multiparameter weights with connections to schauder bases abstract we give several characterizations multiparameter a p t d in terms of strong maximal functions rectangular conjugate functions and rectangular partial sums we also define a way to enumerate z d that corresponds to rectangles we then use these enumerations to give another characterization of rectangular a p t d in terms of trigonometric schauder bases finally we present some applications to principle shift invariant spaces and gabor systems

adjoint complement to viscous finite volume pressure correction methods abstract a hybrid adjoint navier stokes method for the pressure based computation of hydrodynamic objective functional derivatives with respect to the shape is systematically derived in three steps the underlying adjoint partial differential equations and boundary conditions for the frozen turbulence reynolds averaged navier stokes equations are considered in the first step in step two the adjoint discretisation is developed from the primal unstructured finite volume discretisation such that adjoint consistent approximations to the adjoint partial differential equations are obtained following a so called hybrid adjoint approach a unified discrete boundary description is outlined that supports high and low reynolds number turbulent wall boundary treatments for both the adjoint boundary condition and the boundary based gradient formula the third component focused in the development of the industrial adjoint cfd method is the adjoint counterpart to the primal pressure correction algorithm the approach is verified against the direct differentiation method and an application to internal flow problems is presented

bounds in algebraic riccati and lyapunov equations a survey and some new results this paper summarizes and investigates the relations of a number of bounds for the solutions of the algebraic riccati and lyapunov equations that have been reported during the last two decades also presented are bounds for the unified riccati equation using the delta operator and it is shown that some bounds for the continuous and discrete riccati equations can be unified by them

exponential bases paley wiener spaces and applications abstract we investigate the connection between translation bases for paley wiener spaces and exponential fourier bases for a domain we apply these results to the characterization of vector valued time frequency translations of a paley wiener window signal

adjoint sensitivity computations for an embedded boundary cartesian mesh method we present a new approach for the computation of shape sensitivities using the discrete adjoint and flow sensitivity methods on cartesian meshes with general polyhedral cells cut cells at the wall boundaries by directly linearizing geometric constructors of the cut cells an efficient and robust computation of shape sensitivities is achieved for problems governed by the euler equations the accuracy of the linearization is verified by the use of a model problem with an exact solution verification studies show that the convergence rate of gradients is second order for design variables that do not alter the boundary shape and is reduced to first order for shape design problems the approach is applied to several three dimensional problems including inverse design and shape optimization of a re entry capsule in hypersonic flow the results show that reliable approximations of the gradient are obtained in all cases the approach is well suited for geometry control via computer aided design and is especially effective for conceptual design studies with complex geometry where fast turn around time is required

on the proper treatment of grid sensitivities in continuous adjoint methods for shape optimization abstract the continuous adjoint method for shape optimization problems in flows governed by the navier stokes equations can be formulated in two different ways each of which leads to a different expression for the sensitivity derivatives of the objective function with respect to the control variables the first formulation leads to an expression including only boundary integrals it thus has low computational cost but when used with coarse grids its accuracy becomes questionable the second formulation comprises a sum of boundary and field integrals due to the field integrals it has noticeably higher computational cost obtaining though higher accuracy in this paper the equivalence of the two formulations is revisited from the mathematical and particularly the numerical point of view internal and external aerodynamics cases in which the objective function is either the total pressure losses or the force exerted on a solid body are examined and differences in the computed gradients are discussed after identifying the reason behind these discrepancies the adjoint formulation is enhanced by the adjoint to a hypothetical grid displacement model and the new approach is proved to reproduce the accuracy of the second adjoint formulation while maintaining the low cost of the first one

upper matrix bound of the solution for the discrete riccati equation a new upper matrix bound of the solution for the discrete algebraic matrix riccati equation is developed this matrix bound is then used to derive bounds on the eigenvalues trace and determinant of the same solution it is shown that these eigenvalue bounds are less restrictive than previous results

