the largest singular value of e sup x a sub 0 e sup x is convex on convex sets of commuting matrices a short and direct proof of the convexity property is given it is shown that the theorem applies to any convex commuting set of matrices in r sup nxn where a sub 0 in r sup nxn is fixed it is also shown that the result does not hold if x is permitted to be a general square matrix a counterexample is supplied for noncommuting matrices

a finite element method for the simulation of strong and weak discontinuities in solid mechanics in this paper we introduce and analyze a finite element method for elasticity problems with interfaces the method allows for discontinuities internal to the elements in the approximation across the interface we propose a general approach that can handle both perfectly and imperfectly bonded interfaces without modifications of the code for the case of linear elasticity we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh we present numerical examples for the linear case as well as for contact and crack propagation model problems

a trust region spectral bundle method for nonconvex eigenvalue optimization we present a nonsmooth optimization technique for nonconvex maximum eigenvalue functions and for nonsmooth functions which are infinite maxima of eigenvalue functions we prove global convergence of our method in the sense that for an arbitrary starting point every accumulation point of the sequence of iterates is critical the method is tested on several problems in feedback control synthesis

a new computational algorithm for functional inequality constrained optimization problems abstract in this note a computational algorithm is devised for solving a class of functional inequality constrained optimization problems based on a penalty function for illustration a numerical example is solved

a new adaptive backpropagation algorithm based on lyapunov stability theory for neural networks a new adaptive backpropagation bp algorithm based on lyapunov stability theory for neural networks is developed in this paper it is shown that the candidate of a lyapunov function v k of the tracking error between the output of a neural network and the desired reference signal is chosen first and the weights of the neural network are then updated from the output layer to the input layer in the sense that deltav k v k v k 1 0 the output tracking error can then asymptotically converge to zero according to lyapunov stability theory unlike gradient based bp training algorithms the new lyapunov adaptive bp algorithm in this paper is not used for searching the global minimum point along the cost function surface in the weight space but it is aimed at constructing an energy surface with a single global minimum point through the adaptive adjustment of the weights as the time goes to infinity although a neural network may have bounded input disturbances the effects of the disturbances can be eliminated and asymptotic error convergence can be obtained the new lyapunov adaptive bp algorithm is then applied to the design of an adaptive filter in the simulation example to show the fast error convergence and strong robustness with respect to large bounded input disturbances

an immersed discontinuous finite element method for stokes interface problems abstract we present a discontinuous immersed finite element ife method for stokes interface problems on cartesian meshes that do not require the mesh to be aligned with the interface as such the method allows unfitted meshes with elements cut by the interface and thus may contain more than one fluid on these unfitted meshes we construct an immersed q 1 q 0 finite element space according to the location of the interface and pertinent interface jump conditions the proposed q 1 q 0 ife shape functions have several desirable features such as the unisolvence and the partition of unity we present several numerical examples to demonstrate that the proposed ife spaces maintain the optimal approximation capability with respect to the polynomials used we also show that related discontinuous ife solutions of stokes interface problems maintain the optimal convergence rates in both l 2 and broken h 1 norms

nonlinear black box modeling in system identification a unified overview a nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics there has been considerable recent interest in this area with structures based on neural networks radial basis networks wavelet networks and hinging hyperplanes as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules this paper describes all these approaches in a common framework from a user s perspective it focuses on what are the common features in the different approaches the choices that have to be made and what considerations are relevant for a successful system identification application of these techniques it is pointed out that the nonlinear structures can be seen as a concatenation of a mapping form observed data to a regression vector and a nonlinear mapping from the regressor space to the output space these mappings are discussed separately the latter mapping is usually formed as a basis function expansion the basis functions are typically formed from one simple scalar function which is modified in terms of scale and location the expansion from the scalar argument to the regressor space is achieved by a radial or a ridge type approach basic techniques for estimating the parameters in the structures are criterion minimization as well as two step procedures where first the relevant basis functions are determined using data and then a linear least squares step to determine the coordinates of the function approximation a particular problem is to deal with the large number of potentially necessary parameters this is handled by making the number of used parameters considerably less than the number of offered parameters by regularization shrinking pruning or regressor selection

