on the error of linear interpolation and the orientation aspect ratio and internal angles of a triangle in this paper we attempt to reveal the precise relation between the error of linear interpolation on a general triangle and the geometric characters of the triangle taking the model problem of interpolating quadratic functions we derive two exact formulas for the h 1 seminorm and l 2 norm of the interpolation error in terms of the area aspect ratio orientation and internal angles of the triangle these formulas indicate that 1 for highly anisotropic triangular meshes the h 1 seminorm of the interpolation error is almost a monotonically decreasing function of the angle between the orientations of the triangle and the function 2 maximum angle condition is not essential if the mesh is aligned with the function and the aspect ratio is of magnitude sqrt lambda 1 lambda 2 or less where lambda 1 and lambda 2 are the eigenvalues of the hessian matrix of the function with these formulas we identify the optimal triangles which produce the smallest h 1 seminorm of the interpolation error to be the acute isosceles aligned with the solution and of an aspect ratio about 0 8 frac lambda 1 lambda 2 the l 2 norm of the interpolation error depends on the orientation and the aspect ratio of the triangle but not directly on its maximum or minimum angles the optimal triangles for the l 2 norm are those aligned with the solution and of an aspect ratio sqrt lambda 1 lambda 2 these formulas can be used to formulate more accurate mesh quality measures and to derive tighter error bounds for interpolations

deterministic and probabilistic conditions for finite completability of low tucker rank tensor we investigate the fundamental conditions on the sampling pattern i e locations of the sampled entries for finite completability of a low rank tensor given some components of its tucker rank in order to find the deterministic necessary and sufficient conditions we propose an algebraic geometric analysis on the tucker manifold which allows us to incorporate multiple rank components in the proposed analysis in contrast with the conventional geometric approaches on the grassmannian manifold this analysis characterizes the algebraic independence of a set of polynomials defined based on the sampling pattern which is closely related to finite completability of the sampled tensor where finite completability simply means that the number of possible completions of the sampled tensor is finite probabilistic conditions are then studied and a lower bound on the sampling probability is given which guarantees that the proposed deterministic conditions on the sampling patterns for finite completability hold with high probability furthermore using the proposed geometric approach for finite completability we propose a sufficient condition on the sampling pattern that ensures there exists exactly one completion of the sampled tensor

greedy bisection generates optimally adapted triangulations we study the properties of a simple greedy algorithm for the generation of data adapted anisotropic triangulations given a function f the algorithm produces nested triangulations and corresponding piecewise polynomial approximations of f the refinement procedure picks the triangle which maximizes the local lp approximation error and bisect it in a direction which is chosen so to minimize this error at the next step we study the approximation error in the lp norm when the algorithm is applied to c2 functions with piecewise linear approximations we prove that as the algorithm progresses the triangles tend to adopt an optimal aspect ratio which is dictated by the local hessian of f for convex functions we also prove that the adaptive triangulations satisfy a convergence bound which is known to be asymptotically optimal among all possible triangulations

extended optimality properties of the linear quadratic regulator and stationary kalman filter it is shown that the constant gain state feedback solution to the infinite time linear quadratic regulator problem is optimal not only for arbitrarily initial conditions or white noise disturbances but also for worst case l sup 1 disturbances using a similar technique it is shown that in the stationary kalman filter the white disturbance and measurement noise can be replaced by unknown bounded energy signals and that optimality still holds if the performance criterion is a time domain l sub infinity normal of the state estimation errors in the presence of worst case energy signals

the generalized h 2 control problem abstract in this paper we consider the problem of finding an internally stabilizing controller such that the controlled or regulated signals have a guaranteed maximum peak value in response to arbitrary but bounded energy exogenous inputs more specifically we give a complete solution to the problem of finding a stabilizing controller such that the closed loop gain from l 2 0 to l 0 is below a specified level we consider both state feedback and output feedback problems in the state feedback case it is shown that if this synthesis problem is solvable then a solution can be chosen to be a constant state feedback gain necessary and sufficient conditions for the existence of solutions as well as a formula for a state feedback gain that solves this control problem are obtained in terms of a finite dimensional convex feasibility program after showing the separation properties of this synthesis problem the output feedback case is reduced to a state feedback problem it is shown that in the output feedback case generalized h 2 controllers can be chosen to be observer based controllers the theory is demonstrated with a numerical example

