on modified factorizations for large scale linearly constrained optimization we consider the algebraic issues concerning the solution of general large scale linearly constrained nonlinear optimization problems particular attention is given to suitable methods for solving the linear systems that occur at each iteration of such methods the main issue addressed is how to ensure that a quadratic model of the objective function is positive definite in the null space of the constraints while neither adversely affecting the convergence of newton s method nor incurring a significant computational overhead numerical evidence to support the theoretical developments is provided

approximate factorization constraint preconditioners for saddle point matrices we consider the application of the conjugate gradient method to the solution of large symmetric indefinite linear systems special emphasis is put on the use of constraint preconditioners and a new factorization that can reduce the number of flops required by the preconditioning step results concerning the eigenvalues of the preconditioned matrix and its minimum polynomial are given numerical experiments validate these conclusions

error estimates of stochastic optimal neumann boundary control problems we study mathematically and computationally optimal control problems for stochastic partial differential equations with neumann boundary conditions the control objective is to minimize the expectation of a cost functional and the control is of the deterministic boundary value type mathematically we prove the existence of an optimal solution and of a lagrange multiplier we represent the input data in terms of their karhunen loeve expansions and deduce the deterministic optimality system of equations computationally we approximate the finite element solution of the optimality system and estimate its error through the discretizations with respect to both spatial and random parameter spaces

block diagonal and constraint preconditioners for nonsymmetric indefinite linear systems part i theory we study block diagonal preconditioners and an efficient variant of constraint preconditioners for general two by two block linear systems with zero 2 2 block we derive block diagonal preconditioners from a splitting of the 1 1 block of the matrix from the resulting preconditioned system we derive a smaller so called related system that yields the solution of the original problem solving the related system corresponds to an efficient implementation of constraint preconditioning we analyze the properties of both classes of preconditioned matrices in particular their spectra using analytical results we show that the related system matrix has the more favorable spectrum which in many applications translates into faster convergence for krylov subspace methods we show that fast convergence depends mainly on the quality of the splitting a topic for which a substantial body of theory exists our analysis also provides a number of new relations between block diagonal preconditioners and constraint preconditioners for constrained problems solving the related system produces iterates that satisfy the constraints exactly just as for systems with a constraint preconditioner finally for the lagrange multiplier formulation of a constrained optimization problem we show how scaling nonlinear constraints can dramatically improve the convergence for linear systems in a newton iteration our theoretical results are confirmed by numerical experiments on a constrained optimization problem we consider the general nonsymmetric nonsingular case our only additional requirement is the nonsingularity of the schur complement type matrix derived from the splitting that defines the preconditioners in particular the 1 2 block need not equal the transposed 2 1 block and the 1 1 block might be indefinite or even singular this is the first paper in a two part sequence in the second paper we will study the use of our preconditioners in a variety of applications

calculation of the hidden symmetry operator in symmetric quantum mechanics in a recent paper it was shown that if a hamiltonian h has an unbroken symmetry then it also possesses a hidden symmetry represented by the linear operator the operator commutes with both h and the inner product with respect to is associated with a positive norm and the quantum theory built on the associated hilbert space is unitary in this paper it is shown how to construct the operator for the non hermitian symmetric hamiltonian h 1 2p2 1 2x2 ix3 using perturbative techniques it is also shown how to construct the operator for h 1 2p2 1 2x2 x4 using nonperturbative methods

pseudo hermiticity and generalized pt and cpt symmetries we study certain linear and antilinear symmetry generators and involution operators associated with pseudo hermitian hamiltonians and show that the theory of pseudo hermitian operators provides a simple explanation for the recent results of bender brody and jones quant ph 0208076 on the cpt symmetry of a class of pt symmetric non hermitian hamiltonians we present a natural extension of these results to the class of diagonalizable pseudo hermitian hamiltonians h with a discrete spectrum in particular we introduce generalized parity p time reversal t and charge conjugation c operators and establish the pt and cpt invariance of h

karhunen loeve approximation of random fields by generalized fast multipole methods kl approximation of a possibly instationary random field a x l2 dp l d subject to prescribed meanfield ea x a x dp and covariance va x x a x ea x a x ea x dp in a polyhedral domain d rd is analyzed we show how for stationary covariances va x x ga x x with ga z analytic outside of z 0 an m term approximate kl expansion am x of a x can be computed in log linear complexity the approach applies in arbitrary domains d and for nonseparable covariances ca it involves galerkin approximation of the kl eigenvalue problem by discontinuous finite elements of degree p 0 on a quasiuniform possibly unstructured mesh of width h in d plus a generalized fast multipole accelerated krylov eigensolver the approximate kl expansion am x of a x has accuracy o exp bm1 d if ga is analytic at z 0 and accuracy o m k d if ga is ck at zero it is obtained in o mn logn b operations where n o h d

