twistor spaces of hypercomplex manifolds are balanced abstract a hypercomplex structure on a differentiable manifold consists of three integrable almost complex structures that satisfy quaternionic relations if in addition there exists a metric on the manifold which is hermitian with respect to the three structures and such that the corresponding hermitian forms are closed the manifold is said to be hyperkahler in the paper non hermitian yang mills connections 13 kaledin and verbitsky proved that the twistor space of a hyperkahler manifold admits a balanced metric these were first studied in the article on the existence of special metrics in complex geometry 17 by michelsohn in the present article we review the proof of this result and then generalize it and show that twistor spaces of general compact hypercomplex manifolds are balanced

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

percolation in strongly correlated systems abstract we investigate the threshold percolation density pc in strongly correlated lattice systems the probability distribution of the system is the stationary state for a combined kawasaki and glauber dynamics the voter model with flips when the glauber rate goes to zero the pair correlation function of the system at any density decays in three dimensions as r 1 and does not decay at all in two dimensions using monte carlo calculations and finite size scaling we obtain information about pc as a function of the glauber rate percolation in gaussian fields on a lattice which have similar slow decay is also discussed

a semismooth equation approach to the solution of nonlinear complementarity problems in this paper we present a new algorithm for the solution of nonlinear complementarity problems the algorithm is based on a semismooth equation reformulation of the complementarity problem we exploit the recent extension of newton s method to semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild assumptions other interesting features of the new algorithm are an extreme simplicity along with a low computational burden per iteration we include numerical tests which show the viability of the approach

the velocity tracking problem for navier stokes flows with bounded distributed controls we present some systematic approaches to the mathematical analysis and numerical approximation of the time dependent optimal control problem of tracking the velocity for navier stokes flows in bounded two dimensional domains with bounded distributed controls we study the existence of optimal solutions and derive an optimality system from which optimal solutions may be determined we also define and analyze semidiscrete in time and fully space time discrete approximations of the optimality system and a gradient method for the solution of the fully discrete system the results of some computational experiments are provided

active control and drag optimization for flow past a circular cylinder the main objective of this article is to investigate computational methods for the active control and drag optimization of incompressible viscous flow past cylinders using the two dimensional navier stokes equations as the flow model the computational methodology relies on the following ingredients space discretization of the navier stokes equations by finite element approximations time discretization by a second order accurate two step implicit explicit finite difference scheme calculation of the cost function gradient by the adjoint equation approach and minimization of the cost function by a quasi newton method a la bfgs the above methods have been applied to boundary control by rotation of the flow around a circular cylinder and show 30 to 60 drag reduction compared to the fixed cylinder configuration for reynolds numbers in the range from 200 to 1000

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 general framework for robust control in fluid mechanics abstract the application of optimal control theory to complex problems in fluid mechanics has proven to be quite effective when complete state information from high resolution numerical simulations is available p moin t r bewley appl mech rev part 2 47 6 1994 s3 s13 t r bewley p moin r temam j fluid mech 1999 submitted for publication in this approach an iterative optimization algorithm based on the repeated computation of an adjoint field is used to optimize the controls for finite horizon nonlinear flow problems f abergel r temam theoret comput fluid dyn 1 1990 303 325 in order to extend this infinite dimensional optimization approach to control externally disturbed flows in which the controls must be determined based on limited noisy flow measurements alone it is necessary that the controls computed be insensitive to both state disturbances and measurement noise for this reason robust control theory a generalization of optimal control theory has been examined as a technique by which effective control algorithms which are insensitive to a broad class of external disturbances may be developed for a wide variety of infinite dimensional linear and nonlinear problems in fluid mechanics an aim of the present paper is to put such algorithms into a rigorous mathematical framework for it cannot be assumed at the outset that a solution to the infinite dimensional robust control problem even exists in this paper conditions on the initial data the parameters in the cost functional and the regularity of the problem are established such that existence and uniqueness of the solution to the robust control problem can be proven both linear and nonlinear problems are treated and the 2d and 3d nonlinear cases are treated separately in order to get the best possible estimates several generalizations are discussed and an appropriate numerical method is proposed

primal dual strategy for constrained optimal control problems an algorithm for efficient solution of control constrained optimal control problems is proposed and analyzed it is based on an active set strategy involving primal as well as dual variables for discretized problems sufficient conditions for convergence in finitely many iterations are given numerical examples are given and the role of the strict complementarity condition is discussed

