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

renormalization theory in four dimensional scalar fields i we present a renormalization group approach to the renormalization theory of 44 using techniques that have been introduced and used in previous papers and that lead to very simple methods to bound the coefficients of the effective potential and of the schwinger functions the main aim of this paper is to show how one can in this way obtain then bounds

a multigrid optimization algorithm for the numerical solution of quasilinear variational inequalities involving the p laplacian abstract in this paper we propose a multigrid optimization algorithm mg opt for the numerical solution of a class of quasilinear variational inequalities of the second kind this approach is enabled by the fact that the solution of the variational inequality is given by the minimizer of a nonsmooth energy functional involving the p laplace operator we propose a huber regularization of the functional and a finite element discretization for the problem further we analyze the regularity of the discretized energy functional and we are able to prove that its jacobian is slantly differentiable this regularity property is useful to analyze the convergence of the mg opt algorithm in fact we demonstrate that the algorithm is globally convergent by using a mean value theorem for semismooth functions finally we apply the mg opt algorithm to the numerical simulation of the viscoplastic flow of bingham casson and herschel bulkley fluids in a pipe several experiments are carried out to show the efficiency of the proposed algorithm when solving this kind of fluid mechanics problems

renormalization proof for spontaneously broken yang mills theory with flow equations in this paper we present a renormalizability proof for spontaneously broken su 2 gauge theory it is based on flow equations i e on the wilson renormalization group adapted to perturbation theory the power counting part of the proof which is conceptually and technically simple follows the same lines as that for any other renormalizable theory the main difficulty stems from the fact that the regularization violates gauge invariance we prove that there exists a class of renormalization conditions such that the renormalized green functions satisfy the slavnov taylor identities of su 2 yang mills theory on which the gauge invariance of the renormalized theory is based

nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints hamiltoniens sousquadratiques conditions aux limites infinies et donnees d explosion hamiltoniens superquadratiques formulation viscosite des conditions aux limites le probleme ergodique commande stochastique optimale avec contraintes d etat

the repulsive lattice gas the independent set polynomial and the lovasz local lemma we elucidate the close connection between the repulsive lattice gas in equi librium statistical mechanics and the lovasz local lemma in probabilistic com binatorics we show that the conclusion of the lovasz local lemma holds for dependency graph g and probabilities px if and only if the independent set polynomial for g is nonvanishing in the polydisc of radii px furthermore we show that the usual proof of the lovasz local lemma which provides a suffi cient condition for this to occur corresponds to a simple inductive argument for the nonvanishing of the independent set polynomial in a polydisc which was discovered implicitly by shearer 98 and explicitly by dobrushin 37 38 we also present some refinements and extensions of both arguments including a generalization of the lovasz local lemma that allows for soft dependencies in addition we prove some general properties of the partition function of a repulsive lattice gas most of which are consequences of the alternating sign property for the mayer coefficients we conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs

on the large time behavior of solutions of hamilton jacobi equations associated with nonlinear boundary conditions in this article we study the large time behavior of solutions of first order hamilton jacobi equations set in a bounded domain with nonlinear neumann boundary conditions including the case of dynamical boundary conditions we establish general convergence results for viscosity solutions of these cauchy neumann problems by using two fairly different methods the first one relies only on partial differential equations methods which provides results even when the hamiltonians are not convex and the second one is an optimal control dynamical system approach named the weak kam approach which requires the convexity of hamiltonians and gives formulas for asymptotic solutions based on aubry mather sets

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

solution of nonlinear stokes equations discretized by high order finite elements on nonconforming and anisotropic meshes with application to ice sheet dynamics motivated by the need for efficient and accurate simulation of the dynamics of the polar ice sheets we design high order finite element discretizations and scalable solvers for the solution of nonlinear incompressible stokes equations in particular we focus on power law shear thinning rheologies commonly used in modeling ice dynamics and other geophysical flows we use nonconforming hexahedral meshes and the conforming inf sup stable finite element velocity pressure pairings mathbb q k times mathbb q disc k 2 or mathbb q k times mathbb p disc k 1 where k ge 2 is the polynomial order of the velocity space to solve the nonlinear equations we propose a newton krylov method with a block upper triangular preconditioner for the linearized stokes systems the diagonal blocks of this preconditioner are sparse approximations of the 1 1 block and of its schur complement the 1 1 block is approximated using linear finite elements based on the nodes of the high order discretization

