role of the admittance operator in rough surface scattering the linear operator that relates the normal gradient of a wave field to its values on an irregular surface is the basis for a new formulation of the rough surface scattering problem this formulation has led to a class of approximations that surpass traditional methods in accuracy and efficiency the connection between this approach and the standard formulation by means of a boundary integral equation is the subject of this paper while equivalent in all respects the admittance operator formulation provides helpful new perspectives on the physics of surface scattering and lends itself naturally to more elaborate problems for example at a rough boundary between dissimilar media the operators compose themselves into generalized reflection and transmission operators while at a random surface the operator terms lead to analytic forms for the scattered field moments

frequency domain elastic wave modeling by finite differences a tool for crosshole seismic imaging the migration imaging or inversion of wide aperture cross hole data depends on the ability to model wave propagation in complex media for multiple source positions computational costs can be considerably reduced in frequency domain imaging by modeling the frequency domain steady state equations rather than the time domain equations of motion i develop a frequency domain approach in this note that is competitive with time domain modeling when solutions for multiple sources are required or when only a limited number of frequency components of the solution are required

on large deviations in the averaging principle for sdes with a full dependence we establish the large deviation principle for stochastic differential equations with averaging in the case when all coefficients of the fast component depend on the slow one including diffusion

averaging principle for slow fast stochastic differential equations with time dependent locally lipschitz coefficients abstract this paper is devoted to studying the averaging principle for stochastic differential equations with slow and fast time scales where the drift coefficients satisfy local lipschitz conditions with respect to the slow and fast variables and the coefficients in the slow equation depend on time t and making use of the techniques of time discretization and truncation we prove that the slow component strongly converges to the solution of the corresponding averaged equation

a new approach to analyticity of dirichlet neumann operators this paper outlines the theoretical background of a new approach towards an accurate and well conditioned perturbative calculation of dirichlet neumann operators dnos on domains that are perturbations of simple geometries previous work on the analyticity of dnos has produced formulae that as we have found are very ill conditioned we show how a simple change of variables can lead to recursions that satisfy analyticity estimates without relying on subtle cancellation properties at the heart of previous formulae

derivation of closed form expressions for the t matrices of rayleigh rice and extinction theorem perturbation theories closed form expressions are derived for the rayleigh rice and extinction theorem perturbation theories for the case of dirichlet boundary conditions it is shown that the two theories are related through the reciprocity transformation to all orders it follows that a proof of invariance of either of the theories with respect to the reciprocity transformation would imply the formal equivalence of the two theories using the reciprocity relationship between the two theories a recursion relation is derived for an nth order term of the extinction theorem perturbation theory similar to the one for the rayleigh rice perturbation theory in spite of considerable effort no general proof of invariance of the terms of the two perturbation series with respect to the reciprocity transformation was obtained two possible approaches to such a proof are suggested while the question of convergence of the two series is not examined in this work it is possible that the formulas derived here will prove useful in such a study

lp strong convergence of the averaging principle for slow fast spdes with jumps abstract the averaging principle is an important method to extract effective macroscopic dynamic from complex systems with slow component and fast component this paper concerns the l p strong convergence of the averaging principle for two time scales stochastic partial differential equations spdes driven by wiener processes and poisson jumps to achieve this a key step is to show the existence for an invariant measure with exponentially ergodic property for the fast equation where the dissipative conditions are needed furthermore it is shown that under suitable assumptions the slow component l p strongly converges to the solution of the averaged equation the rate of the convergence is also obtained as a byproduct

averaging principle for one dimensional stochastic burgers equation abstract in this paper we consider the averaging principle for one dimensional stochastic burgers equation with slow and fast time scales under some suitable conditions we show that the slow component strongly converges to the solution of the corresponding averaged equation meanwhile when there is no noise in the slow component equation we also prove that the slow component weakly converges to the solution of the corresponding averaged equation with the order of convergence 1 r for any 0 r 1

on analyticity of linear waves scattered by a layered medium abstract the scattering of linear waves by periodic structures is a crucial phenomena in many branches of applied physics and engineering in this paper we establish rigorous analytic results necessary for the proper numerical analysis of a class of high order perturbation of surfaces methods for simulating such waves more specifically we prove a theorem on existence and uniqueness of solutions to a system of partial differential equations which model the interaction of linear waves with a multiply layered periodic structure in three dimensions this result provides hypotheses under which a rigorous numerical analysis could be conducted for recent generalizations to the methods of operator expansions field expansions and transformed field expansions

one dimensional stochastic burgers equation driven by levy processes abstract in this paper we proved the global existence and uniqueness of the strong weak and mild solutions for one dimensional burgers equation perturbed by a poisson form process a poisson form and q wiener process with the dirichlet bounded condition we also proved the existence of the invariant measure of these models