a linearization algorithm for optimizing control systems subject to singular value inequalities the design of feedback linear multivariable systems subject to inequality constraints frequently leads to nonlinear programming problems involving singular values of matrix transfer functions we present a readily implementable and globally convergent algorithm for solving such problems it exploits the structure inherent in singular values differently than do other existing methods

a quadratically convergent local algorithm on minimizing the largest eigenvalue of a symmetric matrix abstract optimization involving eigenvalues arise in many engineering problems we propose a new algorithm on minimizing the largest eigenvalue over an affine family of symmetric matrices under certain assumptions it is shown that if started close enough to the minimizer x the proposed algorithm converges to x quadratically the proposed algorithm can be readily extended to minimizing the largest eigenvalue over an affine family of hermitian matrices also it has been extended to minimizing sums of the largest eigenvalues of a symmetric or hermitian matrix

physical basis of stochastic models of annual flows the main objective of this paper is to demonstrate that annual streamflow time series can be represented by autoregressive and moving average arma processes the method follows the conceptual representation of a watershed given by fiering 1967

an unfitted finite element method based on nitsche s method for elliptic interface problems in this paper we propose a method for the finite element solution of elliptic interface problem using an approach due to nitsche the method allows for discontinuities internal to the elements in the approximation across the interface we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh further we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples

hierarchical fuzzy cmac for nonlinear systems modeling since the fuzzy cerebellar model articulation controller fcmac uses linguistic variables it is highly intuitive and easily comprehended despite the fcmac s good local generalization capability for approximating nonlinear functions and fast learning a normal fcmac requires huge memory and its dimension increases exponentially with the number of inputs in order to overcome the memory explosion problem this paper proposes two types of hierarchical fcmac hfcmac another contribution of the paper is that we give stable learning algorithms for these two hfcmacs backpropagation like approach is applied to train each block with a time varying learning rate which is obtained by the input to state stability technique

a conforming enriched finite element method for stokes interface problems abstract a new conforming enriched finite element method is presented for stokes interface problems with interface unfitted meshes the conforming enriched finite element pair is constructed based on the mini element pair a ghost penalty term is used in the standard discretization form as a stabilization term an inf sup stability result is derived which is uniform with respect to the mesh size our method does not limit the viscosity coefficient to be a piecewise constant finite element errors are proved to be optimal numerical experiments are carried out to validate theoretical results

new cartesian grid methods for interface problems using the finite element formulation new finite element methods based on cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients the triangulations in these methods do not need to fit the interfaces the basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately both non conforming and conforming finite element spaces are considered corresponding interpolation functions are proved to be second order accurate in the maximum norm the conforming finite element method has been shown to be convergent with cartesian triangulations these new methods can be used as finite difference methods numerical examples are provided to support the methods and the theoretical analysis

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

partially penalized immersed finite element methods for elliptic interface problems this article presents new immersed finite element ife methods for solving the popular second order elliptic interface problems on structured cartesian meshes even if the involved interfaces have nontrivial geometries these ife methods contain extra stabilization terms introduced only at interface edges for penalizing the discontinuity in ife functions with the enhanced stability due to the added penalty not only can these ife methods be proven to have the optimal convergence rate in an energy norm provided that the exact solution has sufficient regularity but also numerical results indicate that their convergence rates in both the h 1 norm and the l 2 norm do not deteriorate when the mesh becomes finer which is a shortcoming of the classic ife methods in some situations trace inequalities are established for both linear and bilinear ife functions that are not only critical for the error analysis of these new ife methods but are also of a great potential to be useful in error analysis for other

