robust and adaptive network flows we study network flow problems in an uncertain environment from the viewpoint of robust optimization in contrast to previous work we consider the case that the network parameters e g capacities are known and deterministic but the network structure e g nodes and arcs is subject to uncertainty in this paper we study the robust and adaptive versions of the maximum flow problem and minimum cut problems in networks with node and arc failures and establish structural and computational results the adaptive two stage model adjusts the solution after the realization of the failures in the network this leads to a more flexible model and yields less conservative solutions compared to the robust model we show that the robust maximum flow problem can be solved in polynomial time but the robust minimum cut problem is np hard we also prove that the adaptive versions are np hard we further characterize the adaptive model as a two person zero sum game and prove the existence of an equilibrium in such g

minimax control of nonlinear evolution equations in this paper we study nonlinear parametric optimal control problems we formulate the relevant minimax problem and prove the existence of an optimal control also we derive necessary conditions for saddle point optimality we also associate to the original optimization problem a dual problem and then for the pair we prove a complete duality theorem finally we present two examples of parabolic distributed parameter systems which illustrate the abstract results

sufficient second order optimality conditions for semilinear control problems with pointwise state constraints second order sufficient optimality conditions are established for the optimal control of semilinear elliptic and parabolic equations with pointwise constraints on the control and the state in contrast to former publications on this subject the cone of critical directions is the smallest possible in the sense that the second order sufficient conditions are the closest to the associated necessary ones the theory is developed for elliptic distributed controls in domains up to dimension three moreover problems of elliptic boundary control and parabolic distributed control are discussed in spatial domains of dimension two and one respectively

meg beamforming using bayesian pca for adaptive data covariance matrix regularization beamformers are a commonly used method for doing source localization from magnetoencephalography meg data a key ingredient in a beamformer is the estimation of the data covariance matrix when the noise levels are high or when there is only a small amount of data available the data covariance matrix is estimated poorly and the signal to noise ratio snr of the beamformer output degrades one solution to this is to use regularization whereby the diagonal of the covariance matrix is amplified by a pre specified amount however this provides improvements at the expense of a loss in spatial resolution and the parameter controlling the amount of regularization must be chosen subjectively in this paper we introduce a method that provides an adaptive solution to this problem by using a bayesian principle component analysis pca this provides an estimate of the data covariance matrix to give a data driven non arbitrary solution to the trade off between the spatial resolution and the snr of the beamformer output this also provides a method for determining when the quality of the data covariance estimate maybe under question we apply the approach to simulated and real meg data and demonstrate the way in which it can automatically adapt the regularization to give good performance over a range of noise and signal levels

second order analysis for optimal control problems improving results expected from abstract theory an abstract optimization problem of minimizing a functional on a convex subset of a banach space is considered we discuss natural assumptions on the functional that permit establishing sufficient second order optimality conditions with minimal gap with respect to the associated necessary ones though the two norm discrepancy is taken into account the obtained results exhibit the same formulation as the classical ones known from finite dimensional optimization we demonstrate that these assumptions are fulfilled in particular by important optimal control problems for partial differential equations we prove that in contrast to a widespread common belief the standard second order conditions formulated for these control problems imply strict local optimality of the controls not only in the sense of l infty but also of l 2

minimax control of nonlinear evolution equations in this paper we study nonlinear parametric optimal control problems we formulate the relevant minimax problem and prove the existence of an optimal control also we derive necessary conditions for saddle point optimality we also associate to the original optimization problem a dual problem and then for the pair we prove a complete duality theorem finally we present two examples of parabolic distributed parameter systems which illustrate the abstract results

sufficient second order optimality conditions for a parabolic optimal control problem with pointwise control state constraints an optimal control problem for a semilinear parabolic equation is investigated where pointwise constraints are given on the control and the state the state constraints are of mixed bottleneck type where associated lagrange multipliers can be assumed to be bounded and measurable functions based on this property a second order sufficient optimality condition is established that considers strongly active constraints

a priori error estimates for space time finite element discretization of semilinear parabolic optimal control problems in this paper a priori error estimates for space time finite element discretizations of optimal control problems governed by semilinear parabolic pdes and subject to pointwise control constraints are derived we extend the approach from meidner and vexler siam control optim 47 3 1150 1177 2008 siam control optim 47 3 1301 1329 2008 where linear quadratic problems have been considered discretizing the state equation by usual conforming finite elements in space and a discontinuous galerkin method in time error estimates for controls discretized by piecewise constant functions in time and cellwise constant functions in space are derived in detail and we explain how error estimate for further discretization approaches e g cellwise linear discretization in space the postprocessing approach from meyer and rosch siam j control optim 43 970 985 2004 and the variationally discrete approach from hinze j comput optim appl 30 45 63 2005 can be obtained in addition we derive an estimate for a setting with finitely many time dependent controls