s rock chebyshev methods for stiff stochastic differential equations we present and analyze a new class of numerical methods for the solution of stiff stochastic differential equations sdes these methods called s rock for stochastic orthogonal runge kutta chebyshev are explicit and of strong order 1 and possess large stability domains in the mean square sense for mean square stable stiff sdes they are much more efficient than the standard explicit methods proposed so far for stochastic problems and give significant speed improvement the explicitness of the s rock methods allows one to handle large systems without linear algebra problems usually encountered with implicit methods numerical results and comparisons with existing methods are reported

response to comments on reflection from a corrugated surface revisited j acoust soc am 96 1116 1129 1994 the comments of kazandjian j acoust soc am 98 1813 1814 1995 concerning the paper j acoust soc am 96 1116 1129 1994 are discussed notably those concerning the equivalence of the rayleigh and extinction methods the identity and convergence of perturbation series deriving from these methods and the pertinence of numerical computations as indicators of the validity of theoretical predictions

sensor scheme design for active structural acoustic control efficient sensing schemes for the active reduction of sound radiation from plates are presented based on error signals derived from spatially weighted plate velocity or near field pressure the schemes result in near optimal reductions as compared to weighting procedures derived from eigenvector or singular vector analysis of the radiation operator efficient control configurations are suggested using a possibly analog front end implementing a bank of spatial weighting functions and a digital controller with a minimized number of input and output channels the performance of different weighting functions is compared as well as the performance of different frequency dependent filtering functions design rules are given for the sensor spacing the number of weighting functions the number of actuators and the corresponding controller dimensionality

electromagnetic scattering by rough conducting circular cylinders ii axial corrugation for pt i see ibid vol 36 no 5 p 651 1988 the coherent and incoherent responses of conducting axially corrugated surfaces which are on the average circular cylinders are calculated the calculation encompasses second order perturbation the general approach used is to apply the extinction theorem to write an integral equation for the surface current density and then to use the perturbation theory to solve it the results are recast by using pade approximants to achieve a significant extension of the domain of validity of the original formulation

ultrasound wave propagation through rough interfaces iterative methods two iterative methods for the calculation of acoustic transmission through a rough interface between two media are compared the methods employ a continuous version of the conjugate gradient technique one method is based on plane wave expansions and the other on boundary integral equations and green s functions a preconditioner is presented which improves the convergence for spectra that include evanescent modes the methods are compared with regard to computational efficiency rate of convergence and residual error the sound field differences are determined for a focused ultrasound beam distorted by surfaces having a gaussian roughness spectrum the differences are evaluated from the root mean square differences on the rough surface and in the focal plane

strong convergence rate in averaging principle for stochastic fitzhugh nagumo system with two time scales abstract this article deals with averaging principle for stochastic fitzhugh nagumo system with different time scales under suitable conditions the existence of an averaging equation eliminating the fast variable for this coupled system is proved and as a consequence the system can be reduced to a single stochastic ordinary equation with a modified coefficient moreover the rate of convergence for the slow component towards the solution of the averaging equation is of order 1 2

single point blowup for nonlocal parabolic problems abstract the formation of shear bands in materials has important implications to a variety of physical processes these bands are observed in very thin zones and are generally regarded as a precursor to material failure shear band formation is caused by the heat generated in regions with highest strain rate with insufficient time for diffusion of this heat a localized thermal softening of the material occurs which enhances plastic flow in a thin zone this adiabatic strain localization can be modelled as nonlinear thermally activated reaction diffusion equations this leads to a class of nonlocal parabolic problems and their associated time independent steady state counterparts a discussion of the models and an analysis of the problems will be presented

optimal prediction of underresolved dynamics abstract a method is presented for computing the average solution of problems that are too complicated for adequate resolution but where information about the statistics of the solution is available the method involves computing average derivatives by interpolation based on linear regression and an updating of a measure constrained by the available crude information examples are given

analysis of multiscale methods for stochastic differential equations we analyze a class of numerical schemes proposed in 26 for stochastic differential equations with multiple time scales both advective and diffusive time scales are considered weak as well as strong convergence theorems are proven most of our results are optimal they in turn allow us to provide a thorough discussion on the efficiency as well as optimal strategy for the method c 2005 wiley periodicals inc

a computational strategy for multiscale systems with applications to lorenz 96 model numerical schemes for systems with multiple spatio temporal scales are investigated the multiscale schemes use asymptotic results for this type of systems which guarantee the existence of an effective dynamics for some suitably defined modes varying slowly on the largest scales the multiscale schemes are analyzed in general then illustrated on a specific example of a moderately large deterministic system displaying chaotic behavior due to lorenz issues like consistency accuracy and efficiency are discussed in detail the role of possible hidden slow variables as well as additional effects arising on the diffusive time scale are also investigated as a byproduct we obtain a rather complete characterization of the effective dynamics in lorenz model