on the stability of the reduced basis method for stokes equations in parametrized domains we present an application of reduced basis method for stokes equations in domains with affine parametric dependence the essential components of the method are i the rapid convergence of global reduced basis approximations galerkin projection onto a space wn spanned by solutions of the governing partial differential equation at n selected points in parameter space ii the off line on line computational procedures decoupling the generation and projection stages of the approximation process the operation count for the on line stage in which given a new parameter value we calculate an output of interest depends only on n typically very small and the parametric complexity of the problem the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation design optimization and real time control particular attention is given i to the pressure treatment of incompressible stokes problem ii to find an equivalent inf sup condition that guarantees stability of reduced basis solutions by enriching the reduced basis velocity approximation space with the solutions of a supremizer problem iii to provide algebraic stability of the problem by reducing the condition number of reduced basis matrices using an orthonormalization procedure applied to basis functions iv to reduce computational costs in order to allow real time solution of parametrized problem

on uniform asymptotic stability of time varying parameterized discrete time cascades recently a framework for controller design of sampled data nonlinear systems via their approximate discrete time models has been proposed in the literature in this paper we develop novel tools that can be used within this framework and that are useful for tracking problems in particular results for stability analysis of parameterized time varying discrete time cascaded systems are given this class of models arises naturally when one uses an approximate discrete time model to design a stabilizing or tracking controller for a sampled data plant while some of our results parallel their continuous time counterparts the stability properties that are considered the conditions that are imposed and the the proof techniques that are used are tailored for approximate discrete time systems and are technically different from those in the continuous time context a result on constructing strict lyapunov functions from nonstrict ones that is of independent interest is also presented we illustrate the utility of our results in the case study of the tracking control of a mobile robot this application is fairly illustrative of the technical differences and obstacles encountered in the analysis of discrete time parameterized systems

reduced basis approximation of parametrized optimal flow control problems for the stokes equations this paper extends the reduced basis method for the solution of parametrized optimal control problems presented in negri et al 2013 to the case of noncoercive elliptic equations such as the stokes equations we discuss both the theoretical properties with particular emphasis on the stability of the resulting double nested saddle point problems and on aggregated error estimates and the computational aspects of the method then we apply it to solve a benchmark vorticity minimization problem for a parametrized bluff body immersed in a two or a three dimensional flow through boundary control demonstrating the effectivity of the methodology

on signed incomplete cholesky factorization preconditioners for saddle point systems limited memory incomplete cholesky factorizations can provide robust preconditioners for sparse symmetric positive definite linear systems in this paper the focus is on extending the approach to sparse symmetric indefinite systems in saddle point form a limited memory signed incomplete cholesky factorization of the form ldl t is proposed where the diagonal matrix d has entries pm 1 the main advantage of this approach is its simplicity as it avoids the use of numerical pivoting instead a global shift strategy involving two shifts one for the 1 1 block and one for the 2 2 block of the saddle point matrix is used to prevent breakdown and to improve performance the matrix is optionally prescaled and preordered using a standard sparse matrix ordering scheme that is then postprocessed to give a constrained ordering that reduces the likelihood of breakdown and need for shifts the use of intermediate memory memory used in the construction of the incomplete factorization but subsequently d

least squares finite element methods for optimization and control problems for the stokes equations the approximate solution of optimization and control problems for systems governedby the stokes equations is considered modern computational techniques for such problems are predominantly based on the application of the lagrange multiplier rule while penalty formulations even though widely used in other settings have not enjoyed the same level of popularity for this class of problems a discussion is provided that explains why naively defined penalty methods may not be practical then practical penalty methods are defined using methodologies associated with modern least squares finite element methods the advantages with respect to efficiency of penalty leasts squares methods for optimal control problems compared to methods based on lagrange multipliers are highlighted a tracking problem for the stokes system is used for illustrative purposes

comparison of approaches for random pde optimization problems based on different matching functionals in this article we consider an optimal control problem for an elliptic partial differential equation with random inputs to determine an applicable deterministic control f x we consider the four cases which we compare for efficiency and feasibility we prove the existence of optimal states adjoint states and optimality conditions for each cases we also derive the optimality systems for the four cases the optimality system is then discretized by a standard finite element method and sparse grid collocation method for physical space and probability space respectively the numerical experiments are performed for their efficiency and feasibility

a reduced basis approach for variational problems with stochastic parameters application to heat conduction with variable robin coefficient in this work a reduced basis rb approach is used to solve a large number of boundary value problems bvps parametrized by a stochastic input expressed as a karhunen loeve expansion in order to compute outputs that are smooth functionals of the random solution fields the rb method proposed here for variational problems parametrized by stochastic coefficients bears many similarities to the rb approach developed previously for deterministic systems however the stochastic framework requires the development of new a posteriori estimates for statistical outputs such as the first two moments of integrals of the random solution fields these error bounds in turn permit efficient sampling of the input stochastic parameters and fast reliable computation of the outputs in particular in the many query context