existence uniqueness and behavior of solutions for a class of nonlinear parabolic problems abstract we prove existence uniqueness regularity results and estimates describing the behavior both for large and small times of a solution u of some nonlinear parabolic equations of leray lions type including the p laplacian in particular we show how the summability of the initial datum u 0 and the value of p influence the behavior of the solution u producing ultracontractive or supercontractive estimates or extinction in finite time or different kinds of decay estimates

numerical simulation of air pollution due to traffic flow in urban networks abstract as it is well known traffic flow is the main pollution source in many urban areas where the number of vehicles ranges from many thousands to millions thus estimating the pollution emission rate due to traffic flow in big cities is a very hard task to approach this environmental issue in this paper we propose a methodology that consists of combining the 1d lighthill whitham richards traffic model for road networks with a classical 2d advection diffusion reaction pollution model for the atmosphere here the pollution model uses a source term that takes into account the traffic flow contamination by means of a radon measure supported on a road network within an urban domain furthermore we establish the existence of solution of the coupled model and detail a complete numerical algorithm to compute it mainly interfacing a finite volume scheme based on the supply demand method for the traffic model with a characteristics lagrange finite element method for the pollution model finally several numerical experiences for a real urban domain the guadalajara metropolitan area in mexico are presented

parabolic control problems in measure spaces with sparse solutions optimal control problems in measure spaces lead to controls that have small support which is desirable e g in the context of optimal actuator placement for problems governed by parabolic partial differential equations well posedness is guaranteed in the space of square integrable measure valued functions which leads to controls with a spatial sparsity structure a conforming approximation framework allows one to derive numerically accessible optimality conditions as well as convergence rates in particular although the state is discretized the control problem can still be formulated and solved in the measure space numerical examples illustrate the structural features of the optimal controls

necessary optimality conditions for control problems and the stone cech compactification this paper deals with optimal control problems of parabolic equations in the presence of pointwise state constraints we consider bounded controls which act in the initial condition of the state equation the state variable is a bounded continuous function on the domain q omega times 0 t but is not continuous on overline q in this case the multiplier associated with state constraints is a regular bounded finitely additive measure on q but not a sigma additive one using some properties of the stone u c ech compactification we prove a decomposition theorem for this measure which allows us to interpret the adjoint equation in a classical sense we obtain new optimality conditions for these kinds of problems and we apply these results to the case of bilateral constraints

automated model selection in covariance estimation and spatial whitening of meg and eeg signals abstract magnetoencephalography and electroencephalography m eeg measure non invasively the weak electromagnetic fields induced by post synaptic neural currents the estimation of the spatial covariance of the signals recorded on m eeg sensors is a building block of modern data analysis pipelines such covariance estimates are used in brain computer interfaces bci systems in nearly all source localization methods for spatial whitening as well as for data covariance estimation in beamformers the rationale for such models is that the signals can be modeled by a zero mean gaussian distribution while maximizing the gaussian likelihood seems natural it leads to a covariance estimate known as empirical covariance ec it turns out that the ec is a poor estimate of the true covariance when the number of samples is small to address this issue the estimation needs to be regularized the most common approach downweights off diagonal coefficients while more advanced regularization methods are based on shrinkage techniques or generative models with low rank assumptions probabilistic pca ppca and factor analysis fa using cross validation all of these models can be tuned and compared based on gaussian likelihood computed on unseen data we investigated these models on simulations one electroencephalography eeg dataset as well as magnetoencephalography meg datasets from the most common meg systems first our results demonstrate that different models can be the best depending on the number of samples heterogeneity of sensor types and noise properties second we show that the models tuned by cross validation are superior to models with hand selected regularization hence we propose an automated solution to the often overlooked problem of covariance estimation of m eeg signals the relevance of the procedure is demonstrated here for spatial whitening and source localization of meg signals

nonlinear parabolic equations with measure data in this paper we give summability results for the gradients of solutions of nonlinear parabolic equations whose model isu div u p 2 u on 0 t p with homogeneous cauchy dirichlet boundary conditions wherep 1 and is a bounded measure on 0 t we also study how the summability of the gradient improves if the measure is a function inlm 0 t withm small moreover we give a new proof of the existence of a solution for problem p

state constrained optimal control governed by non well posed parabolic differential equations this paper is concerned with a maximum principle of optimal control problems governed by some parabolic differential equations which could be non well posed both an integral type state constraint and a two point boundary time variable state constraint are considered

second order and stability analysis for optimal sparse control of the fitzhugh nagumo equation optimal sparse control problems are considered for the fitzhugh nagumo system including the so called schlogl model the nondifferentiable objective functional of tracking type includes a quadratic tikhonov regularization term and the l 1 norm of the control that accounts for the sparsity though the objective functional is not differentiable a theory of second order sufficient optimality conditions is established for tikhonov regularization parameter nu 0 and also for the case nu 0 in this context also local minima are discussed that are strong in the sense of the calculus of variations the second order conditions are used as the main assumption for proving the stability of locally optimal solutions with respect to nu to 0 and with respect to perturbations of the desired state functions the theory is confirmed by numerical examples that are resolved with high precision to confirm that the optimal solution obeys the system of necessary optimality conditions

on the lagrange newton sqp method for the optimal control of semilinear parabolic equations a class of lagrange newton sqp methods is investigated for optimal control problems governed by semilinear parabolic initial boundary value problems distributed and boundary controls are given restricted by pointwise upper and lower bounds the convergence of the method is discussed in appropriate banach spaces based on a weak second order sufficient optimality condition for the reference solution local quadratic convergence is proved the proof is based on the theory of newton methods for generalized equations in banach spaces