convergence of mayer expansions the tree graph bound of battle and federbush is extended and used to provide a simple criterion for the convergence of iterated mayer expansions as an application estimates on the radius of convergence of the mayer expansion for the two dimensional yukawa gas nonstable interaction are obtained

monotone multigrid methodsfor elliptic variational inequalities ii we derive globally convergent multigrid methods for discrete elliptic variational inequalities of the second kind as obtained from the approximation of related continuous problems by piecewise linear finite elements the coarse grid corrections are computed from certain obstacle problems the actual constraints are fixed by the preceding nonlinear fine grid smoothing this new approach allows the implementation as a classical v cycle and preserves the usual multigrid efficiency we give 1 o j 3 estimates for the asymptotic convergence rates the numerical results indicate a significant improvement as compared with previous multigrid approaches

smoothing methods and semismooth methods for nondifferentiable operator equations we consider superlinearly convergent analogues of newton methods for nondifferentiable operator equations in function spaces the superlinear convergence analysis of semismooth methods for nondifferentiable equations described by a locally lipschitzian operator in rn is based on rademacher s theorem which does not hold in function spaces we introduce a concept of slant differentiability and use it to study superlinear convergence of smoothing methods and semismooth methods in a unified framework we show that a function is slantly differentiable at a point if and only if it is lipschitz continuous at that point an application to the dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed

continuum theory of epitaxial crystal growth i we present various continuum limits to describe epitaxial thin film growth we consider a hierarchy of models which can take into account the diffusion of terrace adatoms attachment and detachment of edge adatoms vapor phase diffusion and the effect of multiple species the starting point is the burton cabrera frank type step flow model we have obtained partial differential equations in the form of a coupled system of diffusion equation for the adatom density and a hamilton jacobi equation for the film height function this is supplemented with appropriate boundary conditions at the continuum level to describe the growth at the peaks and valleys on the film the results here can be used in a macroscopic description of thin film growth

iterated mayer expansion for classical gases at low temperatures we derive iterated mayer expansions for classical gases and establish recursive bounds which control their convergence these bounds are useful for gases with two body forces which are strong and possibly attractive at distances that are short compared to their range our procedure is based on splitting the potential into pieces of decreasing strength and increasing range this may be called a renormalization group treatment of a classical gas we apply our results to yukawa lattice gas models and obtain convergence of series expansions for the pressure for a range of parameters temperature fugacities range of the interaction that was inaccessible before application to 3 dimensionalu 1 lattice gauge theory coulomb gas z ferromagnet will be made elsewhere

quantization of the nonlinear sigma model revisited we revisit the subject of perturbatively quantizing the nonlinear sigma model in two dimensions from a rigorous mathematical point of view our main contribution is to make precise the cohomological problem of eliminating potential anomalies that may arise when trying to preserve symmetries under quantization the symmetries we consider are twofold i diffeomorphism covariance for a general target manifold ii a transitive group of isometries when the target manifold is a homogeneous space we show that there are no anomalies in case i and that ii is also anomaly free under additional assumptions on the target homogeneous space in agreement with the work of friedan we carry out some explicit computations for the o n model finally we show how a suitable notion of the renormalization group establishes the ricci flow as the one loop renormalization group flow of the nonlinear sigma model

mayer expansions and the hamilton jacobi equation we review the derivation of wilson s differential equation in infinitely many variables which describes the infinitesimal change in an effective potential of a statistical mechanical model or quantum field theory when an infinitesimal integration out is performed we show that this equation can be solved for short times by a very elementary method when the initial data are bounded and analytic the resulting series solutions are generalizations of the mayer expansion in statistical mechanics the differential equation approach gives a remarkable identity for connected parts and precise estimates which include criteria for convergence of iterated mayer expansions applications include the yukawa gas in two dimensions past the 4 threshold and another derivation of some earlier results of gopfert and mack