support and approximation properties of hermite splines abstract in this paper we formally investigate two mathematical aspects of hermite splines that are relevant to practical applications we first demonstrate that hermite splines are maximally localized in the sense that the size of their support is minimal among pairs of functions with identical reproduction properties then we precisely quantify the approximation power of hermite splines for the reconstruction of functions and their derivatives it is known that the hermite and b spline approximation schemes have the same approximation order more precisely their approximation error vanishes as o t 4 when the step size t goes to zero in this work we show that they actually have the same asymptotic approximation error constants too therefore they have identical asymptotic approximation properties hermite splines combine optimal localization and excellent approximation power while retaining interpolation properties and closed form expression in contrast to existing similar functions these findings shed a new light on the convenience of hermite splines in the context of computer graphics and geometrical design

reduction of the adjoint gradient formula in the continuous limit we present a new continuous adjoint method for aerodynamic shape optimization using the euler equations which reduces the computational cost of the gradients by reducing the volume integral part of the adjoint gradient formula to a surface integral the savings are particularly significant for three dimensional aerodynamic shape optimization problems on general unstructured and overset meshes in order to validate the concept the new gradient equations have been tested for various aerodynamic shape optimization problems including an inverse problem for threedimensional wing configurations and drag minimization problems of a single element airfoil and a three dimensional wing fuselage configuration in order to assess their accuracy the results are compared with finite difference gradients complex step gradients and gradients calculated by the previous adjoint method which includes a volume integral thomas v jones professor of engineering department of aeronautics and astronautics stanford university aiaa fellow postdoctoral fellow department of aeronautics and astronautics stanford university aiaa member

the structure of shift invariant subspaces of l2 rn using the range function approach to shift invariant spaces in l2 rn we give a simple characterization of frames and riesz families generated by shifts of a countable set of generators in terms of their behavior on subspaces of l2 zn this in turn gives a simplified approach to the analysis of frames and riesz families done by gramians and dual gramians we prove a decomposition of a shift invariant space into the orthogonal sum of spaces each of which is generated by a quasi orthogonal generator as an application of this fact we characterize shift preserving operators in terms of range operators and prove some facts about the dimension function

vector sampling expansions in shift invariant subspaces multi input multi output mimo sampling scheme which is motivated by applications in multi channel deconvolution and multi source separation has been investigated in many aspects common for most of results on mimo systems is that the input signals are supposed to be band limited in this paper we study vector sampling expansions on general finitely generated shift invariant subspaces necessary and sufficient conditions for a vector sampling theorem to hold are given we also give several examples to illustrate the main result

triangular wavelet based finite elements via multivalued scaling equations this paper develops a class of triangular finite elements based on l2 rn orthonormal compactly supported wavelets as in the case of recently derived tensor product wavelet elements the triangular elements are defined by deriving an algebraic eigenvalue problem characterizing the entries of the elemental matrices these entries are uniquely defined by normalization conditions derived from the polynomial reproducing properties of the wavelet scaling functions the work herein generalizes the development of the tensor product wavelet elements in that the geometry of the triangular elemental domain is characterized using a multivalued scaling equation the multivalued scaling function is used to represent the self similar tiling of rn onto itself by triangular domains

iterative matrix bounds and computational solutions to the discrete algebraic riccati equation bilateral matrix bounds for the solution of the discrete algebraic riccati equation dare are presented they are new or tighter than the existing bound computational algorithms to solve the dare follow

gabor schauder bases and the balian low theorem the balian low theorem is a strong form of the uncertainty principle for gabor systems that form orthonormal or riesz bases for l2 r in this paper we investigate the balian low theorem in the setting of schauder bases we prove that new weak versions of the balian low theorem hold for gabor schauder bases but we constructively demonstrate that several variants of the blt can fail for gabor schauder bases that are not riesz bases we characterize a class of gabor schauder bases in terms of the zak transform and product a2 weights the riesz bases correspond to the special case of weights that are bounded away from zero and infinity