semismooth newton methods for optimal control of the wave equation with control constraints in this paper optimal control problems governed by the wave equation with control constraints are analyzed three types of control action are considered distributed control neumann boundary control and dirichlet control and proper functional analytic settings for them are discussed for treating inequality constraints semismooth newton methods are discussed and their convergence properties are investigated in the case of distributed and neumann control superlinear convergence is shown for dirichlet boundary control superlinear convergence is proved for a strongly damped wave equation for numerical realization a space time finite element discretization is discussed numerical examples illustrate the results

primal dual interior point methods for pde constrained optimization this paper provides a detailed analysis of a primal dual interior point method for pde constrained optimization considered are optimal control problems with control constraints in l p it is shown that the developed primal dual interior point method converges globally and locally superlinearly not only the easier l setting is analyzed but also a more involved l q analysis q is presented in l the set of feasible controls contains interior points and the frechet differentiability of the perturbed optimality system can be shown in the l q setting which is highly relevant for pde constrained optimization these nice properties are no longer available nevertheless a convergence analysis is developed using refined techniques in parti cular two norm techniques and a smoothing step are required the l q analysis with smoothing step yields global linear and local superlinear convergence whereas the l analysis without smoothing step yields only global linear convergence

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

optimal controls of navier stokes equations this paper studies optimal control problems of the fluid flow governed by the navier stokes equations two control problems are formulated in the case of the driven cavity and flow through a channel with sudden expansion and solved successfully using a numerical optimization algorithm based on the augmented lagrangian method existence and the first order optimality condition of the optimal control are established a convergence result on the augmented lagrangian method for nonsmooth cost functional is obtained

optimal control problems with partially polyhedric constraints this paper discusses a class of state constrained optimal control problems for which it is possible to formulate second order necessary or sufficient conditions for local optimality or quadratic growth that do not involve all curvature terms for the constraints this kind of result is classical in the case of polyhedric control constraints our theory of optimization problems with partially polyhedric constraints allows to extend these results to the case when the control constraints are polyhedric in the presence of state constraints satisfying some specific hypotheses the analysis is based on the assumption that some strict semilinearized qualification condition is satisfied we apply the theory to some optimal control problems of elliptic equations with state and control constraints

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

state constrained optimal control of the three dimensional stationary navier stokes equations abstract in this paper an optimal control problem for the stationary navier stokes equations in the presence of state constraints is investigated existence of optimal solutions is proved and first order necessary conditions are derived the regularity of the adjoint state and the state constraint multiplier is also studied lipschitz stability of the optimal control state and adjoint variables with respect to perturbations is proved and a second order sufficient optimality condition for the case of pointwise state constraints is stated

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

optimal control of the stationary navier stokes equations with mixed control state constraints in this paper we consider the distributed optimal control of the navier stokes equations in the presence of pointwise mixed control state constraints after deriving a first order necessary condition the regularity of the mixed constraint multiplier is investigated second order sufficient optimality conditions are studied as well in the last part of the paper a semismooth newton method is applied for the numerical solution of the control problem the convergence of the method is proved and numerical experiments are carried out

second order methods for optimal control of time dependent fluid flow second order methods for open loop optimal control problems governed by the two dimensional instationary navier stokes equations are investigated optimality systems based on a lagrangian formulation and adjoint equations are derived the newton and quasi newton methods as well as various variants of sqp methods are developed for applications to optimal flow control and their complexity in terms of system solves is discussed local convergence and rate of convergence are proved a numerical example illustrates the feasibility of solving optimal control problems for two dimensional instationary navier stokes equations by second order numerical methods in a standard workstation environment

sequential quadratic programming sqp for optimal control in direct numerical simulation of turbulent flow the current work focuses on the development and application of an efficient algorithm for optimization of three dimensional turbulent flows simulated using direct numerical simulation dns or large eddy simulations and further characterized by large dimensional optimization parameter spaces the optimization algorithm is based on sequential quadratic programming sqp in combination with a damped formulation of the limited memory bfgs method the latter is suitable for solving large scale constrained optimization problems whose hessian matrices cannot be computed and stored at a reasonable cost we combine the algorithm with a line search merit function based on an l 1 norm to enforce the convergence from any remote point it is first shown that the proposed form of the damped l bfgs algorithm is suitable for solving equality constrained rosenbrock type functions then we apply the algorithm to an optimal control test problem that consists of finding the optimal initial perturbations to a turbulent temporal mixing layer such that mixing is improved at the end of a simulation time horizon t the controls are further subject to a non linear equality constraint on the total control energy dnss are used to resolve all turbulent scales of motion and a continuous adjoint formulation is employed to calculate the gradient of the cost functionals we compare the convergence speed of the sqp l bfgs algorithm to a conventional non linear conjugate gradient method i e the current standard in dns based optimal control and find that the sqp algorithm is more than an order of magnitude faster than the conjugate gradient method

