uniform convergence to equilibrium for granular media we study the long time asymptotics of a nonlinear nonlocal equation used in the modelling of granular media we prove a uniform exponential convergence to equilibrium for degenerately convex and non convex interaction or confinement potentials improving in particular results by j a carrillo r j mccann and c villani the method is based on studying the dissipation of the wasserstein distance between a solution and the steady state

monge s problem with a quadratic cost by the zero noise limit of h path processes we study the asymptotic behavior in the zero noise limit of solutions to schrodinger s functional equations and that of h path processes and give a new proof of the existence of the minimizer of monge s problem with a quadratic cost

fast calderon preconditioning for helmholtz boundary integral equations abstract calderon multiplicative preconditioners are an effective way to improve the condition number of first kind boundary integral equations yielding provable mesh independent bounds however when discretizing by local low order basis functions as in standard galerkin boundary element methods their computational performance worsens as meshes are refined this stems from the barycentric mesh refinement used to construct dual basis functions that guarantee the discrete stability of l 2 pairings based on coarser quadrature rules over dual cells and h matrix compression we propose a family of fast preconditioners that significantly reduce assembly and computation times when compared to standard versions of calderon preconditioning for the three dimensional helmholtz weakly and hyper singular boundary integral operators several numerical experiments validate our claims and point towards further enhancements

on the efficiency of nested gmres preconditioners for 3d acoustic and elastodynamic h matrix accelerated boundary element methods abstract this article is concerned with the derivation of fast boundary element methods for 3d acoustic and elastodynamic problems in particular we are interested in the acceleration of hierarchical matrix h matrix based iterative solvers while h matrix representations allow to reduce the storage requirements and the cost of a matrix vector product the number of iterations for an iterative solver as the frequency or the problem size increases remains an issue we consider an inner outer preconditioning strategy i e the preconditioner is applied through an iterative solver at the inner level the preconditioner is defined as a h matrix representation of the system matrix with a given accuracy we investigate the influence of various parameters of the preconditioner i e the h matrix accuracy the gmres threshold and the maximum number of iterations of the inner solver different numerical results are presented to compare the efficiency of the preconditioner with respect to the unpreconditioned reference system finally we propose a way to define the optimal setting for this preconditioner

probabilistic analysis of mean field games the purpose of this paper is to provide a complete probabilistic analysis of a large class of stochastic differential games with mean field interactions we implement the mean field game strategy developed analytically by lasry and lions in a purely probabilistic framework relying on tailor made forms of the stochastic maximum principle while we assume that the state dynamics are affine in the states and the controls and the costs are convex our assumptions on the nature of the dependence of all the coefficients upon the statistical distribution of the states of the individual players remains of a rather general nature our probabilistic approach calls for the solution of systems of forward backward stochastic differential equations of a mckean vlasov type for which no existence result is known and for which we prove existence and regularity of the corresponding value function finally we prove that a solution of the mean field game problem as formulated by lasry and lions does indeed provide appr

variational processes and stochastic versions of mechanics the dynamical structure of any reasonable stochastic version of classical mechanics is investigated including the version created by nelson e nelson quantum fluctuations princeton u p princeton nj 1985 phys rev 150 1079 1966 for the description of quantum phenomena two different theories result from this common structure one of them is the imaginary time version of nelson s theory whose existence was unknown and yields a radically new probabilistic interpretation of the heat equation the existence and uniqueness of all the involved stochastic processes is shown under conditions suggested by the variational approach of yasue k yasue j math phys 22 1010 1981

mimicking the one dimensional marginal distributions of processes having an ito differential let t be a stochastic process starting from 0 with ito differential d xi left t right delta left t omega right dw left t right beta left t omega right dt where left w left t right mathfrak f t right is a wiener process and are bounded mathfrak f processes such that t is uniformly positive definite then it is proved that there exists a stochastic differential equation dx left t right sigma left t x left t right right dw left t right b left t x left t right right dt with non random coefficients which admits a weak solutionx t having the same one dimensional probability distribution as t for everyt the coefficients andb have a simple interpretation sigma left t x right left e left delta delta t left t right xi left t right x right right tfrac 1 2 b left t x right e left beta left t right xi left t right x right

