large deviations for stochastic pde with levy noise abstract we prove a large deviation principle result for solutions of abstract stochastic evolution equations perturbed by small levy noise we use general large deviations theorems of varadhan and bryc coupled with the techniques of feng and kurtz 2006 15 viscosity solutions of integro partial differential equations in hilbert spaces and deterministic optimal control methods the laplace limit is identified as a viscosity solution of a hamilton jacobi bellman equation of an associated control problem we also establish exponential moment estimates for solutions of stochastic evolution equations driven by levy noise general results are applied to stochastic hyperbolic equations perturbed by subordinated wiener process

large deviations for the stochastic derivative ginzburg landau equation with multiplicative noise abstract this paper first proves the existence of a unique mild solution to the stochastic derivative ginzburg landau equation the fixed point theorem for the corresponding truncated equation is used as the main tool since we restrict our study to the one dimensional case it is not necessary to introduce another banach space and thus the estimates of the stochastic convolutions in the banach space are avoided secondly we also consider large deviations for the stochastic derivative ginzburg landau equation perturbed by a small noise since the underlying space considered is polish using the weak convergence approach we establish a large deviations principle by proving a laplace principle

a moderate deviation principle for 2 d stochastic navier stokes equations in this paper we prove a central limit theorem and establish a moderate deviation principle for twodimensional stochastic navier stokes equations with multiplicative noise the weak convergence method plays an important role

ergodicity of the stochastic fractional reaction diffusion equation abstract a stochastic reaction diffusion equation with fractional dissipation is considered the smaller the exponent of the equation is the weaker the dissipation of the equation is the equation is discussed in detail when the exponent changes the aim is to prove the well posedness existence and uniqueness of an invariant measure as well as strong law of large numbers and convergence to equilibrium without the analytic property of the semigroup some methods are used to overcome the difficulties to get the energy estimates the results in this paper can be applied to the classic reaction diffusion equation with wiener noise

stochastic partial differential equations in holder spaces we prove the existence and regularity of solutions to stochastic partial differential equations of parabolic ito type in holder spaces under the usual sublinear growth and local lipschitz conditions some examples are given to which our main theorems apply

large deviations for stationary measures of stochastic nonlinear wave equation with smooth white noise we prove the freidlin wentzell type large deviations principle for the family of stationary measures of stochastic nonlinear wave nlw equation with white noise we do not assume that the limiting equation possesses a unique equilibrium and do not impose roughness on the noise this allows to provide the first such result in the pde setting

large deviations for a reaction diffusion equation with non gaussian perturbations in this paper we establish a large deviations principle for the non gaussian stochastic reaction diffusion equation srde partial t nu varepsilon mathscr l nu varepsilon f x nu varepsilon varepsilon sigma x nu varepsilon ddot w tx as a random perturbation of the deterministic rde partial t nu 0 mathscr l nu 0 f x nu 0 here the space variable takes values on the unit circle s 1 and mathscr l is a strongly elliptic second order operator with constant coefficients the functions f and sigma are sufficiently regular so that there is a unique solution to the above srde for any continuous initial condition we also assume that there are positive constants m and m such that m leq sigma x y leq m for all x in s 1 and all y in mathbb r the perturbation ddot w tx is the formal derivative of a brownian sheet it is known that if the initial condition is continuous then the solution will also be continuous and moreover if the initial condition is assumed to be holder continuous of exponent kappa for some 0 kappa frac 1 2 then the solution will be holder continuous of exponent kappa 2 as a function of t x in this paper we establish the large deviations principle for nu varepsilon in the holder norm of exponent kappa 2 when the initial condition is holder continuous of exponent kappa for any 0 kappa frac 1 2 and when the initial condition is assumed only to be continuous we establish the large deviations principle for nu varepsilon in the supremum norm moreover we prove that these large deviations principles are uniform with respect to the initial condition

large deviations for stochastic reaction diffusion systems with multiplicative noise and non lipshitz reaction term following classical work by freidlin trans amer math soc 1988 305 665 657 and subsequent works by sowers ann probab 1992 20 504 537 and peszat probab theory related fields 1994 98 113 136 we prove large deviation estimates for the small noise limit of systems of stochastic reaction diffusion equations with globally lipschitz but unbounded diffusion coefficients however assuming the reaction terms to be only locally lipschitz with polynomial growth this generalizes results of the above mentioned authors our results apply in particular to systems of stochastic ginzburg landau equations with multiplicative noise

