the nonlinear programming method of wilson han and powell with an augmented lagrangian type line search function part 2 an efficient implementation with linear least squares subproblems

a sequential quadratically constrained quadratic programming method with an augmented lagrangian line search function based on an augmented lagrangian line search function a sequential quadratically constrained quadratic programming method is proposed for solving nonlinearly constrained optimization problems compared to quadratic programming solved in the traditional sqp methods a convex quadratically constrained quadratic programming is solved here to obtain a search direction and the maratos effect does not occur without any other corrections the active set strategy used in this subproblem can avoid recalculating the unnecessary gradients and approximate hessian matrices of the constraints under certain assumptions the proposed method is proved to be globally superlinearly and quadratically convergent as an extension general problems with inequality and equality constraints as well as nonmonotone line search are also considered

a superlinearly convergent sequential quadratically constrained quadratic programming algorithm for degenerate nonlinear programming we present an algorithm that achieves superlinear convergence for nonlinear programs satisfying the mangasarian fromovitz constraint qualification and the quadratic growth condition this convergence result is obtained despite the potential lack of a locally convex augmented lagrangian the algorithm solves a succession of subproblems that have quadratic objectives and quadratic constraints both possibly nonconvex by the use of a trust region constraint we guarantee that any stationary point of the subproblem induces superlinear convergence which avoids the problem of computing a global minimum we compare this algorithm with sequential quadratic programming algorithms on several degenerate nonlinear programs

hausdorff moment problem and fractional moments a simplified procedure abstract the purpose of this paper is the recovering of a probability density function with support 0 1 from the knowledge of its sequence of moments i e the classical hausdorff moment problem to avoid the well known ill conditioning firstly the moment curve is calculated from the assigned sequence of moments next the unknown density is approximated by maximum entropy maxent technique selecting some proper points on the moment curve exploiting convergence in entropy a simplified quick procedure is suggested to recover the approximate density an application to laplace transform inversion is illustrated

hausdorff moment problem and maximum entropy a unified approach hausdorff moment problem in the approach of maximum entropy is reconsidered some aspects as existence condition entropy convergence entropy decreasing stability are studied through a unified approach results known in literature and some new ones are illustrated

anisotropic curvature driven flow of convex sets abstract we study in this paper the mean curvature evolution and in particular the anisotropic mean curvature evolution of convex sets in r n without driving forces if the anisotropy is smooth we show that the evolution remains convex if the anisotropy is crystalline we build a convex evolution which satisfies an equation which is a weak form of the crystalline curvature motion equation

the anisotropic total variation flow we prove existence and uniqueness of solutions of the anisotropic total variation flow when the initial data is an l2 function and we give a characterization of such solutions that allows us to find explicit evolutions of sets in the presence of an anisotropy

error analysis of the bjo rck pereyra algorithms for solving vandermonde systems a forward error analysis is presented for the bjorck pereyra algorithms used for solving vandermonde systems of equations this analysis applies to the case where the points defining the vandermonde matrix are nonnegative and are arranged in increasing order it is shown that for a particular class of vandermonde problems the error bound obtained depends on the dimensionn and on the machine precision only being independent of the condition number of the coefficient matrix by comparing appropriate condition numbers for the vandermonde problem with the forward error bounds it is shown that the bjorck pereyra algorithms introduce no more uncertainty into the numerical solution than is caused simply by storing the right hand side vector on the computer a technique for computing running a posteriori error bounds is derived several numerical experiments are presented and it is observed that the ordering of the points can greatly affect the solution accuracy

existence and uniqueness for anisotropic and crystalline mean curvature flows an existence and uniqueness result up to fattening for crystalline mean curvature flows with forcing and arbitrary convex mobilities is proven this is achieved by introducing a new notion of solution to the corresponding level set formulation such a solution satisfies the comparison principle and a stability property with respect to the approximation by suitably regularized problems the results are valid in any dimension and for arbitrary possibly unbounded initial closed sets the approach accounts for the possible presence of a time dependent bounded forcing term with spatial lipschitz continuity as a by product of the analysis the problem of the convergence of the almgren taylor wang minimizing movements scheme to a unique up to fattening flat flow in the case of general possibly crystalline anisotropies is settled

