the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

partial information state based optimization of partially observable markov decision processes and the separation principle we propose a partial information state based approach to the optimization of the long run average performance in a partially observable markov decision process pomdp in this approach the information history is summarized at least partially by a or a few statistic s not necessary sufficient called a partial information state and actions depend on the partial information state rather than system states we first propose the single policy based comparison principle under which we derive an hjb type of optimality equation and policy iteration for the optimal policy in the partial information state based policy space we then introduce the q sufficient statistics and show that if the partial information state is q sufficient then the optimal policy in the partial information state based policy space is optimal in the space of all feasible information state based policies we show that with some further conditions the well known separation principle holds the results are obtained by applying the direct comparison based approach initially developed for discrete event dynamic systems

a solvable stochastic control problem in hyperbolic three space abstract a stochastic optimal control problem in hyperbolic three space is described and explicitly solved the solution is obtained by finding a smooth solution to the hamilton jacobi or dynamic programming equation

fast iterative solution of stabilised stokes systems part i using simple diagonal preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems the use of stabilisation methods for convenient but unstable mixed elements introduces stabilisation parameters we show how these can be chosen to obtain rapid iterative convergence we propose a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems describe the effects of stabilisation on the algebraic structure of the discrete stokes operator and derive estimates of the eigenvalue spectrum of this operator on which the convergence rate of the iteration depends here we discuss the simple case of diagonal preconditioning our results apply to both locally and globally stabilised mixed elements as well as to elements which are inherently stable we demonstrate that convergence rates comparable to that achieved using the diagonally scaled conjugate gradient method applied to the discrete laplacian are approachable for the stokes problem

an all at once approach for the optimal control of the unsteady burgers equation we apply an all at once method for the optimal control of the unsteady burgers equation the nonlinear burgers equation is discretized in time using the semi implicit discretization and the resulting first order optimality conditions are solved iteratively by newton s method the discretize then optimize approach is used because it leads to a symmetric indefinite saddle point problem numerical results for the distributed unconstrained and control constrained problems illustrate the performance of the all at once approach with semi implicit time discretization

on the application of minimum principle for solving partially observable risk sensitive control problems abstract this paper is concerned with the application of a minimum principle derived for general nonlinear partially observable exponential of integral control problems to solve linear exponential quadratic gaussian problems this minimum principle is the stochastic analog of pontryagin s minimum principle for deterministic systems it consists of an information state equation an adjoint process governed by a stochastic partial differential equation with terminal condition and a hamiltonian functional two methods are employed to obtain the optimal control law the first method appeals to the well known approach of completing the squares by first determining the optimal control law that minimizes the hamiltonian functional the second method provides significant insight into relations with the hamiltonijacobi approach associated with completely observable exponential of integral control problems these methods of solution are particularly attractive because they do not assume a certainty equivalence principle hence they can be used to solve nonlinear problems as well

lqg control with an h sup infinity performance bound a riccati equation approach an lqg linear quadratic gaussian control design problem involving a constraint on h sup infinity disturbance attenuation is considered the h sup infinity performance constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on l sub 2 performance in contrast to the pair of separated riccati equations of standard lqg theory the h sup infinity constrained gains are given by a coupled system of three modified riccati equations the coupling illustrates the breakdown of the separation principle for the h sup infinity constrained problem both full and reduced order design problems are considered with an h sup infinity attenuation constraint involving both state and control variables an algorithm is developed for the full order design problem and illustrative numerical results are given

some examples of optimal stochastic controls or the stochastic maximum principle at work the stochastic maximum principle is applied to solve eight stochastic control problems four have been solved previously using much more complicated methods but the last four are new the main thrust of the work is to show that some completely observable stochastic control problems with special structure can be solved quite quickly and easily using the maximum principle

iterative techniques for time dependent stokes problems abstract in this paper we consider solving the coupled systems of discrete equations which arise from implicit time stepping procedures for the time dependent stokes equations using a mixed finite element spatial discretization at each time step a two by two block system corresponding to a perturbed stokes problem must be solved although there are a number of techniques for iteratively solving this type of block system to be effective they require a good preconditioner for the resulting pressure operator schur complement in contrast to the time independent stokes equations where the pressure operator is well conditioned the pressure operator for the perturbed system becomes more ill conditioned as the time step is reduced and or the reynolds number is increased in this paper we shall describe and analyze preconditioners for the resulting pressure systems these preconditioners give rise to iterative rates of convergence which are independent of both the mesh size h as well as the time step and reynolds number parameter k