preconditioned electric field integral equation using calderon identities and dual loop star basis functions an analytic preconditioner for the electric field integral equation based on the calderon identities is considered it is shown based on physical reasoning that rwg elements are not suitable for discretizing the electric field integral operator appearing in the preconditioner instead the geometrically dual basis functions proposed by buffa and christiansen are used however it is found that this preconditioner is vulnerable to roundoff errors at low frequencies a loop star decomposition of the buffa christiansen basis functions is presented along with numerical results demonstrating its effectiveness

volume comparison and the σk yamabe problem abstract in this paper we study the problem of finding a conformal metric with the property that the k th elementary symmetric polynomial of the eigenvalues of its weyl schouten tensor is constant a new conformal invariant involving maximal volumes is defined and this invariant is then used in several cases to prove existence of a solution and compactness of the space of solutions provided the conformal class admits an admissible metric in particular the problem is completely solved in dimension four and in dimension three if the manifold is not simply connected

a second order equation for schrodinger bridges with applications to the hot gas experiment and entropic transportation cost the schrodinger problem is obtained by replacing the mean square distance with the relative entropy in the monge kantorovich problem it was first addressed by schrodinger as the problem of describing the most likely evolution of a large number of brownian particles conditioned to reach an unexpected configuration its optimal value the entropic transportation cost and its optimal solution the schrodinger bridge stand as the natural probabilistic counterparts to the transportation cost and displacement interpolation moreover they provide a natural way of lifting from the point to the measure setting the concept of brownian bridge in this article we prove that the schrodinger bridge solves a second order equation in the riemannian structure of optimal transport roughly speaking the equation says that its acceleration is the gradient of the fisher information using this result we obtain a fine quantitative description of the dynamics and a new functional inequality for the entropic transportation cost that generalize talagrand s transportation inequality finally we study the convexity of the fisher information along schrodigner bridges under the hypothesis that the associated reciprocal characteristic is convex the techniques developed in this article are also well suited to study the feynman kac penalisations of brownian motion

large deviation properties of weakly interacting processes via weak convergence methods we study large deviation properties of systems of weakly interacting particles modeled by ito stochastic differential equations sdes it is known under certain conditions that the corresponding sequence of empirical measures converges as the number of particles tends to infinity to the weak solution of an associated mckean vlasov equation we derive a large deviation principle via the weak convergence approach the proof which avoids discretization arguments is based on a representation theorem weak convergence and ideas from stochastic optimal control the method works under rather mild assumptions and also for models described by sdes not of diffusion type to illustrate this we treat the case of sdes with delay

forward backward stochastic differential equations and controlled mckean vlasov dynamics the purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of nonlinear stochastic dynamical systems of the mckean vlasov type motivated by the recent interest in mean field games we highlight the connection and the differences between the two sets of problems we prove a new version of the stochastic maximum principle and give sufficient conditions for existence of an optimal control we also provide examples for which our sufficient conditions for existence of an optimal solution are satisfied finally we show that our solution to the control problem provides approximate equilibria for large stochastic games with mean field interactions

the mean field schrodinger problem ergodic behavior entropy estimates and functional inequalities we study the mean field schrodinger problem mfsp that is the problem of finding the most likely evolution of a cloud of interacting brownian particles conditionally on the observation of their initial and final configuration its rigorous formulation is in terms of an optimization problem with marginal constraints whose objective function is the large deviation rate function associated with a system of weakly dependent brownian particles we undertake a fine study of the dynamics of its solutions including quantitative energy dissipation estimates yielding the exponential convergence to equilibrium as the time between observations grows larger and larger as well as a novel class of functional inequalities involving the mean field entropic cost i e the optimal value in mfsp our strategy unveils an interesting connection between forward backward stochastic differential equations and the riemannian calculus on the space of probability measures introduced by otto which is of independent interest

loop star basis functions and a robust preconditioner for efie scattering problems an electric field integral equation efie formulation using the loop star basis functions has been developed for modeling plane wave scattering from perfect conducting objects a stability analysis at the dc limit shows that the use of the rao wilton glisson rwg basis functions results in a singular matrix operator however the use of the loop star basis functions results in a well conditioned matrix moreover a preconditioner constructed from a two step process based on near interactions and an incomplete factorization with a heuristic drop strategy has been proposed in conjunction with the conjugate gradient method to solve the resulting matrix equation the approach is shown to be effective for resolving both the low frequency instability and the bad conditioning of the efie method the computational complexity of the proposed approach is shown to be o n sup 2

