average optimal singular control and a related stopping problem we observe a 2 brownian motion xt we can increase or decrease the value of the process paying r times the size of increase and l times the size of decrease holding costs are incurred continuously at a rate h zt where zt is the resulting process we consider a convex holding cost function h which is either finite everywhere or finite inside a finite interval and equal to infinity outside this interval the objective is to minimize average per unit of time expected cost it is shown that the optimal behavior is characterized by two constants a and b a b such that the optimal policy is to keep the process inside the interval a b with minimal efforts it is shown that to find the optimal interval a b one must solve a free boundary problem for the second order differential equation we give a probabilistic solution of the free boundary problem in terms of the value of a special game this game is a generalization of the optimal stopping problem there are two players observing the 2

instantaneous control of brownian motion a controller continuously monitors a storage system such as an inventory or bank account whose content z zt t 0 fluctuates as a 2 brownian motion in the absence of control holding costs are incurred continuously at rate h zt at any time the controller may instantaneously increase the content of the system incurring a proportional cost of r times the size of the increase or decrease the content at a cost of l times the size of the decrease we consider the case where h is convex on a finite interval and h outside this interval the objective is to minimize the expected discounted sum of holding costs and control costs over an infinite planning horizon it is shown that there exists an optimal control limit policy characterized by two parameters a and b a b roughly speaking this policy exerts the minimum amounts of control sufficient to keep zt a b for all t 0 put another way the optimal control limit policy imposes on z a lower reflecting barrier at a an

multivariate approximation theory and applications this introductory paper describes the main topics of this special issue dedicated to leonardo traversoni known at the international level as the promoter of the conference series multivariate approximation theory and applications to celebrate his 60th birthday

optimal consumption and portfolio with both fixed and proportional transaction costs we consider a market model with one risk free and one risky asset in which the dynamics of the risky asset are governed by a geometric brownian motion in this market we consider an investor who consumes from the bank account and who has the opportunity at any time to transfer funds between the two assets we suppose that these transfers involve a fixed transaction cost k 0 independent of the size of the transaction plus a cost proportional to the size of the transaction the objective is to maximize the cumulative expected utility of consumption over a planning horizon we formulate this problem as a combined stochastic control impulse control problem which in turn leads to a nonlinear quasi variational hamilton jacobi bellman inequality qvhjbi we prove that the value function is the unique viscosity solution of this qvhjbi finally numerical results are presented

estimations de l erreur d approximation par splines d interpolation et d ajustement d ordre m s for functions belonging to the sobolev spaceh m s convergence of smoothing m s splines is proved and approximation error estimates for interpolating and smoothing m s splines are established this is a contribution to the m s spline theory of j duchon on montre la convergence des splines d ajustement d ordre m s et on etablit des estimations de l erreur d approximation par splines d interpolation et d ajustement d ordre m s pour des fonctions appartenant a l espace de sobolevh m s ces resultats prolongent ceux de j duchon

global attractors for impulsive dynamical systems a precompact approach abstract in this work we give the definition of a global attractor of an impulsive dynamical system and obtain several important properties for this class of attractors we prove the theorem on existence of such attractors and apply it to chosen ordinary and partial differential equations with impulsive functions

on the impulse control of jump diffusions regularity of the impulse control problem for a nondegenerate n dimensional jump diffusion with infinite activity and finite variation jumps was recently examined in m h a davis x guo and g wu siam j control optim 48 2010 pp 5276 5293 here we extend the analysis to include infinite activity and infinite variation jumps more specifically we show that the value function u of the impulse control problem satisfies u in w 2 p mathrm loc mathbb r n

a unified approach to fast image registration and a new curvature based registration technique abstract image registration is central to many challenges in medical imaging today it has a vast range of applications the purpose of this note is twofold first we review some of the most promising non linear registration strategies currently used in medical imaging we show that all these techniques may be phrased in terms of a variational problem and allow for a unified treatment second we introduce within the variational framework a new non linear registration model based on a curvature type smoother we show that affine linear transformations belong to the kernel of this regularizer as a result the approach becomes more robust against poor initializations of a pre registration step furthermore we develop a stable and fast implementation of the new scheme based on a real discrete cosine transformation we demonstrate the advantages of the new technique for synthetic data sets and present an application of the algorithm for registering mr mammography images

an axiomatic approach to image interpolation we discuss possible algorithms for interpolating data given in a set of curves and or points in the plane we propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations the absolute minimal lipschitz extension model amle is singled out and studied in more detail we show experiments suggesting a possible application the restoration of images with poor dynamic range