certified reduced basis methods for parametrized distributed elliptic optimal control problems with control constraints in this paper we employ the reduced basis method for the efficient and reliable solution of parametrized optimal control problems governed by scalar coercive elliptic partial differential equations we consider the standard linear quadratic problem setting with distributed control and unilateral control constraints for this problem class we propose two different reduced basis approximations and associated error estimation procedures in our first approach we directly consider the resulting optimality system introduce suitable reduced basis approximations for the state adjoint control and lagrange multipliers and use a projection approach to bound the error in the reduced optimal control for our second approach we first reformulate the optimal control problem using a slack variable then develop a reduced basis approximation for the slack problem by suitably restricting the solution space and finally derive error bounds for the slack based optimal control we discuss benefits and drawbacks of b

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

on uniform boundedness of parameterized discrete time systems with decaying inputs applications to cascades a framework for controller design of sampled data nonlinear systems via their approximate discrete time models has been established recently within this framework naturally arises the need to investigate stability properties of parameterized discrete time systems further results that guarantee appropriate stability of the parameterized family of discrete time systems that is used within this framework have been also established for systems with cascaded structure a fundamental condition that is required in this framework is uniform boundedness of solutions of the cascade however this is difficult to check in general in this paper we provide a range of sufficient conditions for uniform boundedness that are easier to check these results further contribute to the toolbox for controller design of sampled data nonlinear systems via their approximate discrete time models

a preconditioner for optimal control problems constrained by stokes equation with a time harmonic control in this article we construct an efficient preconditioner for solving the algebraic systems arising from discretized optimal control problems with time periodic stokes equations based on a preconditioning technique for stationary stokes constrained optimal control problems considered in an earlier paper by the authors a simplified analysis of the derivation of the preconditioner and its properties is presented the preconditioner is fully parameter independent and the condition number of the corresponding preconditioned matrix is bounded by 2 the so constructed preconditioner is favourably compared with another robust preconditioner for the same problem

optimal control with stochastic pde constraints and uncertain controls abstract the optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed the lagrangian that defines the problem is postulated in terms of stochastic functions with the control function possibly decomposed into an unknown deterministic component and a known zero mean stochastic component the extra freedom provided by the stochastic dimension in defining cost functionals is explored demonstrating the scope for controlling statistical aspects of the system response one shot stochastic finite element methods are used to find approximate solutions to control problems it is shown that applying the stochastic collocation finite element method to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional thereby forgoing the primary advantage of the collocation method over the stochastic galerkin method for the considered problem the application of the presented methods is demonstrated through a number of numerical examples the presented framework is sufficiently general to also consider a class of inverse problems and numerical examples of this type are also presented

a general stability criterion for time varying systems using a modified detectability condition uniform asymptotic stability for general nonlinear time varying systems is investigated from the output to state viewpoint a unified criterion is proposed using a modified detectability condition and certain integral inequalities the criterion extends the krasovskii lasalle theorem to nonperiodic time varying systems an example taken from the tracking control of mobile robots is given to illustrate that much as the invariance principle of lasalle is used in studying the stability of time invariant systems our results can be used to study the stability of nonperiodic time varying systems by checking the modified detectability condition

a trust region algorithm with adaptive stochastic collocation for pde optimization under uncertainty the numerical solution of optimization problems governed by partial differential equations pdes with random coefficients is computationally challenging because of the large number of deterministic pde solves required at each optimization iteration this paper introduces an efficient algorithm for solving such problems based on a combination of adaptive sparse grid collocation for the discretization of the pde in the stochastic space and a trust region framework for optimization and fidelity management of the stochastic discretization the overall algorithm adapts the collocation points based on the progress of the optimization algorithm and the impact of the random variables on the solution of the optimization problem it frequently uses few collocation points initially and increases the number of collocation points only as necessary thereby keeping the number of deterministic pde solves low while guaranteeing convergence currently an error indicator is used to estimate gradient errors due to adaptive

generalized pt symmetry and real spectra the fact that eigenvalues of pt symmetric hamiltonians h can be real for some values of ap arameter and complex for others is explained by showing that the matrix elements ofh a ndhence the secular equation are real not only for pt but also for an ya ntiunitary operator a satisfying a 2k 1w ith k odd the argument is illustrated by a 2 2m atrix hamiltonian and two examples of the generalization are given