a simpler approach to matrix completion this paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix these results improve on prior work by candes and recht 2009 candes and tao 2009 and keshavan et al 2009 the reconstruction is accomplished by minimizing the nuclear norm or sum of the singular values of the hidden matrix subject to agreement with the provided entries if the underlying matrix satisfies a certain incoherence condition then the number of entries required is equal to a quadratic logarithmic factor times the number of parameters in the singular value decomposition the proof of this assertion is short self contained and uses very elementary analysis the novel techniques herein are based on recent work in quantum information theory

a lifting technique for linear periodic systems with applications to sampled data control abstract a lifting technique is developed for periodic linear systems and applied to the h and h 2 sampled data control problems

fundamental sampling patterns for low rank multi view data completion abstract we consider the multi view data completion problem i e to complete a matrix u u 1 u 2 where the ranks of u u1 and u2 are given in particular we investigate the fundamental conditions on the sampling pattern i e locations of the sampled entries for finite completability of such a multi view data given the corresponding rank constraints we provide a geometric analysis on the manifold structure for multi view data to incorporate more than one rank constraint we derive a probabilistic condition in terms of the number of samples per column that guarantees finite completability with high probability finally we derive the guarantees for unique completability numerical results demonstrate reduced sampling complexity when the multi view structure is taken into account as compared to when only low rank structure of individual views is taken into account then we propose an apporach using newton s method to almost achieve these information theoretic bounds for mulit view data retrieval by taking advantage of the rank decomposition and the analysis in this work

adaptive multiresolution analysis based on anisotropic triangulations a simple greedy refinement procedure for the generation of data adapted triangulations is proposed and studied given a function of two variables the algorithm produces a hierarchy of triangulations dj j 0 and piecewise polynomial approximations of on these triangulations the refinement procedure consists in bisecting a triangle in a direction which is chosen so as to minimize the local approximation error in some prescribed norm between and its piecewise polynomial approximation after is bisected the hierarchical structure allows us to derive various approximation tools such as multiresolution analysis wavelet bases adaptive triangulations based either on greedy or optimal cart trees as well as a simple encoding of the corresponding triangulations we give a general proof of convergence in the lp norm of all these approximations numerical tests performed in the case of piecewise linear approximation of functions with analytic expressions or of numerical images illustrate the fact that the refinement procedure generates triangles with an optimal aspect ratio which is dictated by the local hessian of in case of c2 functions

on uniqueness in candecomp parafac one of the basic issues in the analysis of three way arrays by candecomp parafac cp has been the question of uniqueness of the decomposition kruskal 1977 has proved that uniqueness is guaranteed when the sum of thek ranks of the three component matrices involved is at least twice the rank of the solution plus 2 since then little has been achieved that might further qualify kruskal s sufficient condition attempts to prove that it is also necessary for uniqueness except for rank 1 or 2 have failed but counterexamples to necessity have not been detected the present paper gives a method for generating the class of all solutions or at least a subset of that class given a cp solution that satisfies certain conditions this offers the possibility to examine uniqueness for a great variety of specific cp solutions it will be shown that kruskal s condition is necessary and sufficient when the rank of the solution is three but that uniqueness may hold even if the condition is not satisfied when the rank is four or higher

sharp asymptotics of the l p approximation error for interpolation on block partitions adaptive approximation or interpolation takes into account local variations in the behavior of the given function adjusts the approximant depending on it and hence yields the smaller error of approximation the question of constructing optimal approximating spline for each function proved to be very hard in fact no polynomial time algorithm of adaptive spline approximation can be designed and no exact formula for the optimal error of approximation can be given therefore the next natural question would be to study the asymptotic behavior of the error and construct asymptotically optimal sequences of partitions in this paper we provide sharp asymptotic estimates for the error of interpolation by splines on block partitions in mathbb r d we consider various projection operators to define the interpolant and provide the analysis of the exact constant in the asymptotics as well as its explicit form in certain cases

continuous mesh framework part i well posed continuous interpolation error in the context of mesh adaptation riemannian metric spaces have been used to prescribe orientation density and stretching of anisotropic meshes but such structures are only considered to compute lengths in adaptive mesh generators in this article a riemannian metric space is shown to be more than a way to compute a length it is proven to be a reliable continuous mesh model in particular we demonstrate that the linear interpolation error can be evaluated continuously on a riemannian metric space from one hand this new continuous framework proves that prescribing a riemannian metric field is equivalent to the local control in the mathbf l 1 norm of the interpolation error this proves the consistency of classical metric based mesh adaptation procedures on the other hand powerful mathematical tools are available and are well defined on riemannian metric spaces calculus of variations differentiation optimization dots whereas these tools are not defined on discrete meshes