modeling water table fluctuations by means of a stochastic differential equation the combined system of soil water and shallow groundwater is modeled with simple mass balance equations assuming equilibrium soil moisture conditions this results in an ordinary but nonlinear differential equation of water table depth at a single location if errors in model inputs errors due to model assumptions and parameter uncertainty are lumped and modeled as a wide band noise process a stochastic differential equation sde results a solution for the stationary probability density function is given through use of the fokker planck equation for the nonstationary case where the model inputs are given as daily time series sample functions of water table depth soil saturation and drainage discharge can be simulated by numerically solving the sde these sample functions can be used for designing drainage systems and to perform risk analyses the parameters and noise statistics of the sde are calibrated on time series of water table depths by embedding the sde in a kaiman filter algorithm and using the filter innovations in a filter type maximum likelihood criterion the stochastic model is calibrated and validated at two locations a peat soil with a very shallow water table and a loamy sand soil with a moderately shallow water table it is shown in both cases that sample functions simulated with the sde are able to reproduce a wide range of statistics of water table depth despite its unrealistic assumption of constant inputs the stationary solution derived from the fokker planck equation gives good results for the peat soil most likely because the characteristic response time of the water table is very small

parametric robust structured control design we present a new approach to parametric robust controller design where we compute controllers of arbitrary order and structure which minimize the worst case h infty norm over a pre specified set of uncertain parameters at the core of our method is a nonsmooth minimization method tailored to functions which are semi infinite minima of smooth functions a rich test bench and a more detailed example illustrate the potential of the technique which can deal with complex problems involving multiple possibly repeated uncertain parameters

when is the greatest eigenvalue of a parametrized symmetric matrix a convex function of the parameter let a aij n n be a real symmetric matrix valued function of u rm let u be the greatest eigenvalue of a u we are concerned with the convexity of when a does not depend affinely on u

space time modeling of water table depth using a regionalized time series model and the kalman filter the spatiotemporal variation of shallow water table depth is modeled with a regionalized version of an autoregressive exogenous arx time series model the arx model relates the temporal variation of the water table depth at a single location to a time series of precipitation surplus the arx model is calibrated first at locations where time series of water table depth are available arx parameters at nonvisited locations are estimated through geostatistical interpolation using auxiliary information resulting in a regionalized arx model or rarx model the parameters of the geostatistical model are estimated by embedding the rarx model in a space time kalman filter and minimization of a maximum likelihood criterion built from the filter innovations the resulting state estimator can be used for optimal space time prediction of water table depth network optimization and space time conditional simulation

graph lyapunov function for switching stabilization and distributed computation abstract this paper studies stabilization of discrete time switched linear systems slss using the notion of graph control lyapunov functions gclfs a gclf is a set of lyapunov functions defined on a weighted digraph where each lyapunov function is represented by a node in the digraph and there is a lyapunov inequality associated with each subgraph consisting of a node and its out neighbors the weight of a directed edge indicates the decay or growth rate of the lyapunov functions it is proved that an sls is switching stabilizable if and only if there exists a gclf the main benefits of gclfs are reduced computational cost and conservatism for stabilizability tests besides we show that the proposed gclf framework unifies several control lyapunov functions and the related stabilization theorems moreover we propose a distributed algorithm to evaluate the stabilizability with reduced computational costs by taking benefits of the graph structure of gclfs several examples are given to demonstrate the efficiency of the algorithm

calibration of transfer function noise models to sparsely or irregularly observed time series a method is presented to calibrate transfer function noise tfn models operating at the same frequency as the input auxiliary variables to sparsely or irregularly observed time series of the output target variable once calibrated the tfn models can be used to predict or simulate the output variable at the same frequency as the input variable consequently the method provides a useful tool for filling in gaps of irregularly or sparsely observed hydrological time series although generic and suitable for any type of time series the method is described through the modeling of a time series of groundwater head data with precipitation surplus precipitation minus potential evapotranspiration as input variable first the tfn model is written in vector notation yielding the state equation of a linear discrete stochastic system subsequently the state equation is embedded in a kalman filter algorithm the kalman filter is then combined with a maximum likelihood criterion to obtain estimates of the parameters of the tfn model for small time steps e g 1 day while using sparsely e g two times a month or even irregularly observed time series of groundwater head data the method is illustrated using subsets of time series of groundwater head data with varying regular and irregular observation intervals