stabilization of nonlinear discrete time systems using state detection some recent results concerning stabilization of continuous time systems using state detection are extended to the case of discrete time nonlinear systems it is shown that if a discrete time locally detectable system can be stabilized by a state feedback law then it can also be stabilized by a feedback law that depends on the output of a weak detector

sparse block factorization of saddle point matrices abstract the factorization method presented in this paper takes advantage of the special structures and properties of saddle point matrices a variant of gaussian elimination equivalent to the cholesky s factorization is suggested and implemented for factorizing the saddle point matrices block wise with small blocks of orders 1 and 2 the gaussian elimination applied to these small blocks on block level also induces a block 3 3 structured factorization of which the blocks have special properties we compare the new block factorization with the schilders factorization in terms of sparsity and computational complexity the factorization can be used as a direct method and also anticipate for preconditioning techniques

solution of indefinite linear systems using an lq decomposition for the linear constraints in this paper indefinite linear systems with linear constraints are considered we present a special decomposition that makes use of the lq decomposition and retains the constraints in the factors the resulting decomposition is of a structure similar to that obtained using the bunch kaufman parlett algorithm the decomposition can be used in a direct solution algorithm for indefinite systems but it can also be used to construct effective preconditioners combinations of the latter with conjugate gradient type methods have been demonstrated to be very useful

sufficient conditions for stabilization of sampled data nonlinear systems via discrete time approximations given a parameterized by sampling period t family of approximate discrete time models of a sampled nonlinear plant and given a family of controllers stabilizing the family of plant models for all t sufficiently small we present conditions which guarantee that the same family of controllers semi globally practically stabilizes the exact discrete time model of the plant for sufficiently small sampling periods when the family of controllers is locally bounded uniformly in the sampling period the inter sample behavior can also be uniformly bounded so that the time varying sampled data model of the plant is uniformly semi globally practically stabilized the result justifies controller design for sampled data nonlinear systems based on the approximate discrete time model of the system when sampling is sufficiently fast and the conditions we propose are satisfied our analysis is applicable to a wide range of time discretization schemes and general plant models

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

fast iterative solution of reaction diffusion control problems arising from chemical processes pde constrained optimization problems and the development of preconditioned iterative methods for the efficient solution of the arising matrix systems is a field of numerical analysis that has recently been attracting much attention in this paper we analyze and develop preconditioners for matrix systems that arise from the optimal control of reaction diffusion equations which themselves result from chemical processes important aspects of our solvers are saddle point theory mass matrix representation and effective schur complement approximation as well as the incorporation of control constraints and application of the outer newton iteration to take into account the nonlinearity of the underlying pdes

a framework for stabilization of nonlinear sampled data systems based on their approximate discrete time models a unified framework for design of stabilizing controllers for sampled data differential inclusions via their approximate discrete time models is presented both fixed and fast sampling are considered in each case sufficient conditions are presented which guarantee that the controller that stabilizes a family of approximate discrete time plant models also stabilizes the exact discrete time plant model for sufficiently small integration and or sampling periods previous results in the literature are extended to cover 1 continuous time plants modeled as differential inclusions 2 general approximate discrete time plant models 3 dynamical discontinuous controllers modeled as difference inclusions and 4 stability with respect to closed arbitrary not necessarily compact sets

on solving block structured indefinite linear systems we consider 2 2 block indefinite linear systems whose 2 2 block is zero such systems arise in many applications we discuss two techniques that are based on modifying the 1 1 block in a way that makes the system easier to solve the main part of the paper focuses on an augmented lagrangian approach a technique that modifies the 1 1 block without changing the system size the choice of the parameter involved the spectrum of the linear system and its condition number are discussed and some analytical observations are provided a technique of deflating the 1 1 block is then introduced finally numerical experiments that validate the analysis are presented

reduced basis method for parametrized elliptic optimal control problems we propose a suitable model reduction paradigm the certied reduced basis method rb for the rapid and reliable solution of parametrized optimal control problems governed by partial dierential equations pdes in particular we develop the methodology for parametrized quadratic optimization problems with elliptic equations as constraint and infinite dimensional control variable firstly we recast the optimal control problem in the framework of saddle point problems in order to take advantage of the already developed rb theory for stokes type problems then the usual ingredients of the rb methodology are called into play a galerkin projection onto a low dimensional space of basis functions properly selected by an adaptive procedure an affine parametric dependence enabling to perform competitive offline online splitting in the computational procedure an efficient and rigorous a posteriori error estimate on the state control and adjoint variables as well as on the cost functional finally we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique

preconditioning optimal control of the unsteady burgers equations with h 1 regularized term in this paper a new preconditioning technique along with minres in nonstandard inner product is designed for the linear system of equations arising from the unsteady burgers equations constrained optimization problems a new schur complement approximation is provided the eigenvalue bounds which verify the effectiveness of the schur complement approximation are given numerical experiments show that the preconditioner using the new approximation schur complement along with minres in nonstandard inner product works well in practice