a new merit function for nonlinear complementarity problems and a related algorithm we investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one assuming that the complementarity problem is defined by a p 0 function we prove that every stationary point of the unconstrained problem is a global solution furthermore if the complementarity problem is defined by a uniform p function the level sets of the merit function are bounded the properties of the new merit function are compared with those of mangasarian solodov s implicit lagrangian and fukushima s regularized gap function we also introduce a new simple active set local method for the solution of complementarity problems and show how this local algorithm can be made globally convergent by using the new merit function

surface order large deviations for high density percolation we derive surface order large deviation estimates for the volume of the largest cluster and for the volume of the largest region surrounded by a cluster of a bernoulli percolation process restricted to a big finite box with sufficiently large parameter we also establish a useful version of the isoperimetric inequality which is the main tool of our proofs

second order sufficient optimality conditions for some state constrained control problems of semilinear elliptic equations this paper deals with a class of optimal control problems governed by elliptic equations with nonlinear boundary condition the case of boundary control is studied pointwise constraints on the control and certain equality and set constraints on the state are considered second order sufficient conditions for local optimality of controls are established

semi discrete a priori error analysis for the optimal control of the unsteady navier stokes equations with variational multiscale stabilization in this work the optimal control problems of the unsteady navier stokes equations with variational multiscale stabilization vms are considered at first the first order continuous optimality conditions are obtained since the adjoint equation of the navier stokes problem is a convection diffusion type system then the same stabilization is applied to it semi discrete a priori error estimates are obtained for the state adjoint state and control variables crank nicholson time discretization is used to get the fully discrete scheme numerical examples verify the theoretical findings and show the efficiency of the stabilization for higher reynolds number

a trust region method for solving generalized complementarity problems based on a semismooth equation reformulation using fischer s function a trust region algorithm is proposed for solving the generalized complementarity problem gcp the algorithm uses a generalized jacobian of the function involved in the semismooth equation and adopts the squared natural residual of the semismooth equation as a merit function the proposed algorithm is applicable to the nonlinear complementarity problem because the latter problem is a special case of the gcp global convergence and under a nonsingularity assumption local q superlinear or quadratic convergence of the algorithm are established moreover calculation of a generalized jacobian is discussed and numerical results are presented

analysis and finite element approximation of optimal control problems for the stationary navier stokes equations with distributed and neumann controls we examine certain analytic and numerical aspects of optimal control problems for the stationary navier stokes equations the controls considered may be of either the distributed or neumann type the functionals minimized are either the viscous dissipation or the l4 distance of candidate flows to some desired flow we show the existence of optimal solutions and justify the use of lagrange multiplier techniques to derive a system of partial differential equations from which optimal solutions may be deduced we study the regularity of solutions of this system then we consider the approximation by finite element methods of solutions of the optimality system and derive optimal error estimates

a simple proof of a theorem by uhlenbeck and yau a subbundle of a hermitian holomorphic vector bundle e h can be metrically and differentially defined by the orthogonal projection onto itself a weakly holomorphic subbundle of e h is by definition an orthogonal projection lying in the sobolev space l21 of l2 sections of end e with l2 first order derivatives in the sense of distributions which satisfies furthermore id d 0 a weakly holomorphic subbundle of e h is shown to define a coherent subsheaf of open image in new window e and implicitly a holomorphic subbundle of e in the complement of an analytic subset of codimension 2 this result provided the key technical argument to the proof given by uhlenbeck and yau for the kobayashi hitchin correspondence on compact kahler manifolds we give here a much simpler proof based on current theory the idea is to construct local meromorphic sections of im which locally span the fibers we first make this construction on one dimensional submanifolds of x and subsequently extend it by means of a hartogs type theorem of shiffman s