model problems for the multigrid optimization of systems governed by differential equations we discuss a multigrid approach to the optimization of systems governed by differential equations such optimization problems appear in many applications and are of a different nature than systems of equations our approach uses an optimization based multigrid algorithm in which the multigrid algorithm relies explicitly on nonlinear optimization models as subproblems on coarser grids our goal is not to argue for a particular optimization based multigrid algorithm but instead to demonstrate how multigrid can be used to accelerate nonlinear programming algorithms furthermore using several model problems we give evidence both theoretical and numerical that the optimization setting is well suited to multigrid algorithms some of the model problems show that the optimization problem may be more amenable to multigrid than the governing differential equation in addition we relate the multigrid approach to more traditional optimization methods as further justification for the use of an optimization based multigrid algorithm

large time asymptotics for one dimensional dirichlet problems for hamilton jacobi equations with noncoercive hamiltonians we study large time asymptotics for a class of noncoercive hamilton jacobi equations with dirichlet boundary condition in one space dimension we prove that the average growth rate of a solution is constant only in a subset of the whole domain and give the asymptotic profile in the subset we show that the large time behavior for noncoercive problems may depend on the space variable in general which is different from the usual results under the coercivity condition this work is an extension with more rigorous analysis of a recent paper by e yokoyama y giga and p rybka in which a growing crystal model is established and the asymptotic behavior described above is first discovered

finite element approximation of the p laplacian in this paper we consider the continuous piecewise linear finite el ement approximation of the following problem given p 1 oo and g find u such that v vu 2vu f iniicr2 u g on a the finite element approximation is defined over i2 a union of regular tri angles yielding a polygonal approximation to q for sufficiently regular so lutions u achievable for a subclass of data g and i2 we prove optimal error bounds for this approximation in the norm wlq q 1 q p for p 2 under the additional assumption that qh c 2 numerical results demonstrating these bounds are also presented

on cell problems for hamilton jacobi equations with non coercive hamiltonians and their application to homogenization problems abstract we study a cell problem arising in homogenization for a hamilton jacobi equation whose hamiltonian is not coercive we introduce a generalized notion of effective hamiltonians by approximating the equation and characterize the solvability of the cell problem in terms of the generalized effective hamiltonian under some sufficient conditions the result is applied to the associated homogenization problem we also show that homogenization for non coercive equations fails in general

stochastic homogenization of interfaces moving with changing sign velocity abstract we are interested in the averaging behavior of interfaces moving in stationary ergodic environments with oscillatory normal velocity which changes sign the problem can be reformulated as the homogenization of a hamilton jacobi equation with a positively homogeneous of degree one non coercive hamiltonian the periodic setting was studied earlier by cardaliaguet lions and souganidis 2009 16 here we concentrate in the random media and show that the solutions of the oscillatory hamilton jacobi equation converge in l weak to a linear combination of the initial datum and the solutions of several initial value problems with deterministic effective hamiltonian s determined by the properties of the random media

hamilton jacobi equations with state constraints in the present paper we consider hamilton jacobi equations of the form h x u vu 0 x e q where q is a bounded open subset of rn h is a given continuous real valued function of x s p e q x r x rn and vu is the gradient of the unknown function u we are interested in particular solutions of the above equation which are required to be supersolutions in a suitable weak sense of the same equation up to the boundary of q this requirement plays the role of a boundary condition the main motivation for this kind of solution comes from deterministic optimal control and differential games problems with constraints on the state of the system as well from related questions in constrained geodesics

monotone multigrid methods for elliptic variational inequalities i we derive fast solvers for discrete elliptic variational inequalities of the first kind obstacle problems as resulting from the approximation of related continuous problems by piecewise linear finite elements using basic ideas of successive subspace correction we modify well known relaxation methods by extending the set of search directions extended underrelaxations are called monotone multigrid methods if they are quasioptimal in a certain sense by construction all monotone multigrid methods are globally convergent we take a closer look at two natural variants the standard monotone multigrid method and a truncated version for the considered model problems the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarse grid transport while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case

large time behavior of solutions of hamilton jacobi equations with periodic boundary data abstract we study the large time behavior of viscosity solutions of hamilton jacobi equations with periodic boundary data on bounded domains we establish a result on convergence of viscosity solutions to state constraint asymptotic solutions or periodic asymptotic solutions depending on the sign of critical value as time goes to infinity