on global uniform asymptotic stability of nonlinear time varying systems in cascade abstract in this short paper we deal with the stability analysis problem of nonautonomous nonlinear systems in cascade in particular we give sufficient conditions to guarantee that i a globally uniformly stable gus nonlinear time varying nltv system remains gus when it is perturbed by the output of a globally uniformly asymptotically stable guas nltv system under the assumption that the perturbing signal is absolutely integrable ii if in addition the perturbed system is guas it remains guas under the cascaded interconnection iii two guas systems yield a guas cascaded system under some growth restrictions over the lyapunov function our proofs rely on the second method of lyapunov roughly speaking on a e stability analysis

certified reduced basis methods for parametrized saddle point problems we present reduced basis approximations and associated rigorous a posteriori error bounds for parametrized saddle point problems first we develop new a posteriori error estimates that unlike earlier approaches provide upper bounds for the errors in the approximations of the primal variable and the lagrange multiplier separately the proposed method is an application of brezzi s theory for saddle point problems to the reduced basis context and exhibits significant advantages over existing methods second based on an analysis of brezzi s theory we compare several options for the reduced basis approximation space from the perspective of approximation stability and computational cost finally we introduce a new adaptive sampling procedure for saddle point problems constructing approximation spaces that are stable and compared to earlier approaches computationally much more efficient the method is applied to a stokes flow problem in a two dimensional channel with a parametrized rectangular obstacle

a natural norm successive constraint method for inf sup lower bounds we present a new approach for the construction of lower bounds for the inf sup stability constants required in a posteriori error analysis of reduced basis approximations to anely parametrized partial dierential equations we combine the linearized inf sup statement of the natural norm approach with the approximation procedure of the successive constraint method scm the former natural norm provides an economical parameter expansion and local concavity in parameter a small er optimization problem which enjoys intrinsic lower bound properties the latter scm provides a systematic optimization framework a linear program lp relaxation which readily incorporates continuity and stability constraints the natural norm scm requires a parameter domain decomposition we propose a greedy algorithm for selection of the scm control points as well as adaptive construction of the optimal subdomains the ecacy of the natural norm scm is illustrated through numerical results for two types of non coercive problems the helmholtz equation for acoustics elasticity and electromagnetics and the convection diusi on equation

an anisotropic sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms input data of the model the method consists of a galerkin approximation in the space variables and a collocation in probability space on sparse tensor product grids utilizing either clenshaw curtis or gaussian knots even in the presence of nonlinearities the collocation approach leads to the solution of uncoupled deterministic problems just as in the monte carlo method this work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem these procedures seem to be very effective for the problems under study the proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation the first approach is effective for problems depending on random variables which weigh approximately equally in the solution while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables as in the case where input random variables are karhunen loeve truncations of smooth random fields this work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates sub exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime with respect to the total number of collocation points it also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems numerical examples illustrate the theoretical results and are used to compare this approach with several others including the standard monte carlo in particular for moderately large dimensional problems the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods

smooth lyapunov functions and robustness of stability for difference inclusions we demonstrate the equivalence of robust global asymptotic stability gas of the origin and the existence of a smooth lyapunov function for difference inclusions defined by upper semicontinuous set valued maps sufficient conditions for robust gas are given as an application of these results we give conditions for robust gas of difference equations defined by discontinuous right hand sides

a sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes a smolyak type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms input data of the model to compute solution statistics the sparse grid stochastic collocation method uses approximate solutions produced here by finite elements corresponding to a deterministic set of points in the random input space this naturally requires solving uncoupled deterministic problems as in the monte carlo method if the number of random variables needed to describe the input data is moderately large full tensor product spaces are computationally expensive to use due to the curse of dimensionality in this case the sparse grid approach is still expected to be competitive with the classical monte carlo method therefore it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than monte carlo this work provides error estimates for the fully discrete solution using l q norms and analyzes the computational efficiency of the proposed method in particular it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem number of input random variables in the final estimates the derived estimates are then used to compare the method with monte carlo indicating for which problems the former is more efficient than the latter computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and monte carlo approaches

threshold incomplete factorization constraint preconditioners for saddle point matrices abstract this paper presents a drop threshold incomplete ld 1lt factorization constraint preconditioner for saddle point systems using a threshold parameter a transformed saddle point matrix is partitioned into a block structure with blocks of order 1 and 2 constituting a priori pivots based on these pivots an incomplete ld 1lt factorization constraint preconditioner is computed that approaches an exact form as approaches zero we prove that both the exact and incomplete factorizations exist such that the entries of the constraint block remain unaltered in the triangular factors numerical results are presented for validation