stochastic volterra equations in banach spaces and stochastic partial differential equation abstract in this paper we study the existence uniqueness and large deviation estimate for stochastic volterra integral equations with singular kernels in 2 smooth banach spaces then we apply them to a large class of semilinear stochastic partial differential equations spde and obtain the existence of unique maximal strong solutions in the sense of sde and pde under local lipschitz conditions moreover stochastic navier stokes equations are also investigated

large deviation for diffusions and hamilton jacobi equation in hilbert spaces large deviation for markov processes can be studied by hamilton jacobi equation techniques the method of proof involves three steps first we apply a nonlinear transform to generators of the markov processes and verify that limit of the transformed generators exists such limit induces a hamilton jacobi equation second we show that a strong form of uniqueness the comparison principle holds for the limit equation finally we verify an exponential compact containment estimate the large deviation principle then follows from the above three verifications this paper illustrates such a method applied to a class of hilbert space valued small diffusion processes the examples include stochastically perturbed allen cahn cahn hilliard pdes and a one dimensional quasilinear pde with a viscosity term we prove the comparison principle using a variant of the tataru method we also discuss different notions of viscosity solution in infinite dimensions in such context

a new algorithm for topology optimization using a level set method the level set method has been recently introduced in the field of shape optimization enabling a smooth representation of the boundaries on a fixed mesh and therefore leading to fast numerical algorithms however most of these algorithms use a hamilton jacobi equation to connect the evolution of the level set function with the deformation of the contours and consequently they can hardly create new holes in the domain at least in 2d in this work we propose an evolution equation for the level set function based on a generalization of the concept of topological gradient this results in a new algorithm allowing for all kinds of topology changes

a staggered approach to shape and topology optimization using the traction method and an evolutionary type advancing front algorithm abstract we investigate a novel approach for structural shape optimization on the basis of complementary shape and topological sensitivity analysis as in early approaches to shape optimization the domain variation is specified by modification of boundary nodal points hence leading to an updated lagrangian description for the course of optimization to overcome the formation of oscillating boundaries in the optimal design trials we employ the traction method to establish smooth descent directions for shape variation therein an auxiliary elastic problem is solved in which the shape sensitivity constitutes the external loading and the deformation of the auxiliary elastic body is used as a descent direction for shape variation rather than the shape sensitivity itself we complement this method through an evolutionary type element removal procedure that is based on the topological sensitivity such that an advancing front algorithm is gradually removing elements from the design boundary of the domain once the minimum topological sensitivity is no longer encountered at the design boundary we create a hole in the domain again using the topological sensitivity to specify its exact location and resume the element removal procedure for the newly established design boundary since this approach yields only a vague estimate of the true optimal shape of newly established holes the traction method is again used for shape variation of the extended design boundary

nonlinear random wave equations nonlinear wave equations describing forced vibrations of a string subject to random perturbations of intensity e are studied herein two kinds of random perturbations are considered one is gaussian in the form of initial white noise and the other is a non gaussian random forcing which involves the formal derivative of a brownian sheet the solutions ue t x of the complete forced equation are a stochastic process in space and time which has a unique stochastic equilibrium the time variable t varies in a finite interval i 0 t where t is a fixed positive time and the space variable x varies in an interval j assuming that the initial conditions are of sufficient regularity the fluctuations ue t x from u0 t x the unperturbed solution are analyzed as the intensity of perturbation e 0

large deviations for infinite dimensional stochastic dynamical systems the large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation the construction and justification of the approximations can be onerous especially in the case where the process state is infinite dimensional in this paper we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of brownian noise the approach is based on a variational representation for functionals of brownian motion proofs of large deviations properties are reduced to demonstrating basic qualitative properties existence uniqueness and tightness of certain perturbations of the original process