on the sequential quadratically constrained quadratic programming methods an iteration of the sequential quadratically constrained quadratic programming method sqcqp consists of minimizing a quadratic approximation of the objective function subject to quadratic approximation of the constraints followed by a line search in the obtained direction methods of this class are receiving attention due to the development of efficient interior point techniques for solving subproblems with this structure via formulating them as second order cone programs recently fukushima et al 2003 proposed a sqcqp method for convex minimization with twice continuously differentiable data their method possesses global and locally quadratic convergence and it is free of the maratos effect the feasibility of subproblems in their method is enforced by switching between the linear and quadratic approximations of the constraints this strategy requires computing a strictly feasible point as well as choosing some further parameters we propose a sqcqp method where feasibility of subproblems is ensured by introducing a slack variable and hence is automatic in addition we do not assume convexity of the objective function or twice differentiability of the problem data while our method has all the desirable convergence properties it is easier to implement among other things it does not require computing a strictly feasible point which is a nontrivial task in addition its global convergence requires weaker assumptions

convergence of a crystalline algorithm for the motion of a simple closed convex curve by weighted curvature approaches based on the object oriented paradigm motion by weighted mean curvature is a geometric evolution law for surfaces and represents steepest descent with respect to anisotropic surface energy it has been proposed that this motion could be computed numerically by using a crystalline approximation to the surface energy in the evolution law in this paper we prove the convergence of this numerical method for the case of simple closed convex curves in the plane

approximation schemes for viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of applications including the calculus of variations control theory and differential games recently m g crandall and p l lions established the correct notion of generalized solutions for these equations this article discusses the convergence of general approximation schemes to this solution and gives under certain hypotheses explicit error estimates these results are then applied to obtain various representations as limits of solutions of general explicit and implicit finite difference schemes with error estimates

on a family of indeterminate distributions abstract we prove that if x is a random variable with a gamma distribution with parameter then x has an indeterminate distribution for max 2 2

existence and uniqueness for a crystalline mean curvature flow an existence and uniqueness result up to fattening for a class of crystalline mean curvature flows with natural mobility is proved the results are valid in any dimension and for arbitrary possibly unbounded initial closed sets the comparison principle is obtained by means of a suitable weak formulation of the flow while the existence of a global in time solution follows via a minimizing movement approach 2016 wiley periodicals inc

on hopf s formulas for solutions of hamilton jacobi equations soit h r n r on considere l equation de hamilton jacobi u t h du 0 dans r n 1 r n 0 on etudie les proprietes des formules de hopf pour les solutions

optimal switching for ordinary differential equations we consider the problem of controlling an ordinary differential equation subject to positive switching costs and show in particular that the value functions form the viscosity solution cf 6 7 of the dynamic programming quasi variational inequalities this interpretation allows for a rigorous application of various dynamic programming techniques

a viscosity solution approach to the asymptotic analysis of queueing systems we consider a system of several interconnected queues with a single class of customers and model the state xt of the system as a jump markov process the problem of interest is to estimate the large deviations behavior of the rescaled system xt ext s corresponding to large time and large excursions of the original unscaled system the techniques employed are those of the theory of viscosity solutions to hamilton jacobi equations from the point of view of large deviation theory the interesting new problem here is the treatment of the process when one or more of the queues are nearly empty since an abrupt change in the jump measure occurs from the point of view of viscosity solutions the discontinuity of the jump measure leads to nonlinear boundary conditions on domains with corners for the associated partial differential equations much of the paper is devoted to proving uniqueness of viscosity solutions for these equations and these sections are of independent interest while our use of test functions in proving the uniqueness is an adaptation of the usual technique the construction of the test functions themselves via the legendre transform is new we obtain a representation for the solution of the equation in terms of a nonstandard optimal control problem which suggests the correct integrand in the large deviation rate functional since it is the treatment of the effects due to the boundaries that is novel we devote the majority of the paper to the detailed development of a simple two dimensional system that exhibits all the essential new features however the arguments may be applied to nueueing systems that are considerably more general and we attempt to indicate this generality as well