preconditioning iterative methods for the optimal control of the stokes equations solving problems regarding the optimal control of partial differential equations pdes also known as pde constrained optimization is a frontier area of numerical analysis of particular interest is the problem of flow control where one would like to effect some desired flow by exerting for example an external force the bottleneck in many current algorithms is the solution of the optimality system a system of equations in saddle point form that is usually very large and ill conditioned in this paper we describe two preconditioners a block diagonal preconditioner for the minimal residual method and a block lower triangular preconditioner for a nonstandard conjugate gradient method which can be effective when applied to such problems where the pdes are the stokes equations we consider only distributed control here although we believe other problems could be treated in the same way we give numerical results and we compare these with those obtained by solving the equivalent forward problem using similar techniques

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

a stochastic collocation method for elliptic partial differential equations with random input data this work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms these input data are assumed to depend on a finite number of random variables the method consists of a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it treats easily a wide range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction of the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method finally we include a section with developments posterior to the original publication of this work there we review sparse grid stochastic collocation methods which are effective collocation strategies for problems that depend on a moderately large number of random variables

formulas relating kl stability estimates of discrete time and sampled data nonlinear systems abstract we provide an explicit kl stability or input to state stability iss estimate for a sampled data nonlinear system in terms of the kl estimate for the corresponding discrete time system and a k function describing inter sample growth it is quite obvious that a uniform inter sample growth condition plus an iss property for the exact discrete time model of a closed loop system implies uniform iss of the sampled data nonlinear system our results serve to quantify these facts by means of comparison functions our results can be used as an alternative to prove and extend results in 1 or extend some results in 4 to a class of nonlinear systems finally the formulas we establish can be used as a tool for some other problems which we indicate

all at once solution of time dependent stokes control the solution of time dependent pde constrained optimization problems subject to unsteady flow equations presents a challenge to both algorithms and computing power in this paper we present an all at once approach where we solve for all time steps of the discretized unsteady stokes problem at once the most desirable feature of this approach is that for all steps of an iterative scheme we only need approximate solutions of the discretized stokes operator this leads to an efficient scheme which exhibits mesh independent behaviour

on uniform global asymptotic stability of nonlinear discrete time systems with applications this paper presents new characterizations of uniform global asymptotic stability for nonlinear and time varying discrete time systems under mild assumptions it is shown that weak zero state detectability is equivalent to uniform global asymptotic stability for globally uniformly stable systems by employing the notion of reduced limiting systems another characterization of uniform global asymptotic stability is proposed on the basis of the detectability for the reduced limiting systems associated with the original system as a by product we derive a generalized discrete time version of the well known krasovskii lasalle theorem for general time varying not necessarily periodic systems furthermore we apply the obtained stability results to analyze uniform asymptotic stability of cascaded time varying systems and show that some technical assumptions in recent papers can be relaxed through a practical application it is shown that our results play a similar role to the classic lasalle invariance principle in guaranteeing attractivity noting that reduced limiting systems are used instead of the original system to validate the conceptual characterizations we study the problem of sampled data stabilization for the benchmark example of nonholonomic mobile robots via the exact discrete time model rather than approximate models this case study also reveals that in general sampled data systems may become non periodic even though their original continuous time system is periodic a novel sampled data stabilizer design is proposed using the new stability results and is supported via simulation results

uniform asymptotic stability of nonlinear switched systems with an application to mobile robots this paper is concerned with the study of both local and global uniform asymptotic stability for general nonlinear and time varying switched systems two concepts of lyapunov functions are introduced and used to establish uniform lyapunov stability and uniform global stability with the help of output functions an almost bounded output energy condition and an output persistent excitation condition are then proposed and employed to guarantee uniform local and global asymptotic stability based on this result a generalized version of krasovskii lasalle theorem in time varying switched systems is proposed for switched systems with persistent dwell time the output persistent excitation condition is guaranteed to hold under a zero state observability condition it is shown that several existing results in past literature can be covered as special cases using the proposed criteria interestingly as opposed to previous work the main results of this paper are applicable to the situation where some switching systems are not asymptotically stable at the origin the robust practical regulation problem of nonholonomic mobile robots is studied as a way of demonstrating the power of the proposed new criteria a novel switching controller is proposed with guaranteed robustness to orientation error and unknown parameters in mobile robots