large deviations for the invariant measure of a reaction diffusion equation with non gaussian perturbations in this paper we establish a large deviations principle for the invariant measure of the non gaussian stochastic partial differential equation spde t v e v e f x v e e x v e ddot w tx here is a strongly elliptic second order operator with constant coefficients h dh xx h and the space variablex takes values on the unit circles 1 the functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic pde and in particular we require that 0 m m wherem andm are some finite positive constants the perturbationw is a brownian sheet it is well known that under some simple assumptions the solutionv 2 is ac k s 1 valued markov process for each 0 1 2 wherec s 1 is the banach space of real valued continuous functions ons 1 which are holder continuous of exponent we prove under some further natural assumptions onf and which imply that the zero element ofc s 1 is a globally exponentially stable critical point of the unperturbed equation t 0 0 f x 0 that e has a unique stationary distributionv k on c s 1 c k s 1 when the perturbation parameter e is small enough some further calculations show that as e tends to zero v k tends tov k 0 the point mass centered on the zero element ofc s 1 the main goal of this paper is to show that in factv k is governed by a large deviations principle ldp our starting point in establishing the ldp forv k is the ldp for the process e which has been shown in an earlier paper our methods of deriving the ldp forv k based on the ldp for e are slightly non standard compared to the corresponding proofs for finite dimensional stochastic differential equations since the state spacec s 1 is inherently infinite dimensional

freidlin wentzell s large deviations for stochastic evolution equations abstract we prove a freidlin wentzell large deviation principle for general stochastic evolution equations with small perturbation multiplicative noises in particular our general result can be used to deal with a large class of quasi linear stochastic partial differential equations such as stochastic porous medium equations and stochastic reaction diffusion equations with polynomial growth zero order term and p laplacian second order term

a fictitious energy approach for shape optimization this paper deals with shape optimization of continuous structures as in early works on shape optimization coordinates of boundary nodes of the fe domain are directly chosen as design variables convergence problems and problems with jagged shapes are eliminated by a new regularization technique an artificial inequality constraint added to the optimization problem limits a fictitious total strain energy that measures the shape change of the design with respect to a reference design the energy constraint defines a feasible design space whose size can be varied by one parameter the upper energy limit by construction the proposed regularization is applicable to a wide range of problems although in this paper the application is restricted to linear elastostatic problems

hessian eigenmaps locally linear embedding techniques for high dimensional data we describe a method for recovering the underlying parametrization of scattered data m i lying on a manifold m embedded in high dimensional euclidean space the method hessian based locally linear embedding derives from a conceptual framework of local isometry in which the manifold m viewed as a riemannian submanifold of the ambient euclidean space r n is locally isometric to an open connected subset of euclidean space r d because does not have to be convex this framework is able to handle a significantly wider class of situations than the original isomap algorithm the theoretical framework revolves around a quadratic form h f m h f m f dm defined on functions f m r here hf denotes the hessian of f and h f averages the frobenius norm of the hessian over m to define the hessian we use orthogonal coordinates on the tangent planes of m the key observation is that if m truly is locally isometric to an open connected subset of r d then h f has a d 1 dimensional null space consisting of the constant functions and a d dimensional space of functions spanned by the original isometric coordinates hence the isometric coordinates can be recovered up to a linear isometry our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the laplacian eigenmaps framework where we substitute a quadratic form based on the hessian in place of one based on the laplacian

rare events in the boussinesq system with fluctuating dynamical boundary conditions abstract the boussinesq system models various phenomena in geophysical and climate dynamics it is a coupled system of the navier stokes equations and the salinity transport equation due to uncertainty in salinity flux on fluid boundary this system is subject to random fluctuations on the boundary this stochastic boussinesq system can be transformed into a random dynamical system rare events or small probability events are investigated in the context of large deviations a large deviations principle is established via a weak convergence approach based on a recently developed variational representation of functionals of infinite dimensional brownian motion

effect of noise on front propagation in reaction diffusion equations of kpp type we consider reaction diffusion equations of kpp type in one spatial dimension perturbed by a fisher wright white noise under the assumption of uniqueness in distribution examples include the randomly perturbed fisher kpp equations partial tu partial x 2u u 1 u epsilon sqrt u 1 u dot w and partial tu partial x 2u u 1 u epsilon sqrt u dot w where dot w dot w t x is a space time white noise we prove the brunet derrida conjecture that the speed of traveling fronts for small e is 2 pi 2 log epsilon 2 2 o log log epsilon log epsilon 3