recovering signals from lowpass data the problem of recovering a signal from its low frequency components occurs often in practical applications due to the lowpass behavior of many physical systems here we study in detail conditions under which a signal can be determined from its low frequency content we focus on signals in shift invariant spaces generated by multiple generators for these signals we derive necessary conditions on the cutoff frequency of the lowpass filter as well as necessary and sufficient conditions on the generators such that signal recovery is possible when the lowpass content is not sufficient to determine the signal we propose appropriate pre processing that can improve the reconstruction ability in particular we show that modulating the signal with one or more mixing functions prior to lowpass filtering can ensure the recovery of the signal in many cases and reduces the necessary bandwidth of the filter

sensor placement for optimal estimation of vector valued diffusion processes abstract diffusion processes are commonplace in many scientific disciplines as they describe a broad range of physical phenomenon consider a diffusion process observed through linear sensors with additive white noise we derive the optimal placement of these sensors for estimating this process where optimality is defined in terms of the mean squared estimation error mse of the state given past observations we consider two cases first we assume the sensors to be orthogonal we show in this case that the minimum mse is related to the nuclear norm of the system matrix of the process second we remove the orthogonality constraint and show that the mse is related to the schatten p norm of the system matrix of the process and the optimal sensors are proportional its matrix cube root we present simulation results illustrating the fact that the gain afforded by optimizing the choice of sensors depends on the ratio p n where n is the dimension of the system and p the dimension of the wiener processes driving it and this gain is in general large especially when p n is small

optimal pruning for multi step sensor scheduling in the considered linear gaussian sensor scheduling problem only one sensor out of a set of sensors performs a measurement to minimize the estimation error over multiple time steps in a computationally tractable fashion the so called information based pruning algorithm is proposed it utilizes the information matrices of the sensors and the monotonicity of the riccati equation this allows ordering sensors according to their information contribution and excluding many of them from scheduling additionally a tight lower is calculated for branch and bound search which further improves the pruning performance

extra invariance of shift invariant spaces on lca groups this article generalizes recent results in the extra invariance for shift invariant spaces to the context of lca groups let g be a locally compact abelian lca group and k a closed subgroup of g a closed subspace of l2 g is called k invariant if it is invariant under translations by elements of k assume now that h is a countable uniform lattice in g and m is any closed subgroup of g containing h in this article we study necessary and sufficient conditions for an h invariant space to be m invariant as a consequence of our results we prove that for each closed subgroup m of g containing the lattice h there exists an h invariant space s that is exactly m invariant that is s is not invariant under any other subgroup m containing h we also obtain estimates on the support of the fourier transform of the generators of the h invariant space related to its m invariance

the structure of shift invariant spaces on a locally compact abelian group abstract we investigate shift invariant subspaces of l 2 g where g is a locally compact abelian group we show among other things that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a parseval frame

the structure of finitely generated shift invariant spaces in l2 rd a simple characterization is given of finitely generated subspaces of l2 rd which are invariant under translation by any multi integer and is used to give conditions under which such a space has a particularly nice generating set namely a basis and more than that a basis with desirable properties such as stability orthogonality or linear independence the last property makes sense only for local spaces i e shift invariant spaces generated by finitely many compactly supported functions and special attention is paid to such spaces as an application we prove that the approximation order provided by a given local space is already provided by the shift invariant space generated by just one function with this function constructible as a finite linear combination of the finite generating set for the whole space hence compactly supported this settles a question of some 20 years standing

upper bounds for the solution of the discrete riccati equation upper bounds for individual eigenvalues and for summations of eigenvalues including the trace of the solution of the discrete algebraic riccati equation are presented some are new and some supplement bounds in the literature

submodularity and greedy algorithms in sensor scheduling for linear dynamical systems this paper focuses on sensor scheduling for state estimation which consists of a network of noisy sensors and a discrete time linear system with process noise as an energy constraint only a subset of sensors can take a measurement at each time step these measurements are fused into a common state estimate using a kalman filter and the goal is to schedule the sensors to minimize the estimation error at a terminal time a simple approach is to greedily choose sensors at each time step to minimize the estimation error at the next time step recent work has shown that this greedy algorithm outperforms other well known approaches results have been established to show that the estimation error is a submodular function of the sensor schedule submodular functions have a diminishing returns property that ensures the greedy algorithm yields near optimal performance as a negative result we show that most commonly used estimation error metrics are not in general submodular functions this disproves an established result we then provide sufficient conditions on the system for which the estimation error is a submodular function of the sensor schedule and thus the greedy algorithm yields performance guarantees