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

karush kuhn tucker conditions for nonsmooth mathematical programming problems in function spaces lagrange multiplier rules for abstract optimization problems with mixed smooth and convex terms in the cost with smooth equality constrained and convex inequality constraints are presented the typical case for the equality constraints that the theory is meant for is given by differential equations applications are given to l 1 minimum norm control problems l infty norm minimization and a class of optimal control problems with distributed state constraints and nonsmooth cost

necessary optimality conditions for infinite dimensional state constrained control problems abstract this paper is concerned with first order necessary optimality conditions for state constrained control problems in separable banach spaces assuming inward pointing conditions on the constraint we give a simple proof of pontryagin maximum principle relying on infinite dimensional neighboring feasible trajectories theorems proved in 20 further we provide sufficient conditions guaranteeing normality of the maximum principle we work in the abstract semigroup setting but nevertheless we apply our results to several concrete models involving controlled pdes pointwise state constraints as positivity of the solutions are allowed

second order analysis for strong solutions in the optimal control of parabolic equations in this paper we provide a second order analysis for strong solutions in the optimal control of parabolic equations we consider the case of box constraints on the control and final integral constraints on the state in contrast to sufficient conditions assuring quadratic growth in the weak sense i e when the cost increases at least quadratically for admissible controls uniformly near to the nominal one see e g 16 26 our main result provides a sufficient condition for quadratic growth of the cost for admissible controls whose associated states are uniformly near to the state of the nominal one as a consequence of our results for qualified problems with a strictly convex and quadratic hamiltonian we prove that both notions of quadratic growth coincide

probabilistic diffusion tractography with multiple fibre orientations what can we gain we present a direct extension of probabilistic diffusion tractography to the case of multiple fibre orientations using automatic relevance determination we are able to perform online selection of the number of fibre orientations supported by the data at each voxel simplifying the problem of tracking in a multi orientation field we then apply the identical probabilistic algorithm to tractography in the multi and single fibre cases in a number of example systems which have previously been tracked successfully or unsuccessfully with single fibre tractography we show that multi fibre tractography offers significant advantages in sensitivity when tracking non dominant fibre populations but does not dramatically change tractography results for the dominant pathways

pontryagin s maximum principle of optimal control governed by some non well posed semilinear parabolic differential equations this paper concerns about necessary conditions for optimal control problems governed by some semilinear parabolic differential equations which may be non well posed the two point boundary time variable state constraint involves the control set may be non convex

optimal boundary control of hyperbolic equations with pointwise state constraints in this paper we consider dynamic optimization problems for hyperbolic systems with boundary controls and pointwise state constraints in contrast to parabolic dynamics such systems have not been sufficiently studied in the literature the reason is the lack of regularity in the case of hyperbolic dynamics we present necessary optimality conditions for both neumann and dirichlet boundary control problems and discuss differences and relationships between them

neumann boundary control of hyperbolic equations with pointwise state constraints we consider optimal control problems for hyperbolic equations with controls in neumann boundary conditions with pointwise constraints on the control and state functions focusing on the multidimensional wave equation with a nonlinear term we derive new necessary optimality conditions in the form of a pointwise pontryagin maximum principle for the state constrained problem under consideration our approach is based on modern methods of variational analysis that allow us to obtain refined necessary optimality conditions with no convexity assumptions on integrands in the minimizing cost functional

adaptive finite element method for parabolic equations with dirac measure abstract in this paper we study the adaptive finite element method for parabolic equations with dirac measure two kinds of problems with separate measure data in time and measure data in space are considered it is well known that the solutions of such kind of problems may exhibit lower regularity due to the existence of the dirac measure and thus fit to adaptive fem for space discretization and variable time steps for time discretization for both cases we use piecewise linear and continuous finite elements for the space discretization and backward euler scheme or equivalently piecewise constant discontinuous galerkin method for the time discretization the a posteriori error estimates based on energy and l 2 norms for the fully discrete problems are then derived to guide the adaptive procedure numerical results are provided at the end of the paper to support our theoretical findings