extensive theoretical numerical comparative studies on h2 and generalised h2 norms in sampled data systems abstractthis paper is concerned with linear time invariant lti sampled data systems by which we mean sampled data systems with lti generalised plants and lti controllers and studies their h2 norms from the viewpoint of impulse responses and generalised h2 norms from the viewpoint of the induced norms from l2 to l a new definition of the h2 norm of lti sampled data systems is first introduced through a sort of intermediate standpoint of those for the existing two definitions we then establish unified treatment of the three definitions of the h2 norm through a matrix function g defined on the sampling interval 0 h this paper next considers the generalised h2 norms in which two types of the l norm of the output are considered as the temporal supremum magnitude under the spatial 2 norm and norm of a vector valued function we further give unified treatment of the generalised h2 norms through another matrix function f which is also defined on 0 h through a close connection between g

tutorial solution to the linear quadratic regulator problem using h2 optimal control theory abstract h 2 optimal control theory is used to derive a simple solution to the linear quadratic regulator lqr problem

tensor completion based on nuclear norm minimization for 5d seismic data reconstruction abstractmany standard seismic data processing and imaging techniques require regularly sampled data prestack seismic data are multidimensional signals that can be represented via low rank fourth order tensors in the frequency space f x domain we propose to adopt tensor completion strategies to recover unrecorded observations and to improve the signal to noise ratio of prestack seismic volumes tensor completion can be posed as an inverse problem and solved by minimizing a convex objective function the objective function contains two terms a data misfit and a nuclear norm the data misfit measures the proximity of the reconstructed seismic data to the observations the nuclear norm constraints the reconstructed data to be a low rank tensor in essence we solve the prestack seismic reconstruction problem via low rank tensor completion the cost function of the problem is minimized using the alternating direction method of multipliers we present synthetic examples to illustrate the behavior of the al

an lqr weight selection approach to the discrete generalized h2 control problem it is known that a generalized h2 gh2 controller for a continuous time system takes the form of a conventional kalman filter together with a state feedback control law and that the feedback gains can be determined using convex programming in this paper the discrete time version of the gh2 control problem and the related convex programmes involving weight selection in a full information lqr problem are considered

on kruskal s uniqueness condition for the candecomp parafac decomposition let x be a real valued three way array the candecomp parafac cp decomposition is written as x under bar y under bar 1 y under bar r e where y r are rank 1 arrays and e under bar is a rest term each rank 1 array is defined by the outer product of three vectors a r b r and c r i e y ijk r a i r b j r c k t these vectors make up the r columns of the component matrices a b and c if 2r 2 is less than or equal to the sum of the k ranks of a b and c then the fitted part of the decomposition is unique up to a change in the order of the rank 1 arrays and rescaling counterscaling of each triplet of vectors a r b r c r forming a rank 1 array this classical result was shown by kruskal his proof is however rather inaccessible and does not seem intuitive in order to contribute to a better understanding of cp uniqueness this paper provides an accessible and intuitive proof of kruskal s condition the proof is both self contained and compact and can easily be adapted for the complex valued cp decomposition c 2006 elsevier inc all rights reserved

optimally adapted meshes for finite elements of arbitrary order and w 1 p norms given a function f defined on a bounded polygonal domain omega subset mathbb r 2 and a number n 0 we study the properties of the triangulation mathcal t n that minimizes the distance between f and its interpolation on the associated finite element space over all triangulations of at most n elements the error is studied in the w 1 p semi norm for 1 p and we consider lagrange finite elements of arbitrary polynomial order m 1 we establish sharp asymptotic error estimates as n when the optimal anisotropic triangulation is used a similar problem has been studied in babenko et al east j approx 12 1 71 101 2006 cao j numer anal 45 6 2368 2391 2007 chen et al math comput 76 179 204 2007 cohen multiscale nonlinear and adaptive approximation springer berlin 2009 mirebeau constr approx 32 2 339 383 2010 but with the error measured in the l p norm the extension of this analysis to the w 1 p norm is required in order to match more closely the needs of numerical pde analysis and it is not straightforward in particular the meshes which satisfy the optimal error estimate are characterized by a metric describing the local aspect ratio of each triangle and by a geometric constraint on their maximal angle a second feature that does not appear for the l p error norm our analysis also provides with practical strategies for designing meshes such that the interpolation error satisfies the optimal estimate up to a fixed multiplicative constant