the errors in variables problem considerations provided by radiation dose response analyses of the a bomb survivor data abstract some basic issues in the errors in variables problem are discussed in terms of considerations that arose in analyses of radiation effects on atomic bomb survivors the setting essentially involves generalized linear models for the response variables a very nonnormal distribution of the true covariable and multiplicative errors in the observed covariable consideration is given to distinctions between structural and functional modeling it is argued that careful attention to the apparent distribution of true covariables is critical in either case and a quasi structural approach to functional models is suggested the focus is on the case in which the expected response is linear in the true covariable and strong assumptions are tentatively made about the model for covariate errors for settings such as just described which differ from that of much of the classical work in the area it is emphasized that an attractive approach is based on weighted regression of the response on the expected value

a branch and cut algorithm for nonconvex quadratically constrained quadratic programming we present a branch and cut algorithm that yields in finite time a globally e optimal solution with respect to feasibility and optimality of the nonconvex quadratically constrained quadratic programming problem the idea is to estimate all quadratic terms by successive linearizations within a branching tree using reformulation linearization techniques rlt to do so four classes of linearizations cuts depending on one to three parameters are detailed for each class we show how to select the best member with respect to a precise criterion the cuts introduced at any node of the tree are valid in the whole tree and not only within the subtree rooted at that node in order to enhance the computational speed the structure created at any node of the tree is flexible enough to be used at other nodes computational results are reported that include standard test problems taken from the literature some of these problems are solved for the first time with a proof of global optimality

stieltjes moment problem via fractional moments stieltjes moment problem is considered to recover a probability density function from the knowledge of its infinite sequence of ordinary moments the approximate density is obtained through maximum entropy technique under the constraint of few fractional moments the latter are numerically obtained from the infinite sequence of ordinary moments and are chosen in such a way as to convey the maximum information content carried by the ordinary moments as a consequence a model with few parameters is obtained and intrinsic numerical instability is avoided it is proved that the approximate density is useful for calculating expected values and some other interesting probabilistic quantities

crystalline mean curvature flow of convex sets we prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in open image in new window this theorem can handle the facet breaking bending phenomena and can be generalized to any anisotropic mean curvature flow the method provides also a generalized geometric evolution starting from any compact convex set existing up to the extinction time satisfying a comparison principle and defining a continuous semigroup in time we prove that when the initial set is convex our evolution coincides with the flat curvature flow in the sense of almgren taylor wang as a by product it turns out that the flat curvature flow starting from a compact convex set is unique

the cube of a normal distribution is indeterminate it is established that if x is a stochastic variable with a normal distribution then x 2n 1 has an indeterminate distribution for n geq 1 furthermore the distribution of x alpha is determinate for 0 4

generalized motion by nonlocal curvature in the plane this paper considers the level set equation for a general planar anisotropic curvature flow equation when the interfacial energy is very singular so that the anisotropic curvature effect is nonlocal a new notion of solutions is introduced to establish an analytic foundation of the level set method including a comparison principle and stability results the main idea behind the proofs is to convert the level sets of solutions into graph like functions this new procedure is called slicing and it is not limited to nonlocal curvature flow equations our theory is useful for establishing the convergence of a crystalline algorithm as well as for justifying the crystalline flow as a limit of anisotropic curvature flow with smooth interfacial energy