ica based artifact correction improves spatial localization of adaptive spatial filters in meg abstract beamformers are one of the most common inverse models currently used in the estimation of source activity from magnetoencephelography meg data they rely on a minimization of total power while constraining the gain in the voxel of interest resulting in the suppression of background noise nonetheless in cases where background noise is strong compared to the source of interest or when many sources are present the ability of the beamformer to detect and accurately localize weak sources is reduced in visual paradigms two main background sources can substantially impact an accurate estimation of weaker sources ocular artifacts are orders of magnitude higher than neural sources making it difficult for the beamformer to effectively suppress them primary visual activations also result in strong signals that can impede localization of weak sources in this paper we systematically evaluated how neural visual and non neural eye heart sources affect the localization accuracy of frontal and medial temporal sources in visual tasks these sources are of tremendous interest in learning and memory studies as well as in clinical settings alzheimer s epilepsy and are typically difficult to localize robustly in meg empirical data from two tasks active learning and control were used to evaluate our analysis techniques global field power calculations showed multiple time periods where active learning was significantly different from response selection with dominant sources converging to the eyes extensive leakage of eye activity into frontal and visual that evoked responses into parietal cortices was also observed contributions from ocular activity to the reconstructed time series were indiscernible from task based recruitment of frontal sources in the original data removing artifacts eye movements cardiac and muscular by means of independent component analysis ica led to a significant improvement in detection and localization of frontal and medial temporal sources we verified our results by using simulations of sources placed in frontal and medial temporal regions with various types of background noise eye heart and visual we report that the detection and localization accuracy of frontal and medial temporal sources with beamformer techniques is highly dependent on the magnitude and location of background sources and that removing artifacts can substantially improve the beamformer s performance

optimal control of semilinear parabolic equations by bv functions optimal control problems for semilinear parabolic equations with control costs involving the total bounded variation seminorm are analyzed this choice of control cost favors optimal controls which are piecewise constant and it penalizes the number of jumps it is an appropriate choice if a simple structure of the optimal controls is desired which however is still sufficiently flexible so that good tracking properties can be maintained existence of optimal controls necessary and sufficient optimality conditions and sparsity properties of the derivatives are obtained convergence of a finite element approximation is analyzed and numerical examples illustrating structural properties of the optimal controls are provided

identification of space time distributed parameters in the gierer meinhardt reaction diffusion system we consider parameter identification for the classic gierer meinhardt reaction diffusion system the original gierer meinhardt model a gierer and h meinhardt kybernetik 12 1972 pp 30 39 was formulated with constant parameters and has been used as a prototype system for investigating pattern formation in developmental biology in our paper the parameters are extended in time and space and used as distributed control variables the methodology employs pde constrained optimization in the context of image driven spatiotemporal pattern formation we prove the existence of optimal solutions derive an optimality system and determine optimal solutions the results of numerical experiments in two dimensions are presented using the finite element method which illustrates the convergence of a variable step gradient algorithm for finding the optimal parameters of the system a practical target function is constructed for the optimal control algorithm corresponding to the actual image of a marine angel

optimal management of an urban road network with an environmental perspective abstract within the framework of numerical simulation and optimal control of partial differential equations in this work we deal with the mathematical modelling and optimal management of urban road networks in particular we are interested in finding the optimal management of the network intersections in order to reduce traffic congestion and atmospheric pollution so we consider two different multi objective control problems the former from a cooperative viewpoint the latter within a hierarchical paradigm propose a complete numerical algorithm to solve them and finally present several numerical tests for a realistic case posed in the guadalajara metropolitan area mexico where the possibilities of our methodology are shown

spike controls for elliptic and parabolic pdes abstract we analyze the use of measures of the minimal norm to control elliptic and parabolic equations we prove the sparsity of the optimal control in the parabolic case we prove that the solution of the optimization problem is a borel measure supported in a set of lebesgue measure zero in both cases the approximate controllability can be achieved efficiently by means of controls that are activated in some finite number of pointwise locations we also analyze the corresponding dual problem

optimal solution for the viscous nonlinear dispersive wave equation in this paper we study the optimal control problem for the viscous nonlinear dispersive wave equation we first prove the existence and uniqueness of a weak solution to this equation in a short interval by using the galerkin method furthermore the existence of an optimal solution to the viscous nonlinear dispersive wave equation is proved

a priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control we consider finite element approximations of parabolic control problems with pointwise control the state equation exhibits low regularity due to the control imposed pointwisely this introduces some difficulties for both theoretical and numerical analysis to discretize the optimal control problem we use variational discretization together with piecewise linear and continuous finite elements for the space discretization of the state and we use the backward euler scheme for time discretization we prove a priori error estimates for the control state and adjoint state numerical experiments are provided which support the theoretical results

robust flows over time models and complexity results we study dynamic network flows with uncertain input data under a robust optimization perspective in the dynamic maximum flow problem the goal is to maximize the flow reaching the sink within a given time horizon t while flow requires a certain travel time to traverse an edge in our setting we account for uncertain travel times of flow we investigate maximum flows over time under the assumption that at most vargamma travel times may be prolonged simultaneously due to delay we develop and study a mathematical model for this problem as the dynamic robust flow problem generalizes the static version it is np hard to compute an optimal flow however our dynamic version is considerably more complex than the static version we show that it is np hard to verify feasibility of a given candidate solution furthermore we investigate temporally repeated flows and show that in contrast to the non robust case that is without uncertainties they no longer provide optimal solutions for the robust problem but rather yield a worst case optimality gap of at least t we finally show that the optimality gap is at most o eta k log t where eta and k are newly introduced instance characteristics and provide a matching lower bound instance with optimality gap varomega log t and eta k 1 the results obtained in this paper yield a first step towards understanding robust dynamic flow problems with uncertain travel times