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

variational mesh adaptation ii error estimates and monitor functions the key to the success of a variational mesh adaptation method is to define a proper monitor function which controls mesh adaptation in this paper we study the choice of the monitor function for the variational adaptive mesh method developed in the previous work j comput phys 174 2001 924 two types of monitor functions scalar matrix and non scalar matrix ones are defined based on asymptotic estimates of interpolation error obtained using the interpolation theory of finite element methods the choice of the adaptation intensity parameter is also discussed for each of these monitor functions asymptotic bounds on interpolation error are obtained for adaptive meshes that satisfy the regularity and equidistribution conditions two dimensional numerical results are given to verify the theoretical findings

optimal anisotropic meshes for minimizing interpolation errors in l p norm in this paper we present a new optimal interpolation error estimate in l p norm 1 p for finite element simplicial meshes in any spatial dimension a sufficient condition for a mesh to be nearly optimal is that it is quasi uniform under a new metric defined by a modified hessian matrix of the function to be interpolated we also give new functionals for the global moving mesh method and obtain optimal monitor functions from the viewpoint of minimizing interpolation error in the l p norm some numerical examples are also given to support the theoretical estimates

convolution and hankel operator norms for linear systems some norms are derived for convolution and hankel operators associated with linear time invariant systems in certain cases these norms are shown to be identical the tightest possible bound has been obtained for the absolute magnitude of the euclidean 2 or infinity norm of the time domain response of a multioutput system to certain classes of input disturbance

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

low rank data completion with very low sampling rate using newton s method newton s method is a widely applicable and empirically efficient method for finding the solution to a set of equations the recently developed algebraic geometry analyses provide information theoretic bounds on the sampling rate to ensure the existence of a unique completion a remained open question from these works is to retrieve the sampled data when the sampling rate is very close to the mentioned information theoretic bounds this paper is concerned with proposing algorithms to retrieve the sampled data when the sampling rate is too small and close to the mentioned information theoretic bounds hence we propose a new approach for recovering a partially sampled low rank matrix or tensor when the number of samples is only slightly more than the dimension of the corresponding manifold by solving a set of polynomial equations using newton s method in particular we consider low rank matrix completion matrix sensing and tensor completion each observed entry contributes one polynomial equation in terms of the factors in the rank factorization of the data by exploiting the specific structures of the resulting set of polynomial equations we analytically characterize the convergence regions of the newton s method for matrix completion and matrix sensing through extensive numerical results we show that the proposed approach outperforms the well known methods such as nuclear norm minimization and alternating minimization in terms of the success rate of data recovery noiseless case and peak signal to noise ratio noisy case especially when the sampling rate is very low

state space formulae for discrete time h optimization abstract state space formulae and proofs are given for ail the important steps in discrete time h 8 optimization the steps closely follow the algorithms for continuous time systems but there are some technically involved differences in the detail that make their derivation non trivial

new anisotropic a priori error estimates we prove a priori anisotropic estimates for the l 2 and h 1 interpolation error on linear finite elements the full information about the mapping from a reference element is employed to separate the contribution to the elemental error coming from different directions this new h 1 error estimate does not require the maximal angle condition the analysis has been carried out for the 2d case but may be extended to three dimensions numerical experiments have been carried out to test our theoretical results

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

on optimal triangular meshes for minimizing the gradient error construction of optimal triangular meshes for controlling the errors in gradient estimation for piecewise linear interpolation of data functions in the plane is discussed using an appropriate linear coordinate transformation rigorously optimal meshes for controlling the error in quadratic data functions are constructed it is shown that the transformation can be generated as a curvilinear coordinate transformation for anyc data function with nonsingular hessian matrix using this transformation a construction of nearly optimal meshes for general data functions is described and the error equilibration properties of these meshes discussed in particular it is shown that equilibration of errors is not a sufficient condition for optimality a comparison of meshes generated under several different criteria is made and their equilibrating properties illustrated