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

impulse control of brownian motion consider a storage system such as an inventory or cash fund whose content fluctuates as a 2 brownian motion in the absence of control holding costs are continuously incurred at a rate proportional to the storage level and we may cause the storage level to jump by any desired amount at any time except that the content must be kept nonnegative both positive and negative jumps entail fixed plus proportional costs and our objective is to minimize expected discounted costs over an infinite planning horizon a control band policy is one that enforces an upward jump to q whenever level zero is hit and enforces a downward jump to q whenever level s is hit 0 q q s we prove the existence of an optimal control band policy and calculate explicitly the optimal values of the critical numbers q q s

trivariate spline approximations of 3d navier stokes equations we present numerical approximations of the 3d steady state navier stokes equations in velocity pressure formulation using trivariate splines of arbitrary degree d and arbitrary smoothness r with r d using functional arguments we derive the discrete navier stokes equations in terms of b coefficients of trivariate splines over a tetrahedral partition of any given polygonal domain smoothness conditions boundary conditions and the divergence free condition are enforced through lagrange multipliers the pressure is computed by solving a poisson equation with neumann boundary conditions we have implemented this approach in matlab and present numerical evidence of the convergence rate as well as experiments on the lid driven cavity flow problem

approximation of vector fields using discrete div rot variational splines in a finite element space this paper deals with an approximation problem concerning vector fields through the new notion of div rot variational splines the minimizing problem is addressed in a finite element space through the choice of some semi norms based on decomposition of the divergence operator and vector fields into a form with a rotational part we study the existence and the uniqueness of the solution of such a problem then a convergence result and an estimation of the error are established some numerical and graphical examples are analyzed in order to prove the validity of our method furthermore we compare and show how our method improves upon one existing in the literature

deformable templates using large deformation kinematics a general automatic approach is presented for accommodating local shape variation when mapping a two dimensional 2 d or three dimensional 3 d template image into alignment with a topologically similar target image local shape variability is accommodated by applying a vector field transformation to the underlying material coordinate system of the template while constraining the transformation to be smooth globally positive definite jacobian smoothness is guaranteed without specifically penalizing large magnitude deformations of small subvolumes by constraining the transformation on the basis of a stokesian limit of the fluid dynamical navier stokes equations this differs fundamentally from quadratic penalty methods such as those based on linearized elasticity or thin plate splines in that stress restraining the motion relaxes over time allowing large magnitude deformations kinematic nonlinearities are inherently necessary to maintain continuity of structures during large magnitude deformations and are included in all results after initial global registration final mappings are obtained by numerically solving a set of nonlinear partial differential equations associated with the constrained optimization problem automatic regridding is performed by propagating templates as the nonlinear transformations evaluated on a finite lattice become singular application of the method to intersubject registration of neuroanatomical structures illustrates the ability to account for local anatomical variability

a variational approach to nonrigid morphological image registration a variational method for nonrigid registration of multimodal image data is presented a suitable deformation will be determined via the minimization of a morphological i e contrast invariant matching functional along with an appropriate regularization energy the aim is to correlate the morphologies of a template and a reference image under the deformation mathematically the morphology of images can be described by the entity of level sets of the image and hence by its gauss map a class of morphological matching functionals is presented which measure the defect of the template gauss map in the deformed state with respect to the deformed gauss map of the reference image the problem is regularized by considering a nonlinear elastic regularization energy existence of homeomorphic minimizing deformation is proved under assumptions on the class of admissible deformations with respect to actual medical applications suitable generalizations of the matching energies and the boundary conditions are pre

biological population management based on a hamilton jacobi bellman equation with boundary blow up establishment of a long term biological population management policy requires balancing its cost and utility disutility we approach this issue from the viewpoint of stochastic control as an effici