state constrained optimal control problems with states of low regularity we consider first order optimality conditions for state constrained optimal control problems in particular we study the case where the state equation does not have enough regularity to admit existence of a slater point in function space we overcome this difficulty by a special transformation under a density condition we show existence of lagrange multipliers which have a representation via measures and additional regularity properties

maximum principle for a hybrid system via singular perturbations we prove a maximum principle of pontryagin type for the time optimal control of a hybrid system the system is nonlinear and consists of a controlled coupled ode pde the control input acts via the boundary and the interior of the domain systems of this type frequently arise in modeling population dynamics in a contaminated environment for the regularization we use ekeland s variational principle along with a singular perturbation technique for this purpose we introduce a new trajectory on an additional exterior domain whose size may be considered as a singular perturbation parameter it turns out that the maximum principle is stable with respect to the size of the additional exterior domain the technique allows us to obtain necessary optimality conditions without involving measure boundary data

first and second order optimality conditions for a class of optimal control problems with quasilinear elliptic equations a class of optimal control problems for quasilinear elliptic equations is considered where the coefficients of the elliptic differential operator depend on the state function first and second order optimality conditions are discussed for an associated control constrained optimal control problem main emphasis is laid on second order sufficient optimality conditions to this aim the regularity of the solutions to the state equation and its linearization is studied in detail and the pontryagin maximum principle is derived one of the main difficulties is the nonmonotone character of the state equation

on the structure of lagrange multipliers for state constrained optimal control problems abstract state constrained optimal control problems represent severe analytical and numerical challenges it is shown that under certain conditions on the active set the measure representing the lagrange multiplier associated to the state constraint can be decomposed in a distributed part in l2 with support on the active set and a boundary measure concentrated on the interface between active and inactive sets

optimal control of parabolic differential equations with two point boundary state constraints this paper deals with the necessary conditions of optimality for some optimal control problems governed by some parabolic differential equations involving monotone graphs the two point boundary time variable state constraints are discussed some analyses on the state equations are given

on regularity of solutions and lagrange multipliers of optimal control problems for semilinear equations with mixed pointwise control state constraints a class of nonlinear elliptic and parabolic optimal control problems with mixed control state constraints is considered extending a method known for the control of ordinary differential equations to the case of pdes the yosida hewitt theorem is applied to show that the lagrange multipliers are functions of certain l p spaces by bootstrapping arguments under natural assumptions optimal controls are shown to be lipschitz continuous in the elliptic case and holder continuous for parabolic problems

pontryagin s principle for state constrained control problems governed by parabolic equations with unbounded controls this paper deals with optimal control problems governed by semilinear parabolic equations with pointwise state constraints and unbounded controls under some strong stability assumption we obtain necessary optimality conditions in the form of a pontryagin s minimum principle in qualified form a pontryagin s principle in nonqualified form is also proved without any stability condition

control of an elliptic problem with pointwise state constraints this paper deals with a quadratic control problem for elliptic equations with pointwise state constraints existence and uniqueness of the solution is proved optimality conditions are given and regularity of the optimal solution is investigated

solvability of a parabolic problem with non smooth data abstract we study a weak solvability of one linear parabolic problem with a summable right hand side and non smooth coefficients we consider this problem as appropriate operator equation in some banach space methods of the semigroups theory fractional powers of operators and integral operators for solvability of this operator equation are used

theoretical and numerical analysis of an optimal control problem related to wastewater treatment in this work we deal with the design and management of wastewater treatment systems mainly the disposal of sea outfalls discharging polluting effluent from a sewerage system this problem is formulated as a pointwise optimal control problem with state and control constraints the main difficulties arise from the lack of regularity of the second member in the state system and from the pointwise constraints on the state variables we develop the theoretical analysis of the problem we propose an algorithm for its numerical resolution and finally we give results for a realistic problem posed in the ria of vigo spain

pontryagin maximum principle for semilinear and quasilinearparabolic equations with pointwise state constraints this paper studies the first order necessary conditions for the optimal controls of semilinear and quasilinear parabolic partial differential equations with pointwise state constraints a pontryagin type maximum principle is obtained

finite element approximations of parabolic optimal control problems with controls acting on a lower dimensional manifold this paper is devoted to the study of finite element approximations to parabolic optimal control problems with controls acting on a lower dimensional manifold the manifold can be a point a curve or a surface which may be independent of time or evolve in the time horizon and is assumed to be strictly contained in the space domain at first we obtain the first order optimality conditions for the control problems and the corresponding regularity results then for the control problems we consider the fully discrete finite element approximations based on the piecewise constant discontinuous galerkin scheme for time discretization and piecewise linear finite elements for space discretization and variational discretization to the control variable a priori error estimates are finally obtained for the fully discretized control problems and supported by numerical examples