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

the equivalence between infinite horizon optimal control of stochastic systems with exponential of integral performance index and stochastic differential games a new method based on the theory of large deviations from the invariant measure is introduced for the analysis of stochastic systems with an infinite horizon exponential of integral performance index it is shown that the infinite horizon optimal exponential of integral stochastic control problem is equivalent to a stationary stochastic differential game for an auxiliary system as an application of the developed technique the infinite horizon risk sensitive lqg problem is analyzed for both the completely observed and partially observed case

the effect of a sparse grad div stabilization on control of stationary navier stokes equations abstract this study presents a reduced sparsity grad div stabilization for the optimal control of an incompressible stationary flow problem the effect of the proposed stabilization is analyzed in terms of stability and velocity a priori error the main difference between this method and the usual grad div stabilization can be observed in numerical computations since the proposed method produces block upper triangular matrices an advantage can be seen in terms of implementation over the common grad div stabilization which produces fully coupled block matrices the stability and the convergence results are given for both the state and the adjoint state variables once the optimality conditions by using the lagrange approach are stated in numerical experiments solutions of the optimal control problem without any stabilization and the proposed method are presented in order to compare the effectiveness of the method the numerical convergence study is also presented

robust stochastic maximum principle for multi model worst case optimization this paper develops a version of the robust maximum principle applied to the minimax mayer problem formulated for stochastic differential equations with a control dependent diffusion term the parametric families of first and second order adjoint stochastic processes are introduced to construct the corresponding hamiltonian formalism the hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic hamiltonians corresponding to each possible value of the parameter the cost function is defined on a finite horizon and contains the mathematical expectation of a terminal term a terminal condition given by a vector function is also considered the optimal control strategies adapted for available information for the wide class of multi model systems given by a stochastic differential equation with parameters from a given finite set are constructed this problem belongs to the class of minimax stochastic optimization problems the p

optimal stochastic regulators with state dependent weights abstract we introduce two optimal regulators for linear stochastic systems the first is of a linear state feedback form and it generalises the linear quadratic regulator by introducing state dependent weights in the cost functional the second is a certain risk sensitive version of the first and it is of a nonlinear state feedback form both regulators are applied to the optimal investment problem

instrumental variable estimation in generalized linear measurement error models abstract instrumental variable estimation in generalized linear measurement error models are studied for models with canonical link functions unbiased estimating equations are derived the maximum likelihood estimator for the normal theory structural linear instrumental variable model is shown to be a solution to the estimating equations derived herein logistic regression is studied in detail an example is given and a simulation study described for the logistic model based on the framingham heart study data

fast iterative solution of stabilised stokes systems part ii using general block preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems part i of this work described a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems a j wathen and d j silvester siam j numer anal 30 1993 pp 630 649 using simple arguments estimates for the eigenvalue distribution of the discrete stokes operator on which the convergence rate of the iteration depends are easily derived part i discussed the important case of diagonal preconditioning scaling this paper considers the more general class of block preconditioners where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables it is shown that provid

a class of iterative methods for solving saddle point problems we consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points we are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations we present an iterative method involving two levels of iteration similar in some respects to the uzawa algorithm we relate the rates of convergence of the outer and inner iterations proving that under natural hypotheses the outer iteration achieves the rate of convergence of the inner iteration the technique is applied to finite element approximations of the stokes equations