second order optimality conditions in sets of sf l infty functions with range in a polyhedron formal extensions of the general second order necessary conditions and sufficient conditions for local optimality in a polyhedral convex set u subset bbb r m are established for sf l infty local optimality and sf l 2 local optimality in the infinite dimensional nonpolyhedral convex set omega of sf l infty functions u cdot 0 1 rightarrow u a more refined analysis for nonconvex cost functions with specially structured differentials yields optimality conditions that apply to an important class of constrained input bolza optimal control problems the gap between the necessary conditions and sufficient conditions in this setting is uncharacteristically small for infinite dimensional problems in the control problem context the sf l infty local optimality conditions and sf l 2 local optimality conditions entail a mild strengthening of a pointwise strict complementarity condition and variants of the legendre clebsch condition and the pontryagin minimum principle in related recent studies similar second order sufficient conditions for the special case u 0 infty are the key hypotheses in corresponding local convergence theories for iterative constrained minimization algorithms

convergence analysis of some algorithms for solving nonsmooth equations this paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally lipschitzian functions for the directional derivative based and the generalized jacobian based newton methods both the iterates and the corresponding function values are locally superlinearly convergent globally a limiting point of the iterate sequence generated by the damped directional derivative based newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one provided that the system is strongly bd regular and semismooth at this point in this case the convergence is superlinear a general attraction theorem is presented which can be applied to two algorithms proposed by han pang and rangaraj a hybrid method which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent is also presented

a primal dual active set algorithm with continuation for compressed sensing the success of compressed sensing relies essentially on the ability to efficiently find an approximately sparse solution to an under determined linear system in this paper we developed an efficient algorithm for the sparsity promoting l 1 regularized least squares problem by coupling the primal dual active set strategy with a continuation technique on the regularization parameter in the active set strategy we first determine the active set from primal and dual variables and then update the primal and dual variables by solving a low dimensional least square problem on the active set which makes the algorithm very efficient the continuation technique globalizes the convergence of the algorithm with provable global convergence under restricted isometry property rip further we adopt two alternative methods i e a modified discrepancy principle and a bayesian information criterion to choose the regularization parameter automatically numerical experiments indicate that our algorithm is very competitive with state of the art algorithms in terms of accuracy and efficiency without a priori information about the regularization parameter

a kobayashi hitchin correspondence for i holomorphic bundles abstract in this paper we introduce the notions of hermitian einstein metric and stability for i holomorphic vector bundles on bi hermitian manifolds moreover we establish a kobayashi hitchin correspondence for i holomorphic vector bundles on bi hermitian manifolds examples of such vector bundles include generalized holomorphic bundles on generalized kahler manifolds we also show that stability of a vector bundle in this sense can depend on the parameter

sufficient second order optimality conditions for convex control constraints abstract in this article sufficient optimality conditions are established for optimal control problems with pointwise convex control constraints here the control is a function with values in r n the constraint is of the form u x u x where u is a set valued mapping that is assumed to be measurable with convex and closed images the second order condition requires coercivity of the lagrange function on a suitable subspace which excludes strongly active constraints together with first order necessary conditions it ensures local optimality of a reference function in an l neighborhood the analysis is done for a model problem namely the optimal distributed control of the instationary navier stokes equations

optimal control for elliptic systems with pointwise euclidean norm constraints on the controls optimal control for an elliptic system with pointwise euclidean norm constraints on the control variables is investigated first order optimality conditions are derived in a manner that is amenable for numerical realisation an efficient semismooth newton algorithm is proposed based on this optimality system numerical examples are given to validate the superlinear convergence of the semismooth newton algorithm

entropic repulsion and the maximum of the two dimensional harmonic crystal we consider the lattice version of the free field in two dimensions also called harmonic crystal the main aim of the paper is to discuss quantitatively the entropic repulsion of the random surface in the presence of a hard wall the basic ingredient of the proof is the analysis of the maximum of the field which requires a multiscale analysis reducing the problem essentially to a problem on a field with a tree structure

preconditioned iterative methods for navier stokes control problems pde constrained optimization problems are a class of problems which have attracted much recent attention in scientific computing and applied science in this paper we discuss preconditioned iterative methods for a class of time independent navier stokes control problems one of the main problems of this type in the field of fluid dynamics having detailed the picard type iteration we use to solve the problems and derived the structure of the matrix system to be solved at each step we utilize the theory of saddle point systems to develop efficient preconditioned iterative solution techniques for these problems we also require theory of solving convection diffusion control problems as well as a commutator argument to justify one of the components of the preconditioner

numerical solution of time dependent component with sparse structure of source term for a time fractional diffusion equation abstract we consider an inverse time dependent component of source term with sparse structure for the time fractional diffusion equation in the present paper we prove the uniqueness of the inverse problem with nonlocal observation data by laplace transform technique concerning the sparsity of the source term we transform the inverse source problem into an elastic net regularization optimization problem the semi smooth newton method is adopted to solve the optimization problem and the superconvergence of the semi smooth newton algorithm is proven several numerical examples are tested to verify the efficiency of the algorithm