analysis of the heterogeneous multiscale method for elliptic homogenization problems the heterogeneous multiscale method hmm is applied to various parabolic problems with multiscale coefficients these problems can be either linear or nonlinear optimal estimates are proved for the error between the hmm solution and the homogenized solution

stability of high order perturbative methods for the computation of dirichlet neumann operators abstract in this paper we present results on the stability of perturbation methods for the evaluation of dirichlet neumann operators dno defined on domains that are viewed as complex deformations of a basic simple geometry in such cases geometric perturbation methods based on variations of the spatial domains of definition have long been recognized to constitute efficient and accurate means for the approximation of dno and in fact several numerical implementations have been previously proposed inspired by our recent analytical work here we demonstrate that the convergence of these algorithms is quite generally limited by numerical instability indeed we show that these standard perturbative methods for the calculation of dno suffer from significant ill conditioning which is manifest even for very smooth boundaries and whose severity increases with boundary roughness moreover and again motivated by our previous work we introduce an alternative perturbative approach that we show to be numerically stable this approach can be interpreted as a reformulation of classical perturbative algorithms in suitable independent variables and thus it allows for both direct comparison and the possibility of analytic continuation of the perturbation series it can also be related to classical preconditioned spectral approaches and as such it retains in finite arithmetic the spectral convergence properties of classical perturbative methods albeit at a higher computational cost as it does not take advantage of possible dimensional reductions still as we show an alternative approach such as the one we propose may be mandated in cases where substantial information is contained in high order harmonics and or perturbation coefficients of the solution

a mathematical framework for stochastic climate models there has been a recent burst of activity in the atmosphere ocean sciences community in utilizing stable linear langevin stochastic models for the unresolved degrees of freedom in stochastic climate prediction here a systematic mathematical strategy for stochastic climate modeling is developed and some of the new phenomena in the resulting equations for the climate variables alone are explored the new phenomena include the emergence of both unstable linear langevin stochastic models for the climate mean variables and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances all of these phenomena are derived from a systematic self consistent mathematical framework for eliminating the unresolved stochastic modes that is mathematically rigorous in a suitable asymptotic limit the theory is illustrated for general quadratically nonlinear equations where the explicit nature of the stochastic climate modeling procedure can be elucidated the feasibility of the approach is demonstrated for the truncated equations for barotropic flow with topography explicit concrete examples with the new phenomena are presented for the stochastically forced three mode interaction equations the conjecture of smith and waleffe phys fluids 11 1999 1608 1622 for stochastically forced three wave resonant equations in a suitable regime of damping and forcing is solved as a byproduct of the approach examples of idealized climate models arising from the highly inhomogeneous equilibrium statistical mechanics for geophysical flows are also utilized to demonstrate self consistency of the mathematical approach with the predictions of equilibrium statistical mechanics in particular for these examples the reduced stochastic modeling procedure for the climate variables alone is designed to reproduce both the climate mean and the energy spectrum of the climate variables c 2001 john wiley sons inc

comments on reflection from a corrugated surface revisited j acoust soc am 96 1116 1129 1994 in the subject paper chesneaux and wirgin use stochastic arithmetic to investigate the equivalence of the rayleigh and extinction theorem methods implemented by the fourier scheme the purpose here is to discuss remarks made by the authors about the following points the interpretation of the result of a previous paper dealing with the problem a conjecture about the identity of the perturbation series derived from the two methods an assertion about the numerical convergence of the waterman fourier method

ergodicity of 2d navier stokes equations with random forcing and large viscosity the stochastically forced two dimensional incompressable navier stokes equations are shown to possess an unique invariant measure if the viscosity is taken large enough this result follows from a stronger result showing that at high viscosity there is a unique stationary solution which attracts solutions started from arbitrary initial conditions that is to say the system has a trivial random attractor along the way results controling the expectation and averaging time of the energy and enstrophy are given

the validity of the perturbation approximation for rough surface scattering using a gaussian roughness spectrum the validity of the perturbation approximation for rough surface scattering is examined 1 by comparison with exact results obtained by solving an integral equation and 2 through comparison of low order perturbation predictions with higher order predictions the pressure release boundary condition is assumed and the field quantity calculated is the bistatic scattering cross section a gaussian roughness spectrum is used and the surfaces have height variations in only one direction it is found in general that the condition kh 1 k is the acoustic wavenumber h is the rms surface height is insufficient to guarantee the accuracy of first order or higher order perturbation theory when the surface correlation length l becomes too large or too small with h held fixed higher order perturbation terms can make larger contributions to the scattering cross section than lower order terms an explanation for this result is given the regions of validity for low order perturbation theory are also given th