on the principle of smooth fit for a class of singular stochastic control problems for diffusions this paper considers the principle of smooth fit for a class of one dimensional singular stochastic control problems allowing the system to be of nonlinear diffusion type the existence and the uniqueness of a convex c 2 solution to the corresponding variational inequality are obtained it is proved that this solution gives the value function of the control problem and the optimal control process is constructed as an example of the degenerate case it is proved that the conclusion is also true for linear systems and the explicit formula for the smooth fit points is derived

snakes shapes and gradient vector flow snakes or active contours are used extensively in computer vision and image processing applications particularly to locate object boundaries problems associated with initialization and poor convergence to boundary concavities however have limited their utility this paper presents a new external force for active contours largely solving both problems this external force which we call gradient vector flow gvf is computed as a diffusion of the gradient vectors of a gray level or binary edge map derived from the image it differs fundamentally from traditional snake external forces in that it cannot be written as the negative gradient of a potential function and the corresponding snake is formulated directly from a force balance condition rather than a variational formulation using several two dimensional 2 d examples and one three dimensional 3 d example we show that gvf has a large capture range and is able to move snakes into boundary concavities

weak almost periodic motions minimality and stability in impulsive semidynamical systems abstract in this paper we study topological properties of semidynamical systems whose continuous dynamics are interrupted by abrupt changes of state first we establish results which relate various concepts as stability of lyapunov weakly almost periodic motions recurrence and minimality in the sequel we study the stability of zhukovskij for impulsive systems and we obtain some results about uniform attractors

pseudo polyharmonic vectorial approximation for div curl and elastic semi norms vector field reconstruction is a problem arising in many scientific applications in this paper we study a div curl approximation of vector fields by pseudo polyharmonic splines this leads to the variational smoothing and interpolating spline problems with minimization of an energy involving the curl and the divergence of the vector field the relationship between the div curl energy and elastic energy is established some examples are given to illustrate the effectiveness of our approach for a vector field reconstruction

ck surface approximation from surface patches we study the problem of constructing a smooth approximant from a finite set of patches given on a surface defined by an equation x3 f x1 x2 as an approximant of f a discrete smoothing spline belonging to a suitable piecewise polynomial space is proposed error results and numerical results are given

vectorial interpolation using radial basis like functions this paper deals with vector field interpolation i e the data are r3 values located in scattered r3 points while the interpolating function is a function from r3 into r3 in order to take into account possible connections between the components of the interpolant we derive it by solving a variational spline problem involving the rotational and the divergence of the interpolant and depending on a parameter significative of the balance of the rotational part and of the divergence part and on the order m of derivatives of the rotational and divergence involved in the minimized seminorm we so obtain interpolants whose expression is x ni 1 x xi ai pm 1 x where is some 3 3 matrix function pm 1 is a degree m 1 vectorial polynomial and where the ai are r3 vectors besides the ai meet a relation generalizing the usual orthogonality to all polynomials of degree at most m 1 for 1 we find the usual m harmonic splines in each component of numerical examples show the interest of the method and we compare the so obtained functions with the ones obtained by matlab s procedures

optimal control with state space constraint i optimal control of piecewise deterministic processes with state space constraint is studied under appropriate assumptions it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain finally the uniform continuity of the value function is obtained under a condition on the deterministic drift

regularity of the optimal stopping problem for jump diffusions the value function of an optimal stopping problem for jump diffusions is known to be a generalized solution of a variational inequality assuming that the diffusion component of the process is nondegenerate and a mild assumption on the singularity of the levy measure this paper shows that the value function of this optimal stopping problem on an unbounded domain with finite infinite variation jumps is in w 2 1 p loc with p in 1 infty as a consequence the smooth fit property holds

impulse control of multidimensional jump diffusions in finite time horizon this paper analyzes a class of impulse control problems for multidimensional jump diffusions in the finite time horizon following the basic mathematical setup from stroock and varadhan multidimensional diffusion processes springer verlag heidelberg 2006 this paper first establishes rigorously an appropriate form of the dynamic programming principle it then shows that the value function is a viscosity solution for the associated hamilton jacobi bellman equation involving integro differential operators finally under additional assumptions that the jumps are of infinite activity but are of finite variation and that the diffusion is uniformly elliptic it proves that the value function is the unique viscosity solution and has w loc 2 1 p regularity for 1 p infty