reduced basis approximation and a posteriori error estimation for stokes flows in parametrized geometries roles of the inf sup stability constants in this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady stokes flows in affinely parametrized geometries focusing on the role played by the brezzi s and babuska s stability constants the crucial ingredients of the methodology are a galerkin projection onto a low dimensional space of basis functions properly selected an affine parametric dependence enabling to perform competitive offline online splitting in the computational procedure and a rigorous a posteriori error estimation on field variables the combinatiofn of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction ideally suited for real time simulation and many query contexts e g optimization control or parameter identification in particular in this work we focus on i the stability of the reduced basis approximation based on the brezzi s saddle point theory and the introduction of a supremizer operator on the pressure terms ii a rigorous a posteriori error estimation procedure for velocity and pressure fields based on the babuska s inf sup constant including residuals calculations iii the computation of a lower bound of the stability constant and iv different options for the reduced basis spaces construction we present some illustrative results for both interior and external steady stokes flows in parametrized geometries representing two parametrized classical poiseuille and couette flows a channel contraction and a simple flow control problem around a curved obstacle

a weighted reduced basis method for elliptic partial differential equations with random input data in this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations pdes with random input data the pdes are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure a priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems

multilevel and weighted reduced basis method for stochastic optimal control problems constrained by stokes equations in this paper we develop and analyze a multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by stokes equations we prove the analytic regularity of the optimal solution in the probability space under certain assumptions on the random input data the finite element method and the stochastic collocation method are employed for the numerical approximation of the problem in the deterministic space and the probability space respectively resulting in many large scale optimality systems to solve in order to reduce the unaffordable computational effort we propose a reduced basis method using a multilevel greedy algorithm in combination with isotropic and anisotropic sparse grid techniques a weighted a posteriori error bound highlights the contribution stemming from each method numerical tests on stochastic dimensions ranging from 10 to 100 demonstrate that our method is very efficient especially for solving high dimensional and large scale optimization problems

non self adjoint hamiltonians with complex eigenvalues motivated by what one observes dealing with pt symmetric quantum mechanics we discuss what happens if a physical system is driven by a diagonalizable hamiltonian with not all real eigenvalues in particular we consider the functional structure related to systems living in finite dimensional hilbert spaces and we show that certain intertwining relations can be deduced also in this case if we introduce suitable antilinear operators we also analyze a simple model computing the transition probabilities in the broken and in the unbroken regime

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

finite element approximations of stochastic optimal control problems constrained by stochastic elliptic pdes abstract in this paper we study mathematically and computationally optimal control problems for stochastic elliptic partial differential equations the control objective is to minimize the expectation of a tracking cost functional and the control is of the deterministic distributed type the main analytical tool is the wiener ito chaos or the karhunen loeve expansion mathematically we prove the existence of an optimal solution we establish the validity of the lagrange multiplier rule and obtain a stochastic optimality system of equations we represent the input data in their wiener ito chaos expansions and deduce the deterministic optimality system of equations computationally we approximate the optimality system through the discretizations of the probability space and the spatial space by the finite element method we also derive error estimates in terms of both types of discretizations

stochastic optimal robin boundary control problems of advection dominated elliptic equations in this work we deal with a stochastic optimal robin boundary control problem constrained by an advection diffusion reaction elliptic equation with advection dominated term we assume that the uncertainty comes from the advection field and consider a stochastic robin boundary condition as control function a stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system a global error estimate in the product of physical space and stochastic space for the numerical approximation is derived illustrative numerical experiments are provided

stochastic collocation for optimal control problems with stochastic pde constraints we discuss the use of stochastic collocation for the solution of optimal control problems which are constrained by stochastic partial differential equations spde thereby the constraining spde depends on data which is not deterministic but random assuming a deter ministic control randomness within the states of the input data will propagate to the states of the system for the solution of spdes there has recently been an increasing effort in the development of efficient numerical schemes based upon the mathematical concept of generalized polynomial chaos modal based stochastic galerkin and nodal based stochastic collocation versions of this methodol ogy exist both of which rely on a certain level of smoothness of the solution in the random space to yield accelerated convergence rates in this paper we apply the stochastic collocation method to develop a gradient descent as well as a sequential quadratic program sqp for the minimization of objective functions constrained by an spde the stochastic function involves several higher order moments of the random states of the system as well as classical regularization of the control in particular we discuss several objective functions of tracking type numerical examples are presented to demonstrate the performance of our new stochastic collocation minimization approach

a generalization of krasovskii lasalle theorem for nonlinear time varying systems converse results and applications this paper presents a practically applicable characterization of uniform global asymptotic stability uas and ugas for general nonlinear time varying systems under certain output dependent conditions in the spirit of the krasovskii lasalle theorem the celebrated krasovskii lasalle theorem is extended from two directions one is using the weak zero state detectability property associated with reduced limiting systems of the system in question to generalize the condition that the maximal invariance set contained in the zero locus of the time derivative of the lyapunov function is the zero set another one is using an almost bounded output energy condition to relax the assumption that the time derivative of the lyapunov function is negative semi definite then the uas and ugas properties of the origin can be guaranteed by employing these two improved conditions related to certain output function for uniformly lyapunov stable systems the proposed conditions turn out to be also necessary under some mild assumptions and thus give a new characterization of ugas and uas through an equivalence relation the proposed detectability condition can also be verified in terms of usual pe condition to validate the proposed results the obtained stability criteria are applied to a class of time varying passive systems and to revisit a tracking control problem of nonholonomic chained systems for the latter under certain persistency of excitation conditions the k exponential stability is achieved based on our approach