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

hamilton jacobi equations in infinite dimensions i uniqueness of viscosity solutions soient un espace de banach v son dual v et des solutions d une equation h x u du 0 dans etablies dans un sous ensemble de v h v r v r est continue on demontre que des solutions classiques sont determinees de facon unique par leurs valeurs limites on montre que l extension de la notion de solutions viscosite aux espaces de banach reussit dans des espaces munis de la propriete de radon nikodym

a note on recovering the distributions from exponential moments the problem of recovering a cumulative distribution function of a positive random variable via the scaled laplace transform inversion is studied the uniform upper bound of proposed approximation is derived the approximation of a compound poisson distribution as well as the estimation of a distribution function of the summands given the sample from a compound poisson distribution are investigated applying the simulation study the question of selecting the optimal scaling parameter of the proposed laplace transform inversion is considered the behavior of the approximants are demonstrated via plots and table

fully nonlinear neumann type boundary conditions for first order hamilton jacobi equations we consider problems of the following type h x u u 0 in where is a smooth bounded open set in r n u is a scalar function from into r u denotes its frechet differential at x and h is a given continuous function on xr r n this equation is nothing but a general form of the classical first order hamilton jacobi equation in order to investigate existence and uniqueness questions the equation has to be complemented with a boundary condition on or on a part of we consider here for reasons described below a fully nonlinear boundary condition of the form f x u u 0 on that we describe as being of the neumann type

numerical aspects of finite hausdorff moment problem by maximum entropy approach some numerical aspects of the finite hausdorff moment problem in the framework of maximum entropy have been reviewed as stability of lagrange s multipliers computation entropy decreasing averages estimate spectral properties of hankel matrices have been considered determinant and conditioning number have been estimated finally entropy estimate has been provided

a crystalline motion uniqueness and geometric properties the author studies a model of crystalline motion in the plane existence and uniqueness of local in time solutions are shown geometric properties of the flow are studied assuming that the wulff shape is a regular n sided polygon the author shows that a small convex grain of ice immersed in the cold melt shrinks provided that the flow does not overly deform the initial interface the author is able to verify the last condition only if the initial interface is a scaled wulff shape in this case the free boundary will remain a scaled wulff shape at later times this is shown by using an isoperimetric inequality

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions part i the case of bounded stochastic evolutions on etudie des equations elliptiques degenerees d ordre 2 non lineaires de la forme f d 2 u du u x 0 dans h ou h est un espace de hilbert separable x un point generique dans h u une fonction de h r

stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise in this work we introduce a new method to prove the existence and uniqueness of a variational solution to the stochastic nonlinear diffusion equation rm d x t rm div left frac nabla x t nabla x t right rm d t x t rm d w t rm in 0 infty times mathcal o where mathcal o is a bounded and open domain in mathbb r n n geqq 1 and w t is a wiener process of the form w t sum infty k 1 mu k e k beta k t e k in c 2 overline mathcal o cap h 1 0 mathcal o and beta k k in mathbb n are independent brownian motions this is a stochastic diffusion equation with a highly singular diffusivity term one main result established here is that for all initial conditions in l 2 mathcal o it is well posed in a class of continuous solutions to the corresponding stochastic variational inequality thus one obtains a stochastic version of the minimal total variation flow the new approach developed here also allows us to prove the finite time extinction of solutions in dimensions 1 leqq n leqq3 which is another main result of this work

facet bending driven by the planar crystalline curvature with a generic nonuniform forcing term abstract we study crystalline driven curvature flow with spatially nonuniform driving force term we assume special monotonicity properties of the driving term which are motivated by our previous work on berg s effect we consider special initial data which we call bent rectangles we prove existence of solutions for a generic forcing term as well as generic subclass of bent rectangles we show the initially flat facets may begin to bend provided loosely speaking they are too large moreover depending on the initial configuration we notice instantaneous loss of regularity of the moving curve