connections between singular control and optimal switching this paper builds a new theoretical connection between singular control of finite variation and optimal switching problems this correspondence provides a novel method for solving high dimensional singular control problems and enables us to extend the theory of reversible investment sufficient conditions are derived for the existence of optimal controls and for the regularity of value functions consequently our regularity result links singular controls and dynkin games through sequential optimal stopping problems

analysis and computation of an optimality equation arising in an impulse control problem with discrete and costly observations abstract the framework of discrete and costly observations provides an attractive mathematical tool for stochastic control of biological and ecological systems in natural environment however this framework is still germinating and its applications in these research areas are still rare in this framework the system dynamics to be controlled is continuous time while the observations by the decision maker are discrete time the control problem of the system dynamics has a time hybridized type nature by which the optimality equation has a different form from the conventional ones the objectives as well as contributions of this paper are formulation of the optimality equation focusing on population management problems its detailed mathematical analysis numerical discretization and a demonstrative application we show that the optimality equation is solvable uniquely and its solution can be approximated recursively a numerical method to discretize the optimality equation is established with a simple finite difference scheme and the recursive formula mathematical analysis results such as monotonicity stability and convergence of the numerical method are discussed as well the present model is then applied to a population management problem finally an advanced problem subject to model ambiguity is also analyzed

optimal impulse control of portfolios an investor has the opportunity of holding shares in n risky assets and one nonrisky asset at every time in a fixed interval t t the risky assets are governed by a stochastic differential equation at random instants of his choice he may intervene in order to rebalance his portfolio and consume a nonnegative amount of money fixed and variable transactions costs are incurred upon intervention at time t all remaining wealth is consumed the solution to the problem of maximizing total utility of consumption is given by way of quasi variational inequalities for the value function with probability one the investor only intervenes finitely many times indication of the solution of the quasi variational inequalities in the case of one risky asset with log normal prices is given together with a description of a discretization procedure

impulse control of multidimensional jump diffusions this paper studies regularity properties of the value function for an infinite horizon discounted cost impulse control problem where the underlying controlled process is a multidimensional jump diffusion with possibly infinite activity jumps surprisingly despite these jumps we obtain the same degree of regularity as for the diffusion case at least when the jump satisfies certain integrability conditions

hamilton jacobi bellman quasi variational inequality arising in an environmental problem and its numerical discretization abstract a hamilton jacobi bellman quasi variational inequality hjbqvi for a river environmental restoration problem with wise use of sediment is formulated and its mathematical properties are analyzed a finite difference scheme with a penalization technique is then established for solving the hjbqvi the scheme is free from any iterative solvers and is unconditionally stable and convergent in the viscosity sense under certain conditions a demonstrative application example of the hjbqvi is finally presented

smooth fit principle for impulse control of multidimensional diffusion processes value functions of impulse control problems are known to satisfy quasi variational inequalities qvis a bensoussan and j l lions impulse control and quasivariational inequalities heyden son philadelphia 1984 translation of controle impulsionnel et inequations quasi variationnelles gauthier villars paris 1982 this paper proves the smooth fit c 1 property of the value function for multidimensional controlled diffusions using a viscosity solution approach we show by examples how to exploit this regularity property to derive explicitly optimal policy and value functions

gradient field approximation application to registration in image processing we study a spline based approximation of vector fields in the conservative case the gradient vector field derives from a potential function we introduce a minimization problem on a hilbert space for which the existence and uniqueness of the solution is given we apply this approach to a registration process in image processing

optimal dividend policies with transaction costs for a class of diffusion processes finding optimal dividend strategies is a classical problem in the financial and actuarial literature the idea is that the company wants to pay some of its surplus as dividends and the problem is to find a dividend strategy that maximizes the expected total discounted dividends received by the shareholders until ruin here we generalize results in j paulsen adv appl probab 39 2007 pp 669 689 in that the rate of growth of the surplus process is assumed to exceed the discounting factor whenever the surplus process is smaller than a fixed number x lambda in j paulsen adv appl probab 39 2007 pp 669 689 it was assumed that this rate of growth is always less than or equal to the discounting factor it turns out that this generalization makes the problem much more complicated and a simple barrier strategy is no longer always optimal