automated divertor target design by adjoint shape sensitivity analysis and a one shot method as magnetic confinement fusion progresses towards the development of first reactor scale devices computational tokamak divertor design is a topic of high priority presently edge plasma codes are used in a forward approach where magnetic field and divertor geometry are manually adjusted to meet design requirements due to the complex edge plasma flows and large number of design variables this method is computationally very demanding on the other hand efficient optimization based design strategies have been developed in computational aerodynamics and fluid mechanics such an optimization approach to divertor target shape design is elaborated in the present paper a general formulation of the design problems is given and conditions characterizing the optimal designs are formulated using a continuous adjoint framework design sensitivities can be computed at a cost of only two edge plasma simulations independent of the number of design variables furthermore by using a one shot method the entire optimization problem can be solved at an equivalent cost of only a few forward simulations the methodology is applied to target shape design for uniform power load in simplified edge plasma geometry

estimate of aliasing error for non smooth signals prefiltered by quasi projections into shift invariant spaces an ideal analog to digital a d converter prefilters a signal by an ideal lowpass filter recent research on a d conversion based on shift invariant spaces reveals that prefiltering signals by quasiprojections into shift invariant spaces provides more flexible choices in designing an a d conversion system of high accuracy this paper focuses on the accuracy of such prefiltering in which the aliasing error e sub f sup spl lambda is found to behave like e sub f sup spl lambda sub 2 o spl lambda sup spl alpha with respect to the dilation spl lambda of the underlying shift invariant space provided that the input signal f is lipschitz spl alpha continuous a formula to calculate the coefficient of the decay rate is also figured out in this paper

near optimal sensor placements in gaussian processes theory efficient algorithms and empirical studies when monitoring spatial phenomena which can often be modeled as gaussian processes gps choosing sensor locations is a fundamental task there are several common strategies to address this task for example geometry or disk models placing sensors at the points of highest entropy variance in the gp model and a d or e optimal design in this paper we tackle the combinatorial optimization problem of maximizing the mutual information between the chosen locations and the locations which are not selected we prove that the problem of finding the configuration that maximizes mutual information is np complete to address this issue we describe a polynomial time approximation that is within 1 1 e of the optimum by exploiting the submodularity of mutual information we also show how submodularity can be used to obtain online bounds and design branch and bound search procedures we then extend our algorithm to exploit lazy evaluations and local structure in the gp yielding significant speedups we also extend our approach to find placements which are robust against node failures and uncertainties in the model these extensions are again associated with rigorous theoretical approximation guarantees exploiting the submodularity of the objective function we demonstrate the advantages of our approach towards optimizing mutual information in a very extensive empirical study on two real world data sets

smooth and robust solutions for dirichlet boundary control of fluid solid conjugate heat transfer problems we study a new optimization scheme that generates smooth and robust solutions for dirichlet velocity boundary control dvbc of conjugate heat transfer cht processes the solutions to the dvbc of the incompressible navier stokes equations are typically nonsmooth due to the regularity degradation of the boundary stress in the adjoint navier stokes equations this nonsmoothness is inherited by the solutions to the dvbc of cht processes since the cht process couples the navier stokes equations of fluid motion with the convection diffusion equations of fluid solid thermal interaction our objective in the cht boundary control problem is to select optimally the fluid inflow profile that minimizes an objective function that involves the sum of the mismatch between the temperature distribution in the fluid system and a prescribed temperature profile and the cost of the control our strategy to resolve the nonsmoothness of the boundary control solution is based on two features namely the objective function with a regularization term on the gradient of the control profile on both the continuous and the discrete levels and the optimization scheme with either explicit or implicit smoothing effects such as the smoothed steepest descent and the limited memory broyden fletcher goldfarb shanno l bfgs methods our strategy to achieve the robustness of the solution process is based on combining the smoothed optimization scheme with the numerical continuation technique on the regularization parameters in the objective function in the section of numerical studies we present two suites of experiments in the first one we demonstrate the feasibility and effectiveness of our numerical schemes in recovering the boundary control profile of the standard case of a poiseuille flow in the second one we illustrate the robustness of our optimization schemes via solving more challenging dvbc problems for both the channel flow and the flow past a square cylinder which use initial control profiles far from optimal and require the numerical continuation technique applied on regularization parameters we believe our solution strategy is general and can be applied to other large scale optimal control problems which involve multiphysics processes and require smooth approximations to the optimal control profile