a semi smooth newton method for control constrained boundary optimal control of the navier stokes equations abstract in this paper we study optimal control of the navier stokes equations when the control acts as a pointwise constrained boundary condition of dirichlet type the problem is analyzed in the control space h 00 1 2 the optimality system and second order sufficient optimality conditions are derived for the numerical solution we apply a semi smooth newton method to a regularized version of the original problem and show convergence properties of the method and of the regularized solutions towards the original one

newton methods for the optimal control of closed quantum spin systems an efficient and robust computational framework for solving closed quantum spin optimal control and exact controllability problems with control constraints is presented closed spin systems are of

balanced metrics on twisted higgs bundles a twisted higgs bundle on a kahler manifold x is a pair e phi consisting of a holomorphic vector bundle e and a holomorphic bundle morphism phi m otimes e rightarrow e for some holomorphic vector bundle m such objects were first considered by hitchin when x is a curve and m is the tangent bundle of x and also by simpson for higher dimensional base the hitchin kobayashi correspondence for such pairs states that e phi is polystable if and only if e admits a hermitian metric solving the hitchin equation this correspondence is a powerful tool to decide whether there exists a solution of the equation but it provides little information as to the actual solution in this paper we study a quantization of this problem that is expressed in terms of finite dimensional data and balanced metrics that give approximate solutions to the hitchin equation motivation for this study comes from work of donagi wijnholt jhep 05 068 2013 concerning balanced metrics for the vafa witten equations

entropic repulsion of the lattice free field consider the massless free field on thed dimensional lattice d d 3 that is the centered gaussian field on open image in new window with covariances given by the green function of the simple random walk on d we show that the probability that all the spins are positive in a box of volumen d decays exponentially at a rate of ordernd 2 logn and compute explicitly the corresponding constant in terms of the capacity of the unit cube the result is extended to a class of transient random walks with transition functions in the domain of the normal and stable law

the velocity tracking problem for navier stokes flows with boundary control we present some systematic approaches to the mathematical formulation and numerical approximation of the time dependent optimal control problem of tracking the velocity for navier stokes flows in a bounded two dimensional domain with boundary control we study the existence of optimal solutions and derive an optimality system from which optimal solutions may be determined we also define and analyze semidiscrete in time and full space time discrete approximations of the optimality system and a gradient method for the solution of the fully discrete system the results of some computational experiments are provided

optimum problems with measurable set valued constraints in this paper we provide a complete analysis of second order admissible variations to inequality type constraints which are given in terms of measurable set valued functions whose images are closed convex sets with nonempty interior as an application we consider optimization problems where such constraints are present and we deduce second order necessary conditions for optimality

entropic repulsion for the free field pathwise characterization in d 3 we study concentration properties of the lattice free field i e the centered gaussian field with covariance given by the green function of the discrete laplacian when constrained to be positive in a region of volume o n d hard wall condition it has been shown in 3 that as n the conditioned field is pushed to infinity more precisely the typical value of the variable to leading order is and the exact value of c was found it was moreover conjectured that the conditioned field once this diverging height is subtracted converges weakly to the lattice free field here we prove this conjecture along with other explicit bounds always in the direction of clarifying the intuitive idea that the free field with hard wall conditioning merely translates away from the hard wall we give also a proof alternative to the one presented in 3 of the lower bound on the probability that the free field is everywhere positive in a region of volume n d

gravitating vortices and the einstein bogomol nyi equations in this work we consider the gravitating vortex equations these equations couple a metric over a compact riemann surface with a hermitian metric over a holomorphic line bundle equipped with a fixed global section the higgs field and have a symplectic interpretation as moment map equations as a particular case of the gravitating vortex equations on mathbb p 1 we find the einstein bogomol nyi equations previously studied in the theory of cosmic strings in physics we prove two main results in this paper our first main result gives a converse to an existence theorem of yang for the einstein bogomol nyi equations establishing in this way a correspondence with geometric invariant theory for these equations in particular we prove a conjecture by yang about the non existence of cosmic strings on mathbb p 1 superimposed at a single point our second main result is an existence and uniqueness result for the gravitating vortex equations in genus greater than one