electromagnetic scattering by rough conduction circular cylinders i angular corrugation the coherent and incoherent responses of perfectly conducting angularly corrugated surfaces which are infinite circular cylinders on the average are calculated using a second order perturbation method the general approach used is to apply the extinction theorem to write an integral equation for the surface current density and then to use perturbation theory to solve it these results are recast by using the only nontrivial pade approximant for this approximation this recasting yield a considerable qualitative improvement which at least in one special case is shown to be quantitative as well

on convergence of higher order schemes for the projective integration method for stiff ordinary differential equations we present a convergence proof for higher order implementations of the projective integration method pi for a class of deterministic multi scale systems in which fast variables quickly settle on a slow manifold the error is shown to contain contributions associated with the length of the microsolver the numerical accuracy of the macrosolver and the distance from the slow manifold caused by the combined effect of micro and macrosolvers respectively we also provide stability conditions for the pi methods under which the fast variables will not diverge from the slow manifold we corroborate our results by numerical simulations

unresolved computation and optimal predictions we present methods for predicting the solution of time dependent partial differential equations when that solution is so complex that it cannot be properly resolved numerically but when prior statistical information can be found the sparse numerical data are viewed as constraints on the solution and the gist of our proposal is a set of methods for advancing the constraints in time so that regression methods can be used to reconstruct the mean future for linear equations we offer general recipes for advancing the constraints the methods are generalized to certain classes of nonlinear problems and the conditions under which strongly nonlinear problems and partial statistical information can be handled are briefly discussed our methods are related to certain data acquisition schemes in oceanography and meteorology c 1999 john wiley sons inc

strong averaging principle for two time scale sdes with non lipschitz coefficients abstract this paper deals with averaging principle for two time scale stochastic differential equations sdes with non lipschitz coefficients which extends the existing results from lipschitz to non lipschitz case under suitable conditions the existence of an averaging equation eliminating the fast variable for coupled system is established and as a result the system can be reduced to a single sdes with a modified coefficient which is also non lipschitz moreover it is shown that the slow variable strongly converges to the solution of the corresponding averaging equation

deciding the nature of the coarse equation through microscopic simulations the baby bathwater scheme recent developments in multiscale computation allow the solution of coarse equations for the expected macroscopic behavior of microscopically evolving particles without ever obtaining these coarse equations in closed form the closure is obtained on demand through appropriately initialized bursts of microscopic simulation the effective coupling of microscopic simulators with macroscopic behavior requires certain decisions about the nature of the unavailable coarse equation such decisions include a the highest spatial derivative active in the coarse equation b whether the equation satisfies certain conservation laws or c whether the coarse dynamics is hamiltonian or dissipative these decisions affect the number and type of boundary conditions as well as the algorithms employed in the absence of an explicit formula for the temporal derivative we propose implement and validate a simple scheme for deciding these and other similar questions about the coarse equation using only the microscopic simulator simulations under periodic boundary conditions are carried out for appropriately chosen families of random initial conditions evaluating the sample variance of certain statistics over the simulation ensemble allows us to infer the highest order of spatial derivatives active in the coarse equation in the same spirit we show how to determine whether a certain coarse conservation law exists or not and we discuss plausibility tests for the existence of a coarse hamiltonian or integrability we believe that such schemes constitute an important part of the equation free approach to multiscale computation

a reduced order representation of the poincare steklov operator an application to coupled multi physics problems this work investigates a model reduction method applied to coupled multi physics systems the case in which a system of interest interacts with an external system is considered an approximation of the poinste operator is computed by simulating in an offline phase the external problem when the inputs are the laplace beltrami eigenfunctions defined at the interface in the online phase only the reduced representation of the operator is needed to account for the influence of the external problem on the main system an online basis enrichment is proposed in order to guarantee a precise reduced order computation several test cases are proposed on different fluid structure couplings

comparison of perturbation theories for rough surface scattering rayleigh rice perturbation theory is compared with the more rigorously derived perturbation theory based on the extinction theorem or extended boundary condition for the case of dirichlet boundary conditions numerical calculations for a sinusoidal surface profile show that the two theories give identical results when carried out to high order this is true even when the two series do not seem to converge numerically numerical convergence of the two series is found in several cases including some cases in which the rayleigh hypothesis is invalid for surfaces of arbitrary shape the two perturbation series appear superficially to differ at third and higher order it is shown however that a reciprocity transformation of one series makes it identical to the other through fifth order finally it is shown that reciprocity holds in both perturbation theories establishing analytically the equivalence of the two theories at least through fifth order these results are surprising given the limited regime