on a stochastic sensor selection algorithm with applications in sensor scheduling and sensor coverage in this note we consider the following problem suppose a set of sensors is jointly trying to estimate a process one sensor takes a measurement at every time step and the measurements are then exchanged among all the sensors what is the sensor schedule that results in the minimum error covariance we describe a stochastic sensor selection strategy that is easy to implement and is computationally tractable the problem described above comes up in many domains out of which we discuss two in the sensor selection problem there are multiple sensors that cannot operate simultaneously e g sonars in the same frequency band thus measurements need to be scheduled in the sensor coverage problem a geographical area needs to be covered by mobile sensors each with limited range thus from every position the sensors obtain a different view point of the area and the sensors need to optimize their trajectories the algorithm is applied to these problems and illustrated through simple examples

eigenvalue inequalities for matrix product we present a family of eigenvalue inequalities for the product of a hermitian matrix and a positive semidefinite matrix our theorem contains or extends some existing results on trace and eigenvalues

systematic continuous adjoint approach to viscous aerodynamic design on unstructured grids doi 10 2514 1 24859 a continuous adjoint approach to aerodynamic design for viscous compressible flow on unstructured grids is developed sensitivity gradients arecomputed using tools ofshapedeformation of boundaryintegrals theresulting expressions involve second order derivatives of the flow variables that require numerical solvers with greater than second order accuracy a systematic way of reducing the order of these terms is presented the accuracy of the sensitivity derivatives is assessed by comparison with finite difference computations and the validity of the overall methodology is illustrated with several design examples

sensor selection for kalman filtering of linear dynamical systems complexity limitations and greedy algorithms abstract we consider the problem of selecting an optimal set of sensors to estimate the states of linear dynamical systems specifically the goal is to choose at design time a subset of sensors satisfying certain budget constraints from a given set in order to minimize the trace of the steady state a priori or a posteriori error covariance produced by a kalman filter we show that the a priori and a posteriori error covariance based sensor selection problems are both np hard even under the additional assumption that the system is stable we then provide bounds on the worst case performance of sensor selection algorithms based on the system dynamics and show that greedy algorithms are optimal for a certain class of systems however as a negative result we show that certain typical objective functions are not submodular or supermodular in general while this makes it difficult to evaluate the performance of greedy algorithms for sensor selection outside of certain special cases we show via simulations that these greedy algorithms perform well in practice

deterministic sensor selection for centralized state estimation under limited communication resource this paper studies a sensor selection problem a group of sensors measure the state of a process and send their measurements to a remote estimator due to communication constraints only limited sensors are allowed to communicate with the estimator the paper intends to answer which sensors should be chosen such that the estimation performance of the estimator is optimized both reliable and packet dropping channels are considered it is required to minimize the steady state estimation error covariance for reliable channels and to minimize the upper bound of the expected estimation error covariance for packet dropping channels for both scenarios the original optimization problems are transformed to problems which can be solved by convex optimization techniques

u invariant sampling and reconstruction in atomic spaces with multiple generators given a u invariant sampling scheme on a hilbert space h this paper characterizes atomic subspaces a of h such that every signal x a can be reconstructed from its generalized samples acquired with the u invariant sampling scheme if signal recovery is possible a linear filter is derived which reconstructs the signal from the samples moreover the paper gives necessary and sufficient conditions on a sequence of sampling functions such that it forms a pseudoframe or pseudo riesz basis for a given atomic subspace a