an iterated mayer expansion for the yukawa gas we construct a convergent expansion for the yukawa gas at small activity and inverse temperature 4 e 2

neumann type boundary conditions for hamilton jacobi equations abstract in this paper we present a notion of viscosity solutions of hamilton jacobi equations for neumann type boundary conditions or more generally oblique derivative in particular we prove the existence uniqueness stability of such solutions and we show that the vanishing viscosity method yields such solutions next we check that value functions of control problems or differential games problems for reflected dynamical processes are solutions in that sense of the associated bellman or isaacs equations finally we consider the ergodic problems

massless sine gordon and massive thirring models proof of coleman s equivalence we prove coleman s conjecture on the equivalence between the massless sine gordon model with finite volume interaction and the thirring model with a finite volume mass term

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

on constrained newton linearization and multigrid for variational inequalities we consider the fast solution of a class of large piecewise smooth minimization problems for lack of smoothness usual newton multigrid methods cannot be applied we propose a new approach based on a combination of convex minization with constrained newton linearization no regularization is involved we show global convergence of the resulting monotone multigrid methods and give polylogarithmic upper bounds for the asymptotic convergence rates efficiency is illustrated by numerical experiments

the two dimensional o n nonlinear σ model renormalisation and effective actions we establish the existence of the wilson renormalised trajectory of theo n sigma model in perturbation theory in the effective charge this yields a proof of perturbative renormalisability and is also relevant in the small field analysis of the rigorous renormalisation group construction of the continuum theory

a multigrid algorithm for the p laplacian we introduce a full approximation storage fas multigrid algorithm to find the finite element solution for a class of nonlinear monotone elliptic problems since the solution of the problem is equivalent to minimize a strictly convex functional we use a polak ribiere conjugate gradient method as the nonlinear smoother in our algorithm the advantage in so doing is that we do not have to calculate derivatives of operators we prove local convergence of our algorithm and illustrate its performance by solving benchmark problems

renormalization group flow of a hierarchical sine gordon model by partial differential equations we use a renormalization group differential equation to rigorously control the renormalization group flow in a hierarchical lattice sine gordon field theory in the kosterlitz thouless phase

loop vertex expansion for φ2k theory in zero dimension in this paper we extend the method of loop vertex expansion to interactions with degree higher than 4 as an example we provide through this expansion an explicit proof that the free energy of 2k scalar theory in zero dimension is borel le roy summable of order k 1 we detail the computations in the case of a 6 interaction in this paper we extend the method of loop vertex expansion to interactions with degree higher than 4 as an example we provide through this expansion an explicit proof that the free energy of 2k scalar theory in zero dimension is borel le roy summable of order k 1 we detail the computations in the case of a 6 interaction

a microscopic time scale approximation to the behavior of the local slope on the faceted surface under a nonuniformity in supersaturation abstract the morphological stability of a growing faceted crystal is discussed we argue that the interplay between nonuniformity in supersaturation on a growing facet and anisotropy of surface kinetics derived from the lateral motion of steps leads to a faceted instability qualitatively speaking as long as the nonuniformity in supersaturation on the facet is not too large it can be compensated for by a variation of step density along the facet and the faceted crystal can grow in a stable manner the problem can be modeled as a hamilton jacobi equation for height of the crystal surface the notion of a maximal stable region of a growing facet is introduced for microscopic time scale approximation of the original hamilton jacobi equation it is shown that the maximal stable region keeps its shape determined by profile of the surface supersaturation with constant growth rate by studying large time behavior of solution of macroscopic time scale approximation a quantitative criterion for the facet stability is given

long time behavior of solutions of hamilton jacobi equations with convex and coercive hamiltonians we investigate the long time behavior of viscosity solutions of hamilton jacobi equations in mathbb r n with convex and coercive hamiltonians and give three general criteria for the convergence of solutions to asymptotic solutions as time goes to infinity we apply the criteria to obtain more specific sufficient conditions for the convergence to asymptotic solutions and then examine them with examples we take a dynamical approach based on tools from weak kam theory such as extremal curves aubry sets and representation formulas for solutions for these investigations