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions iii uniqueness of viscosity solutions for general second order equations we prove here the uniqueness and existence of viscosity solutions for a general class of fully nonlinear second order equations in an infinite dimensional hilbert space the uniqueness is proved using the so called inf and sup convolutions which allow some reduction to finite dimensional problems

multiphase thermomechanics with interfacial structure 2 evolution of an isothermal interface paper 1 1988 g 1 of this series began an investigation whose goal is a thermomechanics of two phase continua based on gibbs s notion of a sharp phase interface endowed with thermomechanical structure in that paper a balance law balance of capillary forces was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics the chief results are thermodynamic restrictions on constitutive equations exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems the simplest versions of these problems the mullins sekerka problems are essentially the classical stefan problem with the free boundary condition u 0 for the temperature replaced by the condition u h k where k is the curvature of the free boundary and h 0 is a material constant this dependence on curvature renders the problem difficult and apart from numerical studies involving linearization stability there are almost no supporting theoretical results

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

a regularity result for viscosity solutions of hamilton jacobi equations in one space dimension on demontre que dans le cas particulier d un probleme stationnaire a une dimension sous des hypotheses tres generales les solutions viscosite continues de lipschitz ont des derivees a droite et a gauche en chaque point

diffuse interfaces with sharp corners and facets phase field models with strongly anisotropic surfaces abstract we provide the general outline of an analysis of the motion of diffuse interfaces in the order parameter phase field formulation which includes nondifferentiable and nonconvex gradient energy terms nondifferentiability leads to equilibrium and motion equations that are apparently undefined and nonconvexity leads to equations that are apparently ill posed the problem of nondifferentiability is resolved by using nonlocal variations to move entire facets or line segments with orientations having such facets or line segments in the underlying wulff shape the problem of ill posedness is resolved by using varifolds infinitesimally corrugated diffuse interfaces constructed from the edges and corners in the underlying wulff shape or equivalently by using the convexification of the gradient energy term and then reinterpreting the solutions as varifolds this is justified on a variational basis we conclude that after an initial transient level sets move by weighted mean curvature in agreement with the sharp interface limit we provide equations for tracking the shocks that develop as edges and corners in level sets

viscosity solutions of hamilton jacobi equations with unbounded nonlinear terms abstract the aim of this paper is to introduce a new definition of viscosity solutions for hamilton jacobi equations in infinite dimensions with unbounded nonlinear terms we prove existence and uniqueness results under usual assumptions regarding the bounded part of the hamiltonian

estimation and accuracy after model selection classical statistical theory ignores model selection in assessing estimation accuracy here we consider bootstrap methods for computing standard errors and confidence intervals that take model selection into account the methodology involves bagging also known as bootstrap smoothing to tame the erratic discontinuities of selection based estimators a useful new formula for the accuracy of bagging then provides standard errors for the smoothed estimators two examples nonparametric and parametric are carried through in detail a regression model where the choice of degree linear quadratic cubic is determined by the cp criterion and a lasso based estimation problem

a nonlinear parabolic equation with discontinuity in the highest order and applications abstract in this paper we establish a viscosity solution theory for a class of nonlinear parabolic equations with discontinuities of the sign function type in the second derivatives of the unknown function we modify the definition of classical viscosity solutions and show uniqueness and existence of the solutions these results are related to the limit behavior for the motion of a curve by a very small power of its curvature which has applications in image processing we also discuss the relation between our equation and the total variation flow in one space dimension

the nonlinear programming method of wilson han and powell with an augmented lagrangian type line search function the paper represents an outcome of an extensive comparative study of nonlinear optimization algorithms this study indicates that quadratic approximation methods which are characterized by solving a sequence of quadratic subproblems recursively belong to the most efficient and reliable nonlinear programming algorithms available at present the purpose of this paper is to analyse the theoretical convergence properties and to investigate the numerical performance in more detail in part 1 the exactl 1 penalty function of han and powell is replaced by a differentiable augmented lagrange function for the line search computation to the able to prove the global convergence and to show that the steplength one is chosen in the neighbourhood of a solution in part 2 the quadratic subproblem is exchanged by a linear least squares problem to improve the efficiency and to test the dependence of the performance from different solution methods for the quadratic or least squares subproblems