matrosov theorem for parameterized families of discrete time systems a version of matrosov s theorem for parameterized discrete time time varying systems is presented the theorem is a discrete time version of the continuous time result in loria et al 2002 d persistency of excitation a necessary and sufficient condition for uniform attractivity 2002 submitted for publication our result facilitates controller design for sampled data nonlinear systems via their approximate discrete time models an application of the theorem to establishing uniform asymptotic stability of systems controlled by model reference adaptive controllers designed via approximate discrete time plant models is presented

multigrid solution of a distributed optimal control problem constrained by the stokes equations in this work we construct multigrid preconditioners to accelerate the solution process of a linear quadratic optimal control problem constrained by the stokes system the first order optimality conditions of the control problem form a linear system the kkt system connecting the state adjoint and control variables our approach is to eliminate the state and adjoint variables by essentially solving two stokes systems and to construct efficient multigrid preconditioners for the schur complement of the block associated with the state and adjoint variables these multigrid preconditioners are shown to be of optimal order with respect to the convergence properties of the discrete methods used to solve the stokes system in particular the number of conjugate gradient iterations is shown to decrease as the resolution increases a feature shared by similar multigrid preconditioners for elliptic constrained optimal control problems

some observations on babu vs ka and brezzi theories summary some observations are made on abstract error estimates for galerkin approximations based on babuska brezzi conditions a basic error estimate due to babuska is sharpened by means of an identity that p i p for any nontrivial idempotent operator p some remarks are also made on the brezzi s theory for mixed variational problems and their galerkin approximations

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

analysis and approximation of the velocity tracking problem for navier stokes flows with distributed control we consider the mathematical formulation analysis and the numerical solution of a time dependent optimal control problem associated with the tracking of the velocity of a navier stokes flow in a bounded two dimensional domain through the adjustment of a distributed control the existence of optimal solutions is proved and the first order necessary conditions for optimality are used to derive an optimality system of partial differential equations whose solutions provide optimal states and controls semidiscrete in time and fully discrete space time approximations are defined and their convergence to the exact optimal solutions is shown a gradient method for the solution of the fully discrete equations is examined as are its convergence properties finally the results of some illustrative computational experiments are presented

erhss iteration method for pde optimal control problem abstract an extended regularized hermitian and skew hermitian splitting iteration method for solving the double saddle point linear system arising from pde optimal control problem is proposed the unconditional convergence of the method is proved furthermore some spectral properties of the preconditioned matrix are investigated numerical experiments show that the iteration method is feasible and efficient

an alternative construction of the positive inner product for pseudo hermitian hamiltonians examples in this paper we build on our earlier proposal for the construction of a positive inner product for pseudo hermitian hamiltonians and present examples to clarify the procedure we focus on two detailed calculations where the method is used namely a simple generalized 2 2 matrix pseudo hermitian hamiltonian which can be diagonalized and a second system where the hamiltonian cannot be diagonalized but can be described as a perturbation of the harmonic oscillator when the quantum mechanical system cannot be diagonalized exactly our construction can be carried out perturbatively and we develop the general formalism for such a perturbative calculation systematically for real eigenvalues in this paper we build on our earlier proposal for the construction of a positive inner product for pseudo hermitian hamiltonians and present examples to clarify the procedure we focus on two detailed calculations where the method is used namely a simple generalized 2 2 matrix pseudo hermitian hamiltonian which can be diagonalized and a second system where the hamiltonian cannot be diagonalized but can be described as a perturbation of the harmonic oscillator when the quantum mechanical system cannot be diagonalized exactly our construction can be carried out perturbatively and we develop the general formalism for such a perturbative calculation systematically for real eigenvalues

crout versions of ilu for general sparse matrices this paper presents an efficient implementation of the incomplete lu ilu factorization derived from the crout version of gaussian elimination at step k of the elimination the kth row of u and the kth column of l are computed using previously computed rows of u and columns of l the data structure and implementation borrow from already known techniques used in developing both sparse direct solution codes and incomplete cholesky factorizations this version of ilu can be computed much faster than standard threshold based ilu factorizations computed rowwise or columnwise in addition the data structure allows efficient implementations of more rigorous and effective dropping strategies