efficient enforcement of far field boundary conditions in the transformed field expansions method the method of transformed field expansions tfe has been demonstrated to be a robust and highly accurate numerical scheme for simulating solutions of boundary value and free boundary problems from the sciences and engineering as a boundary perturbation method it builds highly accurate solutions based upon exact solutions in a simple canonical geometry and corrects these via taylor series to fit the actual geometry at hand the tfe method has significantly enhanced stability properties when compared with other boundary perturbation approaches however this comes at the cost of requiring a full volumetric discretization as opposed the surface formulation that other methods can realize in this paper we outline two techniques for ameliorating this shortcoming first by employing a legendre spectral element method to implement efficient graded meshes and second by utilizing an artificial boundary with a transparent boundary condition placed quite close to the boundary of the domain in this contribution we focus on the specific problem of simulating the dirichlet neumann operator associated to laplace s equation on a periodic cell which arises in the water wave problem while the details of our results are specific to this problem the general conclusions are valid for the wider class of problems to which the tfe method can be applied for each technique we discuss implementation details and display numerical results which support the conclusion that each of these techniques can greatly reduce the computational cost of using the tfe method

nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales we present an efficient numerical algorithm for simulating chemical kinetic systems with multiple time scales this algorithm is an improvement of the traditional stochastic simulation algorithm ssa also known as gillespie s algorithm it is in the form of a nested ssa and uses an outer ssa to simulate the slow reactions with rates computed from realizations of inner ssas that simulate the fast reactions the algorithm itself is quite general and seamless and it amounts to a small modification of the original ssa our analysis of such multi scale chemical kinetic systems allows us to identify the slow variables in the system derive effective dynamics on the slow time scale and provide error estimates for the nested ssa efficiency of the nested ssa is discussed using these error estimates and illustrated through several numerical examples

blow up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition abstract in this paper we consider a semilinear heat equation u t u c x t u p for x t 0 with nonlinear and nonlocal boundary condition u 0 k x y t u l d y and nonnegative initial data where p 0 and l 0 we prove global existence theorem for max p l 1 some criteria on this problem which determine whether the solutions blow up in a finite time for sufficiently large or for all nontrivial initial data or the solutions exist for all time with sufficiently small or with any initial data are also given

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

comparison of the rayleigh fourier and extinction theorem methods applied to scattering and transmission at a rough solid solid interface the rayleigh fourier and extinction theorem mathematical formalisms are established and compared in the case of scattering from and transmission through a rough solid solid interface by using the reciprocity properties of the solution resulting from the extinction theorem formalism it is shown that if they have unique solutions the two formalisms are theoretically equivalent when used to obtain the displacement fields beyond the maximum excursions of the rough interface as a result the perturbation approaches of the two formalisms yield identical results from a numerical point of view it is shown that the two methods can give close results regardless of the verification of the rayleigh hypothesis the closeness of the two approaches which has already been established and numerically confirmed in the case of a fluid solid interface does not seem to be known or even conjectured in the literature in the case of a solid solid interface it is indicated that with the mathematics included in this pap

averaging principle for a class of stochastic reaction diffusion equations we consider the averaging principle for stochastic reaction diffusion equations under some assumptions providing existence of a unique invariant measure of the fast motion with the frozen slow component we calculate limiting slow motion the study of solvability of kolmogorov equations in hilbert spaces and the analysis of regularity properties of solutions allow to generalize the classical approach to finite dimensional problems of this type in the case of spde s

reflection from a corrugated surface revisited the problem of scattering of a plane sonic wave from a soft surface with periodic sinusoidal unevenness along one direction is examined by means of the rayleigh plane wave expansion and the waterman extinction methods numerically implemented by fourier projection and expansion respectively the computations are done with real double precision stochastic arithmetic instead of the usual complex double precision floating point arithmetic in order to precisely evaluate the numerical accuracy of the results conditioned by round off errors it is shown that the low order plane wave coefficients obtained by the rayleigh and waterman methods are identical when obtained from matrix systems that are large enough to give convergence of these coefficients for the same matrix size the higher order coefficients differ the higher the diffraction order it is also shown that the waterman fourier series computation of the near field is generally meaningful whereas that of rayleigh involving summation of t

averaging principle for the higher order nonlinear schrodinger equation with a random fast oscillation this work concerns the problem associated with averaging principle for a higher order nonlinear schrodinger equation perturbed by a oscillating term arising as the solution of a stochastic reaction diffusion equation evolving with respect to the fast time this model can be translated into a multiscale stochastic partial differential equations stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time scales to be more precise under suitable conditions we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the higher order nonlinear schrodinger equation is an average with respect to the stationary measure of the fast varying process finally by using the khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation and as a consequence the system can be reduced to a single higher order nonlinear schrodinger equation with a modified coefficient

acoustic scattering from a rough sphere the scattering of a plane acoustic wave from a random rough sphere is studied using the null field approach the starting point is an integral representation derived from the helmholtz equation the incident field the scattered field and the free space green s function are all expanded in terms of suitably chosen basis functions in this way a relation between the incident and the scattered fields is obtained which is expressed by the transition matrix t matrix the scattered field is expanded in a power series of a small parameter namely the product of the wave number and the root mean square height of the irregularities analytical expressions for the leading terms of the series have been calculated in particular ensemble averages of the far field amplitude and the scattering cross section have been determined as the analytical results are somewhat complicated some numerical results are presented numerical computations of higher order terms indicate that the convergence of the series is satisfactory as long as the root mean square height is small compared to a correlation length describing the average distance between the peaks of the surface