bent rectangles as viscosity solutions over a circle abstract we study the motion of the so called bent rectangles by the singular weighted mean curvature we are interested in the curves which can be rendered as graphs over a smooth one dimensional reference manifold we establish a sufficient condition for that once we deal with graphs we can have the tools of the viscosity theory available like the comparison principle with its help we establish uniqueness of variational solutions constructed by the authors giga et al 2013 in addition we establish a criterion for the mobility coefficient guaranteeing vertex preservation

sequential quadratically constrained quadratic programming norm relaxed algorithm of strongly sub feasible directions abstract in this paper we present a sequential quadratically constrained quadratic programming sqcqp norm relaxed algorithm of strongly sub feasible directions for the solution of inequality constrained optimization problems by introducing a new unified line search and making use of the idea of strongly sub feasible direction method the proposed algorithm can well combine the phase of finding a feasible point by finite iterations and the phase of a feasible descent norm relaxed sqcqp algorithm moreover the former phase can preserve the sub feasibility of the current iteration and control the increase of the objective function at each iteration only a consistent convex quadratically constrained quadratic programming problem needs to be solved to obtain a search direction without any other correctional directions the global superlinear and a certain quadratic convergence which is between 1 step and 2 step quadratic convergence properties are proved under reasonable assumptions finally some preliminary numerical results show that the proposed algorithm is also encouraging

avoiding the maratos effect by means of a nonmonotone line search ii inequality constrained problems feasible iterates when solving inequality constrained optimization problems via sequential quadratic programming sqp it is potentially advantageous to generate iterates that all satisfy the constraints all quadratic programs encountered are then feasible and there is no need for a surrogate merit function feasibility of the successive iterates is in fact required in many contexts such as in real time applications or when the objective function is not defined outside the feasible set it has recently been shown that this is indeed possible by means of a suitable perturbation of the original sqp iteration without losing superlinear convergence in this context the well known maratos effect is compounded by the possible infeasibility of the full step of one even close to a solution these difficulties have been accommodated by making use of a suitable modification of a bending technique proposed by mayne and polak requiring evaluation of the constraints function at an auxiliary point at each iteration in part i

a feasible trust region sequential quadratic programming algorithm an algorithm for smooth nonlinear constrained optimization problems is described in which a sequence of feasible iterates is generated by solving a trust region sequential quadratic programming sqp subproblem at each iteration and by perturbing the resulting step to retain feasibility of each iterate by retaining feasibility the algorithm avoids several complications of other trust region sqp approaches the objective function can be used as a merit function and the sqp subproblems are feasible for all choices of the trust region radius global convergence properties are analyzed under various assumptions on the approximate hessian under additional assumptions superlinear convergence to points satisfying second order sufficient conditions is proved

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

on the moment recovered approximations of regression and derivative functions with applications in this paper three formulas for recovering the conditional mean and conditional variance based on product moments are proposed the upper bounds for the uniform rate of approximations of regression and derivatives of some moment determinate function are derived two cases where the support of underlying functions is bounded and unbounded from above are studied based on the proposed approximations novel nonparametric estimates of the distribution function and its density in multiplicative censoring model are constructed simulation study justifies the consistency of the estimates

a superlinearly convergent sqp method without boundedness assumptions on any of the iterative sequences this paper is aimed to present a new sequential quadratic programming sqp algorithm for finding a solution to nonlinear constrained programming problems with weak conditions where the improved direction can be yielded by solving one quadratic programming qp and the correction direction can be obtained by solving another qp the main characters of the proposed algorithm are as follows first by limiting infeasibility of sqp iterates the boundedness of the iteration sequence can be obtained in the case of the feasible set being nonempty and bounded as well as the constraint functions being convex second global convergence can be proved under slater constraint qualification cq furthermore superlinear convergence can be ensured under suitable conditions third the proposed algorithm is further improved with a bidirectional line search technique finally some numerical experiments are operated to test the proposed algorithms and the results demonstrate that they are promising

convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature motion by weighted mean curvature is a geometric evolution law for surfaces representing steepest descent with respect to an isotropic surface energy it has been proposed that this motion could be computed by solving the analogous evolution law using a crystalline approximation to the surface energy we present the first convergence analysis for a numerical scheme of this type our treatment is restricted to one dimensional surfaces curves in the plane which are graphs in this context the scheme amounts to a new algorithm for solving quasilinear parabolic equations in one space dimension

a comparison principle for singular diffusion equations with spatially inhomogeneous driving force for graphs we introduce the notions of viscosity super and subsolutions suitable for singular diffusion equations of non divergence type with a general spatially inhomogeneous driving term in particular the viscosity super and subsolutions support facets and allow a possible facet bending we prove a comparison principle by a modified doubling variables technique finally we present examples of viscosity solutions our results apply to a general crystalline curvature flow with a spatially inhomogeneous driving term for a graph like curve

curvature driven flows a variational approach this paper introduces a new mathematical approach to the study of time evolutions of solids k t in n space whose boundaries move with velocity equal to the weighted mean curvature derived from the boundary surface energy phi partial k t these flat phi curvature flows are limits of sequences of solutions to variational problems in which a sum of surface and bulk energy is minimized the construction works equally well for smooth elliptic phi s for nondifferentiable crystalline phi s and for anything in between the flows agree with classical smooth flows when the data is smooth and elliptic in any dimension and coincide with motion by crystalline curvature for polyhedral curves in the plane

a note on a model system with sudden directional diffusion we study qualitative properties of solutions to a monodimensional problem u t u x operatorname sgn u x x 0 with the dirichlet boundary conditions such a system presents a key analytical challenge coming from the examination of models of anisotropic phenomena like crystal growth our analysis concentrates on the properties of facets flat regions of solutions typical for this type of problems

on existence and uniqueness of solutions of hamilton jacobi equations abstract the theory of scalar first order nonlinear partial differential equations has been enjoying a rapid development in the last few years this development occurred because the authors established uniqueness criteria for generalized solutions called viscosity solutions which correctly identify the solutions sought in areas of application including control theory differential games and the calculus of variations the concept of viscosity solutions is relatively easy to work with and many formally heuristic or difficult proofs have been made rigorous or simple using this concept a feedback process has begun and the experience recently gained in working with viscosity solution has suggested new existence and uniqueness results the current paper continues this interaction by establishing new existence ane uniqueness results in a natural generality suggested by earlier proofs it is also felt that the presentation of the comparison results which imply uniqueness continuous dependence and are used to estimate moduli of continuity has something to offer over earlier presentations in special cases author

evolving graphs by singular weighted curvature a new notion of solutions is introduced to study degenerate nonlinear parabolic equations in one space dimension whose diffusion effect is so strong at particular slopes of the unknowns that the equation is no longer a partial differential equation by extending the theory of viscosity solutions a comparison principle is established for periodic continuous initial data a unique global continuous solution periodic in space is constructed the theory applies to motion of interfacial curves by crystalline energy or more generally by anisotropic interfacial energy with corners when the curves are the graphs of functions even if the driving force term homogeneous in space exists the initial value problem is solvable for general nonadmissible continuous periodic initial data

viscosity solutions with shocks a solution of single nonlinear first order equations may develop jump discontinuities even if initial data is smooth typical examples include a crude model equation describing some bunching phenomena observed in epitaxial growth of crystals as well as conservation laws where jump discontinuities are called shocks conventional theory of viscosity solutions does not apply we introduce a notion of proper viscosity solutions to track whole evolutions for such equations in multi dimensional spaces we establish several versions of comparison principles we also study the vanishing viscosity method to construct a unique global proper solution at least when the evolution is monotone in time or the initial data is monotone in some sense under additional technical assumptions in fact we prove that the graph of approximate solutions converges to that of a proper solution in the hausdorff distance topology such a convergence is also established for conservation laws with monotone data in particular local uniform convergence outside shocks is proved 2001 john wiley sons inc