the structure of shift modulation invariant spaces the rational case in this paper we study structural properties of shift modulation invariant smi spaces also called gabor subspaces or weyl heisenberg subspaces in the case when shift and modulation lattices are rationally dependent we prove the characterization of smi spaces in terms of range functions analogous to the well known description of shift invariant spaces c de boor r devore a ron the structure of finitely generated shift invariant spaces in l2 r d j funct anal 119 1994 37 78 m bownik the structure of shift invariant subspaces of l2 r n j funct anal 177 2000 282 309 h helson lectures on invariant subspaces academic press new york london 1964 we also give a simple characterization of frames and riesz sequences in terms on their behavior of the fibers of the range function next we prove several orthogonal decomposition results of smi spaces into simpler blocks called principal smi spaces then this is used to characterize operators invariant under both shifts and modulations in terms of families of linear maps acting on the fibers of the range function we also introduce the fundamental concept of the dimension function for smi spaces as a result this leads to the classification of unitarily equivalent smi spaces in terms of their dimension functions finally we show several results illustrating our fiberization techniques to characterize dual gabor frames

on sampling in shift invariant spaces a necessary and sufficient condition for sampling in the general framework of shift invariant spaces is derived then this result is applied respectively to the regular sampling and the perturbation of regular sampling in shift invariant spaces a simple necessary and sufficient condition for regular sampling in shift invariant spaces is attained furthermore an improved estimate for the perturbation is derived for the perturbation of regular sampling in shift invariant spaces the derived estimate is easy to calculate and shown to be optimal in some shift invariant spaces the algorithm to calculate the reconstruction frame is also presented

an endpoint balian low theorem for schauder bases abstract we prove two new results on the validity of the balian low theorem in the setting of schauder bases our first main result proves an endpoint balian low theorem for gabor systems that form schauder bases for l 2 r if g l 2 r is compactly supported and 2 g 2 d then the gabor system g g 1 1 cannot be a schauder basis of so called type our second main result proves that the classical balian low theorem for orthonormal bases and riesz bases fails in the setting of schauder bases in particular given 0 there exists g l 2 r such that g g 1 1 is a schauder basis for l 2 r and such that t 3 g t 2 d t and 3 g 2 d

regular families of kernels for nonlinear approximation abstract this article studies sufficient conditions on families of approximating kernels which provide n term approximation errors from an associated nonlinear approximation space which match the best known orders of n term wavelet expansion these conditions provide a framework which encompasses some notable approximation kernels including splines cardinal functions and many radial basis functions such as the gaussians and general multiquadrics examples of such kernels are given to justify the criteria additionally the techniques involved allow for some new results on n term greedy interpolation of sobolev functions via radial basis functions

on stability of schauder bases of integer translates abstract we apply the garnett jones distance to the analysis of schauder bases of translates a special role is played by periodization functions p with ln p in the closure of l in bmo t in particular for schauder bases with such periodization functions we study the corresponding coefficient space we also use the garnett jones distance approach to show the stability of bases of translates with respect to convolution powers the case of democratic conditional schauder bases of translates is emphasized as well

continuous adjoint method for unstructured grids adjoint based shape optimization methods have proven to be computationally efficient for aerodynamic problems the majority of the studies on adjoint methods have used structured grids to discretize the computational domain because of the potential advantages of unstructured grids for complex configurations in this study we have developed and validated a continuous adjoint formulation for unstructured grids the hurdles posed in the computation of the gradient for unstructured grids are resolved by using a reduced gradient formulation the methods to impose thickness constraints on unstructured grids are also discussed the results for two and three dimensional simulations of airfoils and wings in inviscid transonic flow are used to validate the design procedure finally the design procedure is applied to redesign the shape of a transonic business jet configuration we were able to reduce the inviscid drag of the aircraft from 235 to 216 counts resulting in a shock free wing although the euler equations are the focus of the study in this paper of the adjoint based approach the solution of the adjoint system and gradient formulation can be conceptually extended to viscous flows the approach presented in this study has been successfully used by the first and third authors for viscous flows using structured grids however particular aspects of the design process such as the robustness of the mesh deformation process for unstructured grids need more attention for viscous flows and are therefore the subject of ongoing research