probabilistic independent component analysis for functional magnetic resonance imaging we present an integrated approach to probabilistic independent component analysis ica for functional mri fmri data that allows for nonsquare mixing in the presence of gaussian noise in order to avoid overfitting we employ objective estimation of the amount of gaussian noise through bayesian analysis of the true dimensionality of the data i e the number of activation and non gaussian noise sources this enables us to carry out probabilistic modeling and achieves an asymptotically unique decomposition of the data it reduces problems of interpretation as each final independent component is now much more likely to be due to only one physical or physiological process we also describe other improvements to standard ica such as temporal prewhitening and variance normalization of timeseries the latter being particularly useful in the context of dimensionality reduction when weak activation is present we discuss the use of prior information about the spatiotemporal nature of the source processes and an alternative hypothesis testing approach for inference using gaussian mixture models the performance of our approach is illustrated and evaluated on real and artificial fmri data and compared to the spatio temporal accuracy of results obtained from classical ica and glm analyses

robust discrete optimization and network flows we propose an approach to address data uncertainty for discrete optimization and network flow problems that allows controlling the degree of conservatism of the solution and is computationally tractable both practically and theoretically in particular when both the cost coefficients and the data in the constraints of an integer programming problem are subject to uncertainty we propose a robust integer programming problem of moderately larger size that allows controlling the degree of conservatism of the solution in terms of probabilistic bounds on constraint violation when only the cost coefficients are subject to uncertainty and the problem is a 0 1 discrete optimization problem on n variables then we solve the robust counterpart by solving at most n 1 instances of the original problem thus the robust counterpart of a polynomially solvable 0 1 discrete optimization problem remains polynomially solvable in particular robust matching spanning tree shortest path matroid intersection etc are polynomially solvable we also show that the robust counterpart of an np hard approximable 0 1 discrete optimization problem remains approximable finally we propose an algorithm for robust network flows that solves the robust counterpart by solving a polynomial number of nominal minimum cost flow problems in a modified network

a priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time in this paper we consider an optimal control problem which is governed by a linear parabolic equation and is subject to state constraints pointwise in time optimal order error estimates are developed for a space time finite element discretization of this problem numerical examples confirm the theoretical results as a by product of our analysis we derive a new regularity result for the optimal control

optimal control of the viscous weakly dispersive degasperis procesi equation abstract in this paper we study the optimal control problem for the viscous weakly dispersive degasperis procesi equation we deduce the existence and uniqueness of a weak solution to this equation in a short interval by using the galerkin method then according to optimal control theories and distributed parameter system control theories the optimal control of the viscous weakly dispersive degasperis procesi equation under boundary conditions is given and the existence of an optimal solution to the viscous weakly dispersive degasperis procesi equation is proved

an extension of pontryagin s principle for state constrained optimal control of semilinear elliptic equations and variational inequalities this paper deals with state constrained optimal control problems governed by semilinear elliptic equations or variational inequalities by using ekeland s principle a minimum principle of pontryagin s type under some stability conditions of the optimal cost with respect to the state constraints is derived

implicit taylor methods for parabolic problems with nonsmooth data and applications to optimal heat control as a rule parabolic problems with nonsmooth data show rapid changes of its solution or even possess solutions of reduced smoothness while for smooth data various time integration methods e g the trapezoidal rule or the euler backwards scheme work efficiently but in case of jumps effects of high frequency oscillations are observable over a long time horizon or steep changes are smeared out implicit taylor methods itm which are mostly applied in specific applications like interval methods but not commonly used for general cases combine high accuracy with strong damping of unwanted oscillations these properties make them a good choice in case of nonsmooth data in the present paper itm are investigated in detail for semi discrete linear parabolic problems in itm at each time level a large scale linear system has to be solved and preconditioned conjugate gradient methods pcg can efficiently be applied here adapted preconditioners are constructed and tight spectral bounds are derived which are independent of the discretization parameters of the parabolic problem as an important application itm are considered in case of boundary heat control occurring control constraints are involved by means of penalty functions to solve the completely discretized problem gradient based numerical algorithms are used where the gradient of the objective is partially evaluated via discrete adjoints and partially by explicitly available terms corresponding to the penalties some test examples illustrate the efficiency of the considered algorithms