entropic repulsion for the gaussian free field conditioned on disconnection by level sets we investigate level set percolation of the discrete gaussian free field on mathbb z d d ge 3 in the strongly percolative regime we consider the event that the level set of the gaussian free field below a level alpha disconnects the discrete blow up of a compact set a subseteq mathbb r d from the boundary of an enclosing box we derive asymptotic large deviation upper bounds on the probability that the local averages of the gaussian free field deviate from a specific multiple of the harmonic potential of a when disconnection occurs these bounds combined with the findings of the recent work by duminil copin goswami rodriguez and severo show that conditionally on disconnection the gaussian free field experiences an entropic push down proportional to the harmonic potential of a in particular due to the slow decay of correlations the disconnection event affects the field on the whole lattice furthermore we provide a certain profile description for the field in the presence of disconnection we show that while on a macroscopic scale the field is pinned around a level proportional to the harmonic potential of a it locally retains the structure of a gaussian free field shifted by a constant value our proofs rely crucially on the solidification estimates developed in nitzschner and sznitman to appear in j eur math soc arxiv 1706 07229

constrained optimal control of navier stokes flow by semismooth newton methods abstract we propose and analyze a semismooth newton type method for the solution of a pointwise constrained optimal control problem governed by the time dependent incompressible navier stokes equations the method is based on a reformulation of the optimality system as an equivalent nonsmooth operator equation we analyze the flow control problem and prove q superlinear convergence of the method in the numerical implementation adjoint techniques are combined with a truncated conjugate gradient method numerical results are presented that support our theoretical results and confirm the viability of the approach

critical large deviations for gaussian fields in the phase transition regime i we investigate large deviations for the empirical distribution functional of a gaussian random field on mathbb r mathbb z d d geq 3 in the phase transition regime we first prove that the specific entropy governs an n d volume order large deviation principle outside the gibbsian class within the gibbsian class we derive an n d 2 capacity order large deviation principle with exact rate function and we apply this result to the asymptotics of microcanonical ensembles we also give a spins profile description of the field and show that smooth profiles obey n d 2 order large deviations whereas discontinuous profiles obey n d 1 surface order large deviations

adaptive finite elements for elliptic optimization problems with control constraints in this paper we develop a posteriori error estimates for finite element discretization of elliptic optimization problems with pointwise inequality constraints on the control variable we derive error estimators for assessing the discretization error with respect to the cost functional as well as with respect to a given quantity of interest these error estimators provide quantitative information about the discretization error and guide an adaptive mesh refinement algorithm allowing for substantial saving in degrees of freedom the behavior of the method is demonstrated on numerical examples

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

numerical solution of a flow control problem vorticity reduction by dynamic boundary action in order to laminarize an unsteady internal flow the vorticity field is minimized in a least squares sense using an optimal control approach the flow model is the navier stokes equation for a viscous incompressible fluid and the flow is controlled by suction and blowing on a part of the boundary a quasi newton method is used for the minimization of a quadratic objective function involving a measure of the vorticity and a regularization term the navier stokes equations are approximated using a finite difference scheme in time and finite element approximations in space accurate expressions for the gradient of the discrete objective function are needed to obtain a satisfactory convergence rate of the minimization algorithm therefore first order necessary conditions for a minimizer of the objective function are derived in the fully discrete case a memory saving device is discussed without which problems of any realistic size especially in three space dimensions would remain computationally intractable the feasibility of the optimal control approach for flow control problems is demonstrated by numerical experiments for a two dimensional flow in a rectangular cavity at a reynolds number high enough for nonlinear effects to be important

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

global convergence of trust region interior point algorithms for infinite dimensional nonconvex minimization subject to pointwise bounds a class of interior point trust region algorithms for infinite dimensional nonlinear optimization subject to pointwise bounds in l p banach spaces 2 le p le infty is formulated and analyzed the problem formulation is motivated by optimal control problems with l p controls and pointwise control constraints the interior point trust region algorithms are generalizations of those recently introduced by coleman and li siam j optim 6 1996 pp 418 445 for finite dimensional problems many of the generalizations derived in this paper are also important in the finite dimensional context all first and second order global convergence results known for trust region methods in the finite dimensional setting are extended to the infinite dimensional framework of this paper

the primal dual active set method for nonlinear optimal control problems with bilateral constraints the primal dual active set method has proved to be an efficient numerical tool in the context of diverse applications so far it has been investigated mainly for linear problems this paper is devoted to the study of global convergence of the primal dual active set method for nonlinear problems with bilateral constraints utilizing the close relationship between the primal dual active set method and semismooth newton methods local superlinear convergence of the method is investigated as well