stochastic burgers equations and their semi classical expansions in this paper we use the hopf cole logarithmic transformation and the stochastic hamilton jacobi theory to study stochastic heat equations and burgers equations before the caustics the stochastic inviscid burgers equation gives the first term of our semi classical expansion i e the inviscid limit of the viscous stochastic burgers equation in order to push our results beyond the inviscid limit we construct solutions for iterated stochastic hamilton jacobi continuity equations then we give the semi classical asymptotic expansions for stochastic heat equations and burgers equations by using nelson s stochastic mechanical processes with drifts given by the solution of the iterated stochastic hamilton jacobi continuity equations the explicit formula for the remainder term is given by a path integral

strong convergence in stochastic averaging principle for two time scales stochastic partial differential equations abstract the theory of stochastic averaging principle provides an effective approach for the qualitative analysis of stochastic systems with different time scales and is relatively mature for stochastic ordinary differential equations in this paper we study the averaging principle for a class of stochastic partial differential equations with two separated time scales driven by scalar noises under suitable assumptions it is shown that the slow component strongly converges to the solution of the corresponding averaged equation

projective methods for stiff differential equations problems with gaps in their eigenvalue spectrum we show that there exist classes of explicit numerical integration methods that can handle very stiff problems if the eigenvalues are separated into two clusters one containing the stiff or fast components and one containing the slow components these methods have large average step sizes relative to the fast components conventional implicit methods involve the solution of nonlinear equations at each step which for large problems requires significant communication between processors on a multiprocessor machine for such problems the methods proposed here have significant potential for speed improvement

on stochastic diffusion equations and stochastic burgers equations in this paper we construct a strong solution for the stochastic hamilton jacobi equation by using stochastic classical mechanics before the caustics we thereby obtain the viscosity solution for a certain class of inviscid stochastic burgers equations this viscosity solution is not continuous beyond the caustics of the corresponding hamilton jacobi equation the hopf cole transformation is used to identify the stochastic heat equation and the viscous stochastic burgers equation the exact solutions for the above two equations are given in terms of the stochastic hamilton jacobi function under a no caustic condition we construct the heat kernel for the stochastic heat equation for zero potentials in hyperbolic space and for harmonic oscillator potentials in euclidean space thereby obtaining the stochastic mehler formula

a multiscale method for highly oscillatory ordinary differential equations with resonance a multiscale method for computing the effective behavior of a class of stiff and highly oscillatory ordinary differential equations odes is presented the oscillations may be in resonance with on

traveling water waves we present a numerical continuation method for traveling wave solutions to the full water wave problem using a spectral collocation discretization the water wave problem is reformulated in terms of surface variables giving rise to the zakharov craig sulem formulation and traveling waves are studied by introducing a phase velocity vector as a parameter we follow non trivial solution branches bifurcating from the trivial solution branch via numerical continuation methods techniques such as projections and filtering allow the computation to proceed for greater distances up the branch and parallelism allows the computation of larger problems we conclude with results including the formation of hexagonal patterns for the three dimensional problem

general perturbative solution to wave scattering from a soft random cylindrical surface a perturbative solution to the problem of plane wave scattering from a soft randomly rough cylindrical surface is obtained by use of the rayleigh hypothesis and the assumption that coupled radiation modes can be present in the scattered wave the wave amplitudes of the cylindrical wave expansion of the scattered field are expressed as an asymptotic series in the root mean square of surface irregularities the end result of the analysis is a concise recursive formula for the coefficients of that series the solution is energy consistent up to fourth order in the smallness parameter specific information about the validity and applicability of the solution is given by use of the mean boundary condition error which is defined in this paper with a view to assessing the conformity of near field results with the boundary condition the numerical results show that the fourth order approximation is more accurate than the second order one but it may under certain conditions exhibit higher mean boundary condition error

importance sampling in path space for diffusion processes with slow fast variables importance sampling is a widely used technique to reduce the variance of a monte carlo estimator by an appropriate change of measure in this work we study importance sampling in the framework of diffusion process and consider the change of measure which is realized by adding a control force to the original dynamics for certain exponential type expectation the corresponding control force of the optimal change of measure leads to a zero variance estimator and is related to the solution of a hamilton jacobi bellmann equation we focus on certain diffusions with both slow and fast variables and the main result is that we obtain an upper bound of the relative error for the importance sampling estimators with control obtained from the limiting dynamics we demonstrate our approximation strategy with an illustrative numerical example