transport inequalities gradient estimates entropy and ricci curvature we present various characterizations of uniform lower bounds for the ricci curvature of a smooth riemannian manifold m in terms of convexity properties of the entropy considered as a function on the space of probability measures on m as well as in terms of transportation inequalities for volume measures heat kernels and brownian motions and in terms of gradient estimates for the heat semigroup 2004 wiley periodicals inc

long time average of mean field games with a nonlocal coupling we study the long time average as the time horizon tends to infinity of the solution of a mean field game system with a nonlocal coupling we show an exponential convergence to the solution of the associated stationary ergodic mean field game proofs rely on semiconcavity estimates and smoothing properties of the linearized system

a numerical implementation of a modified form of the electric field integral equation the details of a galerkin discretization scheme for a modified form of the electric field integral equation are outlined for smooth three dimensional perfectly conducting scatterers limitations of the divergence conforming finite element bases in preserving the self stabilizing properties of the electric field integral equation operator are indicated a numerically efficient alternative is outlined which relies on an operator based helmholtz decomposition the condition number of the resulting matrix equation is demonstrated to be frequency independent for scattering from a perfectly conducting sphere at various frequencies

probabilistic approach for granular media equations in the non uniformly convex case we use here a particle system to prove both a convergence result with convergence rate and a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex the proof of convergence is simpler than the one in carrillo mccann villani rev mat iberoamericana 19 971 1018 2003 arch rat mech anal 179 217 263 2006 all the results complete former results of malrieu ann appl probab 13 540 560 2003 in the uniformly convex case the main tool is an uniform propagation of chaos property and a direct control in wasserstein distance of solutions starting with different initial measures the deviation inequality is obtained via a t 1 transportation cost inequality replacing the logarithmic sobolev inequality which is no more clearly dimension free

mesh independent operator preconditioning for boundary elements on open curves boundary value problems for the poisson equation in the exterior of an open bounded lipschitz curve mathcal c can be recast as first kind boundary integral equations featuring weakly singular or hypersingular boundary integral operators bios based on the recent discovery in c jerez hanckes and j nedelec siam j math anal 44 2012 pp 2666 2694 of inverses of these bios for mathcal c 1 1 we pursue operator preconditioning of the linear systems of equations arising from galerkin petrov discretization by means of zeroth and first order boundary elements the preconditioners rely on boundary element spaces defined on dual meshes and they can be shown to perform uniformly well independently of the number of degrees of freedom even for families of locally refined meshes

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

sufficient conditions for the invertibility of adapted perturbations of identity on the wiener space let w h be the classical wiener space assume that u iw u is an adapted perturbation of identity i e u w h is adapted to the canonical filtration of w we give some sufficient analytic conditions on u which imply the invertibility of the map u in particular it is shown that if u in rm id p 1 h is adapted and if exp frac 1 2 nabla u 2 2 delta u in l q mu where p 1 q 1 1 then iw u is almost surely invertible with the help of this result it is shown that if nabla u in l infty mu h otimes h then the girsanov exponential of u times the wiener measure satisfies the logarithmic sobolev inequality and this implies the invertibility of u iw u as a consequence if there exists an integer k 1 such that nabla k u h otimes k 1 in l infty mu then iw u is again almost surely invertible under the almost sure continuity hypothesis of t to nabla i dot u t for i k 1

uniform preconditioners for problems of positive order uniform preconditioners for operators of negative order discretized by dis continuous piecewise polynomials of any order are constructed from a boundedly invertible operator of opposite order discretized by continuous piecewise linears besides the cost of the application of the latter discretized operator the other cost of the preconditioner scales linearly with the number of mesh cells compared to earlier proposals the preconditioner has the following advantages it does not require the inverse of a non diagonal matrix it applies without any mildly grading assumption on the mesh and it does not require a barycentric refinement of the mesh underlying the trial space