second order nonsmooth optimization for h synthesis the standard way to compute h feedback controllers uses algebraic riccati equations and is therefore of limited applicability here we present a new approach to the h output feedback control design problem which is based on nonlinear and nonsmooth mathematical programming techniques our approach avoids the use of lyapunov variables and is therefore flexible in many practical situations

an analysis of a broken p_1 nonconforming finite element method for interface problems we study some numerical methods for solving a second order elliptic problem with interface we introduce an immersed finite element method based on the broken p 1 nonconforming piecewise linear polynomials on interface triangular elements having edge averages as degrees of freedom these linear polynomials are broken to match the homogeneous jump condition along the interface which is allowed to cut through the element we prove optimal orders of convergence in the h 1 and l 2 norm next we propose a mixed finite volume method in the context introduced in s h chou d y kwak and k y kim math comp 72 2003 pp 525 539 using the raviart thomas mixed finite element and this broken p 1 nonconforming element the advantage of this mixed finite volume method is that once we solve the symmetric positive definite pressure equation without lagrangian multiplier the velocity can be computed locally by a simple formula this procedure avoids solving the saddle point problem furthermore we show optimal error estimates of velocity and pressure in our mixed finite volume method numerical results show optimal orders of error in the l 2 norm and broken h 1 norm for the pressure and in the h mathrm div norm for the velocity

a primal dual potential reduction method for problems involving matrix inequalities we describe a potential reduction method for convex optimization problems involving matrix inequalities the method is based on the theory developed by nesterov and nemirovsky and generalizes gonzaga and todd s method for linear programming a worst case analysis shows that the number of iterations grows as the square root of the problem size but in practice it appears to grow more slowly as in other interior point methods the overall computational effort is therefore dominated by the least squares system that taust be solved in each iteration a type of conjugate gradient algorithm can be used for this purpose which results in important savings for two reasons first it allows us to take advantage of the special structure the problems often have e g lyapunov or algebraic riccati inequalities second we show that the polynomial bound on the number ofiterations remains valid even if the conjugate gradient algorithm is not run until completion which in practice can greatly reduce the computational effort per iteration we describe in detail how the algorithm works for optimization problems with l lyapunov inequaliues each of size m we prove an overall worst case operation count of o mssll5 the average case complexity appears to be closer to o m4ll5 this estimate is justified by extensive numerical experimentation and is consistent with other researchers experience with the practical performance of interior point algorithms for linear programming this result means that the computational cost of extending current control theory based on the solution of lyapunov or riccati equations to a theory that is based on the solution of multiple coupled lyapunov or riccati inequalities is modest

physical basis of time series models for water table depths the relationship between precipitation excess and water table depth can be described by empirical time series models such as transfer function noise models autoregressive exogenous variable models arx and threshold autoregressive self exciting open loop models in this paper these models are interpreted in terms of the water balance of a soil column a physically based arx model is used to predict the effect of an intervention on the water table dynamics at two locations it is shown that the physically based arx model predicts the effect of interventions reasonably well

analysis of an xfem discretization for stokes interface problems we consider a stationary stokes interface problem in the discretization the interface is not aligned with the triangulation for the discretization we use the p 1 extended finite element space p 1 xfem for the pressure and the standard conforming p 2 finite element space for the velocity since this pair is not necessarily lbb stable a consistent stabilization term known from the literature is added for the discrete bilinear form an inf sup stability result is derived which is uniform with respect to h mesh size parameter the viscosity quotient mu 1 mu 2 and the position of the interface in the triangulation based on this discretization error bounds are derived an optimal preconditioner for the stiffness matrix corresponding to this pair p 1 xfe for pressure and p 2 fe for velocity is presented the preconditioner has block diagonal form with a multigrid preconditioner for the velocity block and a new schur complement preconditioner optimality of this block preconditioner

nondifferentiable optimization algorithm for designing control systems having singular value inequalities it has been known for some time that many control system design requirements can be expressed as differentiable inequalities more recently it has been shown that important structural properties such as robustness and low noise sensitivity can be expressed as nondifferentiable inequalities involving the singular values of a system or return difference transfer function matrix this paper presents an optimization algorithm which permits all these constraints to be considered