on a crystalline variational problem part ii bv regularity and structure of minimizers on facets for a nonsmooth positively one homogeneous convex function n 0 it is possible to introduce the class n of smooth boundaries with respect to to define their mean curvature and to prove that for e n l e 9 based on these results we continue the analysis on the structure of e and on the regularity properties of we prove that a facet f of e is lipschitz up to negligible sets and that has bounded variation on f further properties of the jump set of are inspected in particular in three space dimensions we relate the sublevel sets of on f to the geometry of the wulff shape 1

a semidefinite programming approach to the generalized problem of moments we consider the generalized problem of moments gpm from a computational point of view and provide a hierarchy of semidefinite programming relaxations whose sequence of optimal values converges to the optimal value of the gpm we then investigate in detail various examples of applications in optimization probability financial economics and optimal control which all can be viewed as particular instances of the gpm

a working set sqcqp algorithm with simple nonmonotone penalty parameters in this paper we present a new sequential quadratically constrained quadratic programming sqcqp algorithm in which a simple updating strategy of the penalty parameter is adopted this strategy generates nonmonotone penalty parameters at early iterations and only uses the multiplier corresponding to the bound constraint of the quadratically constrained quadratic programming qcqp subproblem instead of the multipliers of the quadratic constraints which will bring some numerical advantages furthermore by using the working set technique we remove the constraints of the qcqp subproblem that are locally irrelevant and thus the computational cost could be reduced without assuming the convexity of the objective function or the constraints the algorithm is proved to be globally superlinearly and quadratically convergent preliminary numerical results show that the proposed algorithm is very promising when compared with the tested sqp algorithms

hausdorff moment problem via fractional moments we outline an efficient method for the reconstruction of a probability density function from the knowledge of its infinite sequence of ordinary moments the approximate density is obtained resorting to maximum entropy technique under the constraint of some fractional moments the latter ones are obtained explicitly in terms of the infinite sequence of given ordinary moments it is proved that the approximate density converges in entropy to underlying density so that it turns out to be useful for calculating expected values

reconstruction of conditional expectations from product moments with applications in this paper it is shown under conditions associated with the moment problem that a sequence of product moments uniquely determines a conditional expectation of two random variables then a numerical procedure is derived to reconstruct a conditional expectation in terms of a sequence of its product moments

on a crystalline variational problem part i first variation and global l regularity let n 0 be a given positively one homogeneous convex function and let 1 pursuing our interest in motion by crystalline mean curvature in three dimensions we introduce and study the class n of smooth boundaries in the relative geometry induced by the ambient banach space n it can be seen that even when is a polytope n cannot be reduced to the class of polyhedral boundaries locally resembling curved portions must be necessarily included and this fact as well as the nonsmoothness of is the source of several technical difficulties related to the geometry of lipschitz manifolds given a boundary e in the class n we rigorously compute the first variation of the corresponding anisotropic perimeter which leads to a variational problem on vector fields defined on e it turns out that the minimizers have a uniquely determined intrinsic tangential divergence on e we define such a divergence to be the mean curvature of e the function is expected to be the initial velocity of e whenever e is considered as the initial datum for the corresponding anisotropic mean curvature flow we prove that is bounded on e and that its sublevel sets are characterized through a variational inequality

hyperdisk based large margin classifier we introduce a large margin linear binary classification framework that approximates each class with a hyperdisk the intersection of the affine support and the bounding hypersphere of its training samples in feature space and then finds the linear classifier that maximizes the margin separating the two hyperdisks we contrast this with support vector machines svms which find the maximum margin separator of the pointwise convex hulls of the training samples arguing that replacing convex hulls with looser convex class models such as hyperdisks provides safer margin estimates that improve the accuracy on some problems both the hyperdisks and their separators are found by solving simple quadratic programs the method is extended to nonlinear feature spaces using the kernel trick and multi class problems are dealt with by combining binary classifiers in the same ways as for svms experiments on a range of data sets show that the method compares favourably with other popular large margin classifiers