an improved formalism for wave scattering from rough surfaces the scattered field and its normal gradient obey a mutual linear relation at the scattering surface that is distinct from the physical boundary condition connecting either quantity to the incident radiation on a moderately rough surface this relation can be represented by an operator whose series expansion in surface slope converges nicely even for large values of the rayleigh parameter this allows the helmholtz integral for scattering amplitude to be written as a series of readily computable terms one or two of which provide good approximations for surfaces too irregular for the usual bragg expansion this formulation reproduces the bragg series for small surface elevations and in the limit of low roughness wavenumber gives the kirchhoff approximation with an explicit correction term in surface curvature the usual results for a composite surface also emerge naturally on several test profiles the method produces better overall accuracy than other multiscale approximations at comparable efficiency

strong averaging principle for two time scale non autonomous stochastic fitzhugh nagumo system with jumps in this paper we study an averaging principle for stochastic fitzhugh nagumo system with different time scales driven by cylindrical wiener processes and poisson jumps where the slow equation is non autonomous and the fast equation is autonomous case under suitable assumptions we show that the slow component mean square strongly converges to the solution of the corresponding averaging equation and the rate of the convergence as a by product is also affirmed finally we give some open problems which are derived from this paper

a dirichlet problem under integral boundary condition abstract in this paper we establish some results concerning existence and multiplicity of solution for a elliptic problem under integral dirichlet boundary condition we mainly use a lower and upper solution method and a result due to rabinowitz 11 related to the existence of a continuum of solutions of an abstract nonlinear eigenvalue problem

model reduction of multi scale chemical langevin equations this paper addresses the model reduction problem for a class of stiff chemical langevin equations that arise as models of biomolecular networks with fast and slow reactions and can be described as continuous markov processes initially a coordinate transformation is sought that allows the decoupling of fast and slow variables in the model equations necessary and sufficient conditions are derived for such a linear transformation to exist along with an explicit change of variables which achieves the desired decoupling for the systems for which this step is applicable the method of adiabatic elimination is applied to determine a representation of the slow dynamics theoretical concepts and results are illustrated with simple examples

averaging principle and systems of singularly perturbed stochastic differential equations by developing certain auxiliary results a modified version of the stochastic averaging principle is developed to investigate dynamical systems consisting of fast and slow phenomena moreover an attempt is made to establish a relationship between the averaging assumption and certain ergodic type properties of the random process determined by an auxiliary system of stochastic differential equations finally an example is given to illustrate the scope of the results

a new framework for extracting coarse grained models from time series with multiscale structure in many applications it is desirable to infer coarse grained models from observational data the observed process often corresponds only to a few selected degrees of freedom of a high dimensional dynamical system with multiple time scales in this work we consider the inference problem of identifying an appropriate coarse grained model from a single time series of a multiscale system it is known that estimators such as the maximum likelihood estimator or the quadratic variation of the path estimator can be strongly biased in this setting here we present a novel parametric inference methodology for problems with linear parameter dependency that does not suffer from this drawback furthermore we demonstrate through a wide spectrum of examples that our methodology can be used to derive appropriate coarse grained models from time series of partial observations of a multiscale system in an effective and systematic fashion

numerical simulation of gravity waves abstract we present a new spectral method to simulate numerically the waterwave problem in a channel for a fluid of finite or infinite depth it is formulated in terms of the free surface elevation and the velocity potential the numerical method is based on the reduction of this problem to a lower dimensional computation involving surface variables alone to accomplish this we describe the taylor expansion of the dirichlet neumann operator in homogeneous powers of the surface elevation each term is a concatenation of fourier multipliers with powers of and its derivatives and is valid uniformly in wavenumber these are easily calculated using the fast fourier transform the method is illustrated by computing the long time evolution of modulated wave packets and of approximations to the stokes steady wave train by imposing a surface pressure we observe surface steepening in large amplitude evolution and wake and bow wave development for flows with a close to critical froude number finally we give an example of nonlinear evolution of the distribution of energy among normal modes

variance reduction for particle filters of systems with time scale separation we present a particle filter construction for a system that exhibits time scale separation the separation of time scales allows two simplifications that we exploit 1 the use of the averaging principle for the dimensional reduction of the dynamics for each particle during the prediction step and 2 the factorization of the transition probability for the rao blackwellization of the update step the resulting particle filter is faster and has smaller variance than the particle filter based on the original system the method is tested on a multiscale stochastic differential equation and on a multiscale pure jump diffusion motivated by chemical reactions