on the extension of constrained optimization algorithms from differentiable to nondifferentiable problems this paper presents three general schemes for extending differentiable optimization algorithms to nondifferentiable problems it is shown that the armijo gradient method phase i phase ii methods of feasible directions and exact penalty function methods have conceptual analogs for problems with locally lipschitz functions and implementable analogs for problems with semismooth functions the exact penalty method has required the development of a new optimality condition

robust local stability of multilayer recurrent neural networks we derive a condition for robust local stability of multilayer recurrent neural networks with two hidden layers the stability condition follows from linking theories about linearization robustness analysis of linear systems under nonlinear perturbation and matrix inequalities a characterization of the basin of attraction of the origin is given in terms of the level set of a quadratic lyapunov function similar to the nl sub q theory the local stability is imposed around the origin and the apparent basin of attraction is made large by applying the criterion while the proven basin of attraction is relatively small due to conservatism of the criterion modification of the dynamic backpropagation by the new stability condition is discussed and illustrated by simulation examples

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

analysis of a stokes interface problem we consider a stationary stokes problem with a piecewise constant viscosity coefficient for the variational formulation of this problem we prove a well posedness result in which the constants are uniform with respect to the jump in the viscosity coefficient we apply a standard discretization with a pair of lbb stable finite element spaces the main result of the paper is an infsup result for the discrete problem that is uniform with respect to the jump in the viscosity coefficient from this we derive a robust estimate for the discretization error we prove that the mass matrix with respect to some suitable scalar product yields a robust preconditioner for the schur complement results of numerical experiments are presented that illustrate this robustness property

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

optimal l sub infinity bounds for disturbance robustness robust disturbance rejection bounds obtained recently by zhu et al are tight with respect to the disturbance set that has a specified l sub 2 norm bound it is shown that the bounds are also tight with respect to a weighted l sub 2 disturbance set that has a specified bound on its outer product by choosing a special weight matrix in the l sub 2 norm the matrix can computed by numerical iteration

tarso modeling of water table depths threshold autoregressive self exciting open loop tarso models are fitted to six time series of water table depths with precipitation excess as input variable basically these models are nonlinear in structure because they incorporate several regimes which are separated by so called thresholds for each well a subset tarso ss tarso model is selected using a bayes information criterion bic ss tarso models are used to simulate realizations of water table depths with lengths of 30 years from which characteristics such as durations of exceedance are computed the simulation performance of the fitted ss tarso models is compared with results obtained from transfer function noise tfn models dynamic regression dr models and with a physical descriptive model called swatre extended with additional autoregressive moving average arma processes for the noise swatre arma as compared to the linear tfn and dr models the ss tarso models perform better because they incorporate several regimes these regimes are the result of different soil layers or drainage levels furthermore it is interesting that ss tarso models show a good relative performance as compared to the swatre arma models a possible reason may be that inputs of swatre are uncertain

nl sub q theory checking and imposing stability of recurrent neural networks for nonlinear modeling it is known that many discrete time recurrent neural networks such as e g neural state space models multilayer hopfield networks and locally recurrent globally feedforward neural networks can be represented as nl sub q systems sufficient conditions for global asymptotic stability and input output stability of nl sub q systems are available including three types of criteria 1 diagonal scaling 2 criteria depending on diagonal dominance 3 condition number factors of certain matrices the paper discusses how narendra s 1990 1991 dynamic backpropagation procedure which is used for identifying recurrent neural networks from i o measurements can be modified with an nl sub q stability constraint in order to ensure globally asymptotically stable identified models an example illustrates how system identification of an internally stable model corrupted by process noise may lead to unwanted limit cycle behavior and how this problem can be avoided by adding the stability constraint

nonlinear system identification using neural state space models applicable to robust control design prediction error learning algorithms for neural state space models are developed both for the deterministic case and the stochastic case with measurement and process noise for the stochastic case a predictor with direct parametrization of the kalman gain by a neural net architecture is proposed expressions for the gradients are derived by applying narendra s sensitivity model approach finally a linear fractional transformation representation is given for neural state space models which makes it possible to use these models obtained from input output measurements on a plant in a standard robust performance control scheme