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

on the adjoint process for optimal control of diffusion processes for the markovian control problem begin gathered mathop min limits u e left int 0 t l t x u dt c x t right hfill dx f t x u dt sigma t x dw hfill end gathered it is shown that the adjoint process appearing in the maximum principle has the form p t x e tx left frac partial c partial x x t phi t t int t t frac partial l partial x s x s hat u s x s phi s t ds right where hat u is the optimal feedback control e tx denotes conditional expectation and phi is the fundamental matrix solution of dy frac partial f partial x t x t hat u t x t ydt sum k frac partial sigma k partial x t x y dw k here sigma k is the kth column of sigma and w k is thekth component of the brownian motion w it is also shown that p t x t satisfies denotes transpose begin gathered dp left frac partial f partial x t x t hat u t x t p

analysis of iterative methods for the steady and unsteady stokes problem application to spectral element discretizations a new and detailed analysis of the basic uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady stokes operator is presented the paper focuses on the following new aspects explicit construction of the uzawa pressure operator spectrum for a semiperiodic model problem general relationship of the convergence rate of the uzawa procedure to classical inf sup discretization analysis and application of the method to high order variational discretization

anamorphic analysis sampling and estimation for covariate effects when both exposure and disease are known when study subjects are obtained through sampling procedures based on the level of known characteristics covariate data can be used to adjust effect estimates this principle which is well understood for both exposure based sampling cohort studies and disease based sampling case control studies can also be applied to cross classifications of the source population an opportunity for exposure disease based sampling arises whenever data on exposure and disease are both available on a group of subjects who are incompletely characterized in terms of other covariates this is a common intermediate stage in epidemiologic studies of many kinds

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

semiexplicit solutions to some nonlinear nonquadratic mean field type games a direct method this article examines mean field type game problems by means of a direct method we provide various solvable examples beyond the classical linear quadratic game problems these include quadratic quadratic games and games with power logarithmic sine square hyperbolic sine square payoffs nonlinear state dynamics such as log state control dependent regime switching quadratic state cotangent state and hyperbolic cotangent state are considered we identify equilibrium strategies and equilibrium payoffs in state and conditional mean field type feedback form it is shown that a simple direct method can be used to solve broader classes of nonquadratic mean field type games under jump diffusion regime switching gauss volterra processes which include fractional brownian motions and multifractional brownian motions we provide semiexplicit solutions to the fully cooperative noncooperative nonzero sum and adversarial game problems

optimal control of partially observable stochastic systems with an exponential of integral performance index the stochastic control problem with linear stochastic differential equations driven by brownian motion processes and as cost functional the exponential of a quadratic form is considered the soluti

robust optimal control for minimax stochastic linear quadratic problem the robust maximum principle applied to the minimax linear quadratic problem is derived for stochastic differential equations containing a control dependent diffusion term the parametric families of the first and second order adjoint stochastic processes are obtained to construct the corresponding hamiltonian formalism the hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic hamiltonians corresponding to each value of the uncertain parameter from a given finite set the cost function is considered on a finite horizon contains the mathematical expectation of both an integral and a terminal term and on an infinite one a time averaged losses function these problems belong to the class of minimax stochastic optimization problems it is shown that the construction of the minimax optimal controller can be reduced to an optimization problem on a finitedimensional simplex and consists in the analysis of the dependence of ric

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

survey paper a partial history of the early development of continuous time nonlinear stochastic systems theory this article is a survey of the early development of selected areas in nonlinear continuous time stochastic control key developments in optimal control and the dynamic programming principle existence of optimal controls under complete and partial observations nonlinear filtering stochastic stability the stochastic maximum principle and ergodic control are discussed issues concerning wide bandwidth noise for stability modeling filtering and ergodic control are dealt with the focus is on the earlier work but many important topics are omitted for lack of space