topological sensitivity analysis the so called topological derivative concept has been seen as a powerful framework to obtain the optimal topology for several engineering problems this derivative characterizes the sensitivity of the problem when a small hole is created at each point of the domain however the greatest limitation of this methodology is that when a hole is created it is impossible to build a homeomorphic map between the domains in study because they have not the same topology therefore some specific mathematical framework should be developed in order to obtain the derivatives this work proposes an alternative way to compute the topological derivative based on the shape sensitivity analysis concepts the main feature of this methodology is that all the mathematical procedure already developed in the context of shape sensitivity analysis may be used in the calculus of the topological derivative this idea leads to a more simple and constructive formulation than the ones found in the literature further to point out the straightforward use of the proposed methodology it is applied for solving some design problems in steady state heat conduction

large deviations and approximations for slow fast stochastic reaction diffusion equations abstract a large deviation principle is derived for a class of stochastic reaction diffusion partial differential equations with slow fast components the result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small gaussian perturbation this result also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow fast stochastic partial differential equations

small perturbation of stochastic parabolic equations a power series analysis abstract a semi linear second order stochastic parabolic equation is considered with coefficients free terms and initial condition depending on a parameter it is shown that under some natural conditions the solution can be written as a power series in the parameter the equations for the coefficients in the power series expansion are derived and the convergence of the power series is studied an example from nonlinear filtering of diffusion process is discussed

topological derivative based topology optimization of structures subject to drucker prager stress constraints abstract an algorithm for topology optimization of elastic structures under plane stress subject to the drucker prager stress constraint is presented the algorithm is based on the use of the topological derivative of the associated objective functional in conjunction with a level set representation of the structure domain in this context a penalty functional is proposed to enforce the point wise stress constraint and a closed formula for its topological derivative is derived the resulting algorithm is of remarkably simple computational implementation it does not require post processing procedures of any kind and features only a minimal number of user defined algorithmic parameters this is in sharp contrast with current procedures for topological structural optimization with local stress constraints the effectiveness and efficiency of the algorithm presented here are demonstrated by means of numerical examples the examples show in particular that it can easily handle structural optimization problems with underlying materials featuring strong asymmetry in their tensile and compressive yield strengths

a probabilistic decomposition synthesis method for the quantification of rare events due to internal instabilities we consider the problem of the probabilistic quantification of dynamical systems that have heavy tailed characteristics these heavy tailed features are associated with rare transient responses due to the occurrence of internal instabilities systems with these properties can be found in a variety of areas including mechanics fluids and waves here we develop a computational method a probabilistic decomposition synthesis technique that takes into account the nature of internal instabilities to inexpensively determine the non gaussian probability density function for any arbitrary quantity of interest our approach relies on the decomposition of the statistics into a non extreme core typically gaussian and a heavy tailed component this decomposition is in full correspondence with a partition of the phase space into a stable region where we have no internal instabilities and a region where non linear instabilities lead to rare transitions with high probability we quantify the statistics in the stable region using a gaussian approximation approach while the non gaussian distribution associated with the intermittently unstable regions of phase space is inexpensively computed through order reduction methods that take into account the strongly nonlinear character of the dynamics the probabilistic information in the two domains is analytically synthesized through a total probability argument the proposed approach allows for the accurate quantification of non gaussian tails at more than 10 standard deviations at a fraction of the cost associated with the direct monte carlo simulations we demonstrate the probabilistic decomposition synthesis method for rare events for two dynamical systems exhibiting extreme events a two degree of freedom system of nonlinearly coupled oscillators and in a nonlinear envelope equation characterizing the propagation of unidirectional water waves

large deviations for a class of stochastic partial differential equations we consider the random fields x e t q t 0 q o governed by stochastic partial differential equations driven by a gaussian white noise in space time where o is a bounded domain in lid with regular boundary to study the continuity of the random fields x e in space and time variables we prove an analogue of garsia s theorem we then derive the large deviation results based on the methods used by the second author in another paper this article provides an alternative proof of sowers result for the case of d 1