analytic continuation of dirichlet neumann operators the analytic dependence of dirichlet neumann operators dno with respect to variations of their domain of definition has been successfully used to devise diverse computational strategies for their estimation these strategies have historically proven very competitive when dealing with small deviations from exactly solvable geometries as in this case the perturbation series of the dno can be easily and recursively evaluated in this paper we introduce a scheme for the enhancement of the domain of applicability of these approaches that is based on techniques of analytic continuation we show that in fact dno depend analytically on variations of arbitrary smooth domains in particular this implies that they generally remain analytic beyond the disk of convergence of their power series representations about a canonical separable geometry and this in turn guarantees that alternative summation mechanisms such as pade approximation can be effectively used to numerically access this extended domain of analyticity our method of proof is motivated by our recent development of stable recursions for the coefficients of the perturbation series here we again utilize this recursion as we compare and contrast the performance of our new algorithms with that of previously advanced perturbative methods the numerical results clearly demonstrate the beneficial effect of incorporating analytic continuation procedures into boundary perturbation methods moreover the results also establish the superior accuracy and applicability of our new approach which as we show allows for precise calculations corresponding to very large perturbations of a basic geometry

limit behavior of two time scale diffusions revisited abstract this work is concerned with diffusions with two time scales or singularly perturbed diffusions asymptotic expansions of the solution of the associated cauchy problem for parabolic partial differential equation are obtained and the desired error bounds are derived these asymptotic expansions are then used to analyze related limit distributions of normalized integral functionals

numerical methods for stochastic partial differential equations with multiple scales a new method for solving numerically stochastic partial differential equations spdes with multiple scales is presented the method combines a spectral method with the heterogeneous multiscale method hmm presented in w e d liu e vanden eijnden analysis of multiscale methods for stochastic differential equations commun pure appl math 58 11 2005 1544 1585 the class of problems that we consider are spdes with quadratic nonlinearities that were studied in d blomker m hairer g a pavliotis multiscale analysis for stochastic partial differential equations with quadratic nonlinearities nonlinearity 20 7 2007 1721 1744 for such spdes an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales our method based on micro and macro solvers allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem numerical experiments illustrate the behavior of the proposed method

parameter estimation for multiscale diffusions we study the problem of parameter estimation for time series possessing two widely separated characteristic time scales the aim is to understand situations where it is desirable to fit a homogenized single scale model to such multiscale data we demonstrate numerically and analytically that if the data is sampled too finely then the parameter fit will fail in that the correct parameters in the homogenized model are not identified we also show numerically and analytically that if the data is subsampled at an appropriate rate then it is possible to estimate the coefficients of the homogenized model correctly the ideas are studied in the context of thermally activated motion in a two scale potential however the ideas may be expected to transfer to other situations where it is desirable to fit an averaged or homogenized equation to multiscale data

temporal upscaling in micromagnetism via heterogeneous multiscale methods abstract we consider a multiscale strategy addressing the disparate scales in the landau lifschitz equations in micromagnetism at the microscopic scale the dynamics of magnetic moments are driven by a high frequency field on the macroscopic scale we are interested in simulating the dynamics of the magnetisation without fully resolving the microscopic scales the method follows the framework of heterogeneous multiscale methods and it has two main ingredients a micro and a macroscale model the microscopic model is assumed to be known exactly whereas the macromodel is incomplete as it lacks effective quantities the two models use different temporal and spatial scales and effective parameter values for the macromodel are computed on the fly allowing for improved efficiency over traditional one scale schemes for the analysis we consider a single spin under a high frequency field and show that effective quantities can be obtained accurately with step sizes much larger than the size of the microscopic scales required to resolve the microscopic features numerical results both for a single magnetic particle as well as a chain of interacting magnetic particles are given to validate the theory

an operator expansion method for computing nonlinear surface waves on a ferrofluid jet we present a new numerical method to simulate the time evolution of axisymmetric nonlinear waves on the surface of a ferrofluid jet it is based on the reduction of this problem to a lower dimensional computation involving surface variables alone to do so we describe the associated dirichlet neumann operator in terms of a taylor series expansion where each term can be efficiently computed by a pseudo spectral scheme using the fast fourier transform we show detailed numerical tests on the convergence of this operator and to illustrate the performance of our method we simulate the long time propagation and pairwise collisions of axisymmetric solitary waves both depression and elevation waves are examined by varying the magnetic field comparisons with weakly nonlinear predictions are also provided accurate and efficient numerical solver for the full nonlinear problem reduction of the original problem to a lower dimensional system of equations series expansion for the dirichlet neumann operator in the axisymmetric case study of head on and overtaking collisions of axisymmetric solitary waves

an operator expansions method for computing dirichlet neumann operators in linear elastodynamics abstract the propagation of linear elastic waves arises in a wide array of applications for instance in mechanical engineering materials science and the geosciences many configurations of interest can be effectively modeled as layers of isotropic homogeneous materials separated by thin interfaces across which material properties vary rapidly in the frequency domain one must solve a system of coupled elliptic partial differential equations however this can be greatly simplified in the instance of layered media by considering interface unknowns to realize this one must be able to produce normal stresses tractions at these interfaces and dirichlet neumann operators accomplish this in this contribution we discuss a novel boundary perturbation approach to compute these operators in a rapid high order and robust fashion