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

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

optimal solvers for pde constrained optimization optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering particularly in problems of design the solution of such pde constrained optimization problems is usually a major computational task here we consider simple problems of this type distributed control problems in which the 2 and 3 dimensional poisson problem is the pde the large dimensional linear systems which result from discretization and which need to be solved are of saddle point type we introduce two optimal preconditioners for these systems which lead to convergence of symmetric krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem these preconditioners are block structured and involve standard multigrid cycles the optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases the theoretical proof indicates that these approaches may have much broader applicability for other pdes

risk sensitive and leg filtering problems are not equivalent filtering problems with general exponential quadratic criteria are investigated for gauss markov processes in this setting the linear exponential gaussian and risk sensitive filtering problems are solved and it is shown that they may have different solutions

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

designing a logistic regression study using surrogate measures for exposure and outcome summary suppose that a binary or polytomous outcome variable is related to a possibly continuous exposure variable through a logistic regression model and that prior to sample selection subjects can be characterized according to surrogates for outcome and exposure the surrogate variables are assumed to satisfy certain conditional independence properties commonly used in errors in variables models this paper considers the problem of using the surrogate variables to select subjects for measurement of the true values with the goal of minimizing the variance of an estimate of the log odds ratio we describe three possible sampling plans and show that the log odds ratio parameter can be estimated in each plan using ordinary logistic regression

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

low rank solvers for unsteady stokes brinkman optimal control problem with random data abstract we consider the numerical simulation of an optimal control problem constrained by the unsteady stokes brinkman equation involving random data more precisely we treat the state the control the target or the desired state as well as the viscosity as analytic functions depending on uncertain parameters this allows for a simultaneous generalized polynomial chaos approximation of these random functions in the stochastic galerkin finite element method discretization of the model the discrete problem yields a prohibitively high dimensional saddle point system with kronecker product structure we develop a new alternating iterative tensor method for an efficient reduction of this system by the low rank tensor train representation besides we propose and analyze a robust schur complement based preconditioner for the solution of the saddle point system the performance of our approach is illustrated with extensive numerical experiments based on two and three dimensional examples where the full problem size exceeds one billion degrees of freedom the developed tensor train scheme reduces the solution storage by two three orders of magnitude depending on discretization parameters

design of validation studies for estimating the odds ratio of exposure disease relationships when exposure is misclassified summary we compared several validation study designs for estimating the odds ratio of disease with misclassified exposure we assumed that the outcome and misclassified binary covariate are available and that the error free binary covariate is measured in a subsample the validation sample we considered designs in which the total size of the validation sample is fixed and the probability of selection into the validation sample may depend on outcome and misclassified covariate values design comparisons were conducted for rare and common disease scenarios where the optimal design is the one that minimizes the variance of the maximum likelihood estimator of the true log odds ratio relating the outcome to the exposure of interest misclassification rates were assumed to be independent of the outcome we used a sensitivity analysis to assess the effect of misspecifying the misclassification rates under the scenarios considered our results suggested that a balanced design which allocates equal numbers of validation subjects into each of the four outcome mismeasured covariate categories is preferable for its simplicity and good performance a user friendly fortran program is available from the second author which calculates the optimal sampling fractions for all designs considered and the efficiencies of these designs relative to the optimal hybrid design for any scenario of interest

efficient solution of large scale saddle point systems arising in riccati based boundary feedback stabilization of incompressible stokes flow we investigate numerical methods for solving large scale saddle point systems which arise during the feedback control of flow problems we focus on the instationary stokes equations that describe instationary incompressible flows for moderate viscosities after a mixed finite element discretization we get a differential algebraic system of differential index two j weickert navier stokes equations as a differential algebraic system preprint sfb393 96 08 department of mathematics chemnitz university of technology chemnitz germany 1996 to reduce this index we follow the analytic ideas of j p raymond siam j control optim 45 2006 pp 790 828 coupled with the projection idea of m heinkenschloss d c sorensen and k sun siam j sci comput 30 2008 pp 1038 1063 avoiding this explicit projection leads to solving a series of large scale saddle point systems in this paper we construct iterative methods to solve such saddle point systems by deriving efficient preconditioners b