a pontryagin maximum principle for infinite dimensional problems a basic idea of the classical approach for obtaining necessary optimality conditions in optimal control is to construct suitable needle like control variations we use this idea to prove the main result of the present paper a pontryagin maximum principle for infinite dimensional optimal control problems with pointwise terminal constraints in arbitrary banach state space by refining the classical variational technique we are able to replace the differentiability of the norm of the state space guaranteed by the strict convexity of its dual norm which is assumed in the known results by a separation argument we also drop another key assumption which is common in the existing literature on infinite dimensional control problems that the set of variations in the state space of the state trajectory s endpoint resulting from the control variations be finite codimensional instead we require only that it has nonempty interior in its closed affine hull as an application of the abstract result we present an illustrative example an optimal control problem for an age structured system with pointwise terminal state constraints

optimal control problems for semilinear non well posed parabolic differential equations this paper deals with necessary conditions for optimal control problem governed by some semilinear parabolic differential equation which may be non well posed state constrained problem is considered finally under some suitable assumptions we obtain the existence of optimal pairs

boundary feedback stabilization of the schlogl system the schlogl system is governed by a nonlinear reaction diffusion partial differential equation with a cubic nonlinearity that determines three constant equilibrium states it is a classical example of a chemical reaction system that is bistable the constant equilibrium that is enclosed by the other two constant equilibrium points is unstable in this paper robin boundary feedback laws are presented that stabilize the system in a given stationary state or more generally in a given time dependent desired system orbit the exponential stability of the closed loop system with respect to the l 2 norm is proved in particular it is shown that with the boundary feedback law the unstable constant equilibrium point can be stabilized

optimal control of a parabolic equation with time dependent state constraints in this paper we study the optimal control problem of the heat equation with a cubic nonlinearity by a distributed control over a subset of the domain in the presence of a state constraint the latter is integral over the space and has to be satisfied each time using for the first time the technique of alternative optimality systems in the context of optimal control of partial differential equations we show that both the control and multiplier are continuous in time under some natural geometric hypotheses we can prove that extended polyhedricity holds allowing us to obtain no gap second order optimality conditions that characterize quadratic growth an expansion of the value function and of approximate solutions can be computed for a directional perturbation of the right hand side of the state equation

optimality conditions for stochastic boundary control problems governed by semilinear parabolic equations abstract we study the boundary control problems for stochastic parabolic equations with neumann boundary conditions imposing super parabolic conditions we establish the existence and uniqueness of the solution of state and adjoint equations with non homogeneous boundary conditions by the galerkin approximations method we also find that in this case the adjoint equation bspde has two boundary conditions one is non homogeneous the other is homogeneous by these results we derive necessary optimality conditions for the control systems under convex state constraints by the convex perturbation method

pontryagin s principle for state constrained boundary control problems of semilinear parabolic equations this paper deals with state constrained optimal control problems governed by semilinear parabolic equations we establish a minimum principle of pontryagin s type to deal with the state constraints we introduce a penalty problem by using ekeland s principle the key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are defined conditions for normality of optimality conditions are given

state constrained optimal control problems governed by coupled nonlinear wave equations with memory this work considers the optimal control problems governed by coupled nonlinear wave equations with memory where the state constraint is a type of integral first we investigate the existence of the optimal solution for the optimal control problem and then we deduce the maximum principle for the optimal solution with state constraint moreover we give a concrete example which is covered by the main result

optimal control of the viscous degasperis procesi equation this paper studies the problem for optimal control of the viscous degasperis procesi equation the existence and uniqueness of weak solution to the equation are proved in a short interval the optimal control of the viscous degasperis procesi equation under boundary condition is given and the existence of optimal solution to the equation is proved

global attractors for the p laplacian equations with nonregular data abstract let be a smooth bounded domain in r n n 3 we consider the long time behavior of solutions to the p laplacian equation u t p u f u g where 2 p n p div u p 2 u is the p laplace operator assume that g and the initial condition u 0 lie in l 1 f r r is of class c 1 and satisfies proper growth conditions firstly we prove the uniqueness of the entropy solution and establish some regularity results then we show the existence of a global attractor a in l r 1 w 0 1 s with s max n p 1 n 1 p r 1 r to obtain the results a decomposition method and a bootstrap technique are used

second order necessary and sufficient optimality conditions for optimization problems and applications to control theory this paper deals with a class of nonlinear optimization problems in a function space where the solution is restricted by pointwise upper and lower bounds and by finitely many equality and inequality constraints of functional type second order necessary and sufficient optimality conditions are established where the cone of critical directions is arbitrarily close to the form which is expected from the optimization in finite dimensional spaces the results are applied to some optimal control problems for ordinary and partial differential equations

optimal control of semilinear parabolic equations with k approximate periodic solutions in this paper we study some optimal control problems governed by certain semilinear parabolic equations with mathcal k approximate periodic solutions we first prove the existence and uniqueness theorems for mathcal k approximate periodic solutions of the equations we then use these results to establish the qualified pontryagin maximum principle the existence for such optimal controls is also investigated in the paper

