are derived and translated in a nonlinear least squares optimization problem solved by optimization algorithms from tensorlab numerical experiments illustrate the validity of this approach

l 1 optimal compensators for continuous time systems previous work has been concerned with minimizing the l 1 norm of an error pulse response for discrete time siso 1 and mimo 2 systems in this paper we study the problem of minimizing the l1 norm of the error impulse response for siso continuous time systems this problem is quite different from the discrete time problem in that irrational solutions are obtained even when the problem data are rational two methods are suggested for the solution of the continuous time problem an exact method which leads to a finite dimensional nonlinear programming problem and an approximate method which leads to a linear programming problem