a phase transition for a stochastic pde related to the contact process we consider the one dimensional heat equation with a semilinear term and with a nonlinear white noise term r durrett conjectured that this equation arises as a weak limit of the contact process with longrange interactions we show that our equation possesses a phase transition to be more precise we assume that the initial function is nonnegative with bounded total mass if a certain parameter in the equation is small enough then the solution dies out to 0 in finite time with probability 1 if this parameter is large enough then the solution has a positive probability of never dying out to 0 this result answers a question of durett

large deviation principle for stochastic evolution equations the large deviation principle obtained by freidlin and wentzell for measures associated with finite dimensional diffusions is extended to measures given by stochastic evolution equations with non additive random perturbations the proof of the main result is adopted from the priouret paper concerning finite dimensional diffusions exponential tail estimates for infinite dimensional stochastic convolutions are used as main tools

structural optimization using sensitivity analysis and a level set method in the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level set method for front propagation we implement this method in two and three space dimensions for a model of linear or nonlinear elasticity we consider various objective functions with weight and perimeter constraints the shape derivative is computed by an adjoint method the cost of our numerical algorithm is moderate since the shape is captured on a fixed eulerian mesh although this method is not specifically designed for topology optimization it can easily handle topology changes however the resulting optimal shape is strongly dependent on the initial guess

a gradient based parameter free approach to shape optimization in shape optimization the independent node movement approach wherein finite element node coordinates are used directly as design variables allows the most freedom for shape change and avoids the time consuming parameterization preprocess however this approach lacks a length scale control that is necessary to ensure a well posed shape optimization problem and avoid numerical instability motivated by the success of filtering techniques that impose minimum length scales in topology optimization we propose a scheme with consistent filtering to introduce a length scale and thereby ensure smoothness in shape optimization while preserving the advantages of the independent node movement approach

large deviations for stochastic nonlinear beam equations abstract we establish a large deviation principle for the solutions of stochastic partial differential equations for nonlinear vibration of elastic panels also called stochastic nonlinear beam equations

exit from a basin of attraction for stochastic weakly damped nonlinear schrodinger equations we consider weakly damped nonlinear schrodinger equations perturbed by a noise of small amplitude the small noise is either complex and of additive type or real and of multiplicative type it is white in time and colored in space zero is an asymptotically stable equilibrium point of the deterministic equations we study the exit from a neighborhood of zero invariant under the flow of the deterministic equation in l 2 or in h 1 due to noise large fluctuations from zero occur thus on a sufficiently large time scale exit from these domains of attraction occur a formal characterization of the small noise asymptotic of both the first exit times and the exit points is given

a moderate deviation principle for 2 d stochastic navier stokes equations driven by multiplicative levy noises abstract in this paper we establish a moderate deviation principle for two dimensional stochastic navier stokes equations driven by multiplicative levy noises the weak convergence method introduced by budhiraja dupuis and ganguly in 3 plays a key role

large deviations for the stochastic quasigeostrophic equation with multiplicative noise this paper proves a large deviation principle for the stochastic quasigeostrophic equation with multiplicative noise the proof is based on the laplace principle and on weak convergence methods we also give a rigorous mathematical result on the existence of a strong solution and uniqueness of mild solutions to the stochastic quasigeostrophic equation in the dimension d 2 3 finally large deviations for short time are considered

stochastic 3d tamed navier stokes equations existence uniqueness and small time large deviation principles abstract in this paper we first give a direct approach to the existence and uniqueness of strong solutions to stochastic 3d tamed navier stokes equations in case of dirichlet boundary conditions for the stokes laplacian then we prove a small time large deviation principle for the solutions

blowup for the heat equation with a noise term in this paper we study blow up of the equation u t u xx u gamma dot w tx where dot w tx is a two dimensional white noise field and where dirichlet boundary conditions are enforced it is known that if 3 2 then the solution exists for all time in this paper we show that if is much larger than 3 2 then the solution blows up in finite time with positive probability we prove this by considering how peaks in the solution propagate if a peak of high mass forms we rescale the equation and divide the mass of the peak into a collection of peaks of smaller mass and these peaks evolve almost independently in this way we compare the evolution ofu to a branching process large peaks are regarded as particles in this branching process offspring are peaks which are higher by some factor we show that the expected number of offspring is greater than one when is much larger than 3 2 and thus the branching process survives with positive probability corresponding to blowup in finite time