an admm numerical approach to linear parabolic state constrained optimal control problems optimal control problems arising from systems modeled by linear parabolic equations may be difficult for both theoretical analysis and algorithmic design for the case where there are additional constraints on the state variables restrictive regularity assumptions are usually required to guarantee the existence of the associated lagrange multiplier and thus some regularization type methods such as the moreau yosida and lavrentiev methods have been discussed in the literature in this article we study the application of the alternating direction method of multipliers admm to linear parabolic state constrained optimal control problems and propose an admm numerical approach we prove the convergence of the admm algorithm without any existence or regularity assumption on the lagrange multiplier and estimate its worst case convergence rate in both the ergodic and nonergodic senses an important feature of the admm approach is that it decouples the state constraints and the parabolic optimal control problems inside each iteration we show the efficiency of the admm approach by testing some control problems in two space dimensions

flows over time with load dependent transit times more than forty years ago ford and fulkerson studied maximum s t flows over time also called dynamic flows in networks with fixed transit times on the arcs and a fixed time horizon here flow on arcs may change over time and transit times specify the amount of time it takes for flow to travel through a particular arc ford and fulkerson proved that there always exists an optimal solution which sends flow on certain s t paths at a constant rate as long as there is enough time left for the flow along a path to arrive at the sink a flow over time featuring this simple structure is called temporally repeated although this result does not hold for the more general and also more realistic setting where transit times depend on the current flow situation we show that there always exists a provably good temporally repeated solution moreover such a solution can be determined very efficiently by only one minimum convex cost flow computation our results rest upon a new model of flow dependent transit times it is based on two assumption on the pace of flow on a particular arc first the pace of flow on an arc is assumed to be uniform for all flow units on an arc for each point in time second this uniform pace is for each moment determined by the actual amount of flow on this arc finally we show that the resulting flow over time problem is strongly np hard and cannot be approximated with arbitrary precision in polynomial time unless p np

optimal problems for nonlinear parabolic boundary control systems this paper considers optimal problems for boundary control systems with the control acting through a general nonlinear boundary condition the problems include constraints on the control and target conditions the final result is a version of pontryagin s maximum principle this setup covers such physical situations as stefan boltzmannboundary conditions and is based on an abstract theory of nonlinear programming problems

necessary conditions for optimal control of distributed parameter systems the ekeland variational principle is used to prove a maximum principle for the optimal controls of a general semilinear evolutionary control system in a banach space with strictly convex dual the boundary constraint is of general type which includes the optimal periodic control problem as a special case

constrained linear basis sets for hrf modelling using variational bayes abstract fmri modelling requires flexible haemodynamic response function hrf modelling with the hrf being allowed to vary spatially and between subjects to achieve this flexibility voxelwise parameterised hrfs have been proposed however inference on such models is very slow an alternative approach is to use basis functions allowing inference to proceed in the more manageable general linear model glm framework however a large amount of the subspace spanned by the basis functions produces nonsensical hrf shapes in this work we propose a technique for choosing a basis set and then the means to constrain the subspace spanned by the basis set to only include sensible hrf shapes penny et al neuroimage 2003 showed how variational bayes can be used to infer on the glm for fmri here we extend the work of penny et al to give inference on the glm with constrained hrf basis functions and with spatial markov random fields on the autoregressive noise parameters constraining the subspace spanned by the basis set allows for far superior separation of activating voxels from nonactivating voxels in fmri data we use spatial mixture modelling to produce final probabilities of activation and demonstrate increased sensitivity on an fmri dataset

optimal control of fluid dynamic systems with state constraint of pointwise type abstract the aim of this work is to obtain the maximum principle by spike perturbation for the optimal control problem with pointwise type state constraint governed by 3 dimensional fluid dynamic systems

optimal control for semilinear evolutionary variational bilateral problem this paper is concerned with an optimal control problem for some semilinear evolutionary variational inequalities associated with bilateral constraints the control domain is a general separable metric space and has no algebraic structure in particular it is not necessarily convex existence and optimality conditions of optimal pairs are established

boundary control of semilinear elliptic equations with pointwise state constraints this paper is concerned with state constrained optimal control problems of semilinear elliptic equations the control being on the boundary optimality conditions are derived and regularity of the optimal solution is investigated

pontryagin s principle for local solutions of control problems with mixed control state constraints this paper deals with optimal control problems of semilinear parabolic equations with pointwise state constraints and coupled integral state control constraints we obtain necessary optimality conditions in the form of a pontryagin s minimum principle for local solutions in the sense of l p p leq infty

multiple sparse priors for the m eeg inverse problem this paper describes an application of hierarchical or empirical bayes to the distributed source reconstruction problem in electro and magnetoencephalography eeg and meg the key contribution is the automatic selection of multiple cortical sources with compact spatial support that are specified in terms of empirical priors this obviates the need to use priors with a specific form e g smoothness or minimum norm or with spatial structure e g priors based on depth constraints or functional magnetic resonance imaging results furthermore the inversion scheme allows for a sparse solution for distributed sources of the sort enforced by equivalent current dipole ecd models this means the approach automatically selects either a sparse or a distributed model depending on the data the scheme is compared with conventional applications of bayesian solutions to quantify the improvement in performance