brief paper a formula for the optimal cost in the general discrete time leqg problem this paper is aimed at deriving an explicit formula for the optimal cost for discrete time linear exponential of quadratic gaussian leqg control problems we make direct calculations for the general case with cross terms in the cost and noise covariance matrices using an information state approach

risk sensitive mean field games in this paper we study a class of risk sensitive mean field stochastic differential games we show that under appropriate regularity conditions the mean field value of the stochastic differential game with exponentiated integral cost functional coincides with the value function satisfying a hamilton jacobi bellman hjb equation with an additional quadratic term we provide an explicit solution of the mean field best response when the instantaneous cost functions are log quadratic and the state dynamics are affine in the control an equivalent mean field risk neutral problem is formulated and the corresponding mean field equilibria are characterized in terms of backward forward macroscopic mckean vlasov equations fokker planck kolmogorov equations and hjb equations we provide numerical examples on the mean field behavior to illustrate both linear and mckean vlasov dynamics

a low rank in time approach to pde constrained optimization the solution of time dependent pde constrained optimization problems is a challenging task in numerical analysis and applied mathematics all at once discretizations and corresponding solvers provide efficient methods to robustly solve the arising discretized equations one of the drawbacks of this approach is the high storage demand for the vectors representing the discrete space time cylinder here we introduce a low rank in time technique that exploits the low rank nature of the solution the theoretical foundations for this approach originate in the numerical treatment of matrix equations and can be carried over to pde constrained optimization we illustrate how three different problems can be rewritten and used within a low rank krylov subspace solver with appropriate preconditioning

a risk sensitive maximum principle abstract a stochastic maximum principle is derived which is formally of general application and so not restricted to the diffusion case it moreover differs from the classic deterministic principle only by a modification of the hamiltonian and so requires no solution of stochastic differential equations or calculation of conditional expectations the two essential points in its construction are that one considers the family of criterion functions exponential in an additive cost function of which the conventional expected cost case is a degenerate member and that in the non lqg case appeal to large deviation theory must be valid

logistic regression for two stage case control data summary samples of diseased cases and nondiseased controls are drawn at random from the population at risk after classification according to the exposure of interest subsamples of cases and controls are selected for purposes of covariable ascertainment a modification of the usual logistic regression analysis yields consistent estimates of covariable adjusted relative risks and their standard errors by balancing the numbers of exposed and nonexposed for whom covariable inforniation is ascertained within case and control samples some efficiency may be gained over the usual single stage design particularly when the exposure is rare and the relative risks associated with the covariables are large the procedure may be useful also when covariable information is missing for a large part of the sample

low rank solution to an optimization problem constrained by the navier stokes equations the numerical solution of pde constrained optimization problems subject to the nonstationary navier stokes equation is a challenging task while space time approaches often show favorable convergence properties they often suffer from storage problems here we propose to approximate the solution to the optimization problem in a low rank form which is similar to the model order reduction mor approach however in contrast to classical mor schemes we do not compress the full solution at the end of the algorithm but start our algorithm with low rank data and maintain this form throughout the iteration numerical experiments indicate that this approach reduces the computational costs by two orders of magnitude

lagrange lemma and the optimal control of diffusions ii nonlinear lagrange functionals abstract in kosmol 1991 kosmol and pavon 1993 a new elementary approach to optimal control problems relying on the lagrange lemma was described which appears to be technically and conceptually much simpler than existing methods and furthermore provides a unified variational approach in kosmol and pavon 1992 this method was further clarified and developed for linear lagrange functionals we devote this second paper to nonlinear lagrange functionals the power of this approach is here clearly demonstrated in particular it is shown that the nonlinear lagrange functional induced by the value function of the problem is just one of the many functionals which may effectively be employed to solve control problems see examples 1 and 2 in section 3 moreover in our approach we can deal in the same framework with problems with state constraints or nonsmooth problems hence the lagrange functionals approach provides new tools to solve control problems which may be readily applied even when existing approaches fail