Search

Dust grains play a significant role in several astrophysical processes, including gas/dust dynamics, chemical reactions, and radiative transfer. Replenishment of small-grain populations is mainly governed by fragmentation during pair-wise collisions between grains. The wide spectrum of fragmentation outcomes, from complete disruption to erosion and/or mass transfer, can be modelled by the general non-linear fragmentation equation. Efficiently solving this equation is crucial for an accurate treatment of the dust fragmentation in numerical modelling. However, similar to dust coagulation, numerical errors in current fragmentation algorithms employed in astrophysics are dominated by the numerical over-diffusion problem -- particularly in 3D hydrodynamic simulations where the discrete resolution of the mass density distribution tends to be highly limited. With this in mind, we have derived the first conservative form of the general non-linear fragmentation with a mass flux highlighting the mass transfer phenomenon. Then, to address cases of limited mass density resolution, we applied a high-order discontinuous Galerkin scheme to efficiently solve the conservative fragmentation equation with a reduced number of dust bins. An accuracy of 0.1 -1% is reached with 20 dust bins spanning a mass range of 9 orders of magnitude., Comment: Accepted for publication in MNRAS

Motivated by the modeling of the spatial structure of the velocity field of three-dimensional turbulent flows, and the phenomenology of cascade phenomena, a linear dynamics has been recently proposed able to generate high velocity gradients from a smooth-in-space forcing term. It is based on a linear Partial Differential Equation (PDE) stirred by an additive random forcing term which is delta-correlated in time. The underlying proposed deterministic mechanism corresponds to a transport in Fourier space which aims at transferring energy injected at large scales towards small scales. The key role of the random forcing is to realize these transfers in a statistically homogeneous way. Whereas at finite times and positive viscosity the solutions are smooth, a loss of regularity is observed for the statistically stationary state in the inviscid limit. We here present novel simulations, based on finite volume methods in the Fourier domain and a splitting method in time, which are more accurate than the pseudo-spectral simulations. We show that the novel algorithm is able to reproduce accurately the expected local and statistical structure of the predicted solutions. We conduct numerical simulations in one, two and three spatial dimensions, and we display the solutions both in physical and Fourier spaces. We additionally display key statistical quantities such as second-order structure functions and power spectral densities at various viscosities., Comment: 22 pages, 7 figures

We consider a class of linear Vlasov partial differential equations driven by Wiener noise. Different types of stochastic perturbations are treated: additive noise, multiplicative It\^o and Stratonovich noise, and transport noise. We propose to employ splitting integrators for the temporal discretization of these stochastic partial differential equations. These integrators are designed in order to preserve qualitative properties of the exact solutions depending on the stochastic perturbation, such as preservation of norms or positivity of the solutions. We provide numerical experiments in order to illustrate the properties of the proposed integrators and investigate mean-square rates of convergence.

We study a class of stochastic semilinear damped wave equations driven by additive Wiener noise. Owing to the damping term, under appropriate conditions on the nonlinearity, the solution admits a unique invariant distribution. We apply semi-discrete and fully-discrete methods in order to approximate this invariant distribution, using a spectral Galerkin method and an exponential Euler integrator for spatial and temporal discretization respectively. We prove that the considered numerical schemes also admit unique invariant distributions, and we prove error estimates between the approximate and exact invariant distributions, with identification of the orders of convergence. To the best of our knowledge this is the first result in the literature concerning numerical approximation of invariant distributions for stochastic damped wave equations.

We introduce a positivity-preserving numerical scheme for a class of nonlinear stochastic heat equations driven by a purely time-dependent Brownian motion. The construction is inspired by a recent preprint by the authors where one-dimensional equations driven by space-time white noise are considered. The objective of this paper is to illustrate the properties of the proposed integrators in a different framework, by numerical experiments and by giving convergence results.

We construct a positivity-preserving Lie--Trotter splitting scheme with finite difference discretization in space for approximating the solutions to a class of nonlinear stochastic heat equations with multiplicative space-time white noise. We prove that this explicit numerical scheme converges in the mean-square sense, with rate $1/4$ in time and rate $1/2$ in space, under appropriate CFL conditions. Numerical experiments illustrate the superiority of the proposed numerical scheme compared with standard numerical methods which do not preserve positivity.

In this article, we study numerically the dispersion of colloids in a two-dimensional cellular flow in the presence of an imposed mean salt gradient. Owing to the additional scalar, the colloids do not follow exactly the Eulerian flow field, but have a (small) extra-velocity proportional to the salt gradient, $\mathbf{v}_\mathrm{dp}=\alpha\nabla S$, where $\alpha$ is the phoretic constant and $S$ the salt concentration. We study the demixing of an homogenous distribution of colloids and how their long-term mean velocity $\mathbf{V_m}$ and effective diffusivity $D_\mathrm{eff}$ are influenced by the phoretic drift. We observe two regimes of colloids dynamics depending on a blockage criterion $R=\alpha G L/\sqrt{4 D_cD_s}$, where $G$ is the mean salt gradient amplitude, $L$ the length scale of the flow and $D_c$ and $D_s$ the molecular diffusivities of colloids and salt. When $R<1$, the mean velocity is strongly enhanced with $V_m \propto \alpha G \sqrt{Pe_s}$, $Pe_s$ being the salt P\'eclet number. When $R> 1$, the compressibility effect due to the phoretic drift is so strong that a depletion of colloids occurs along the separatrices inhibiting cell-to-cell transport.

We study a family of numerical schemes applied to a class of multiscale systems of stochastic differential equations. When the time scale separation parameter vanishes, a well-known homogenization or Wong--Zakai diffusion approximation result states that the slow component of the considered system converges to the solution of a stochastic differential equation driven by a real-valued Wiener process, with Stratonovich interpretation of the noise. We propose and analyse schemes for effective approximation of the slow component. Such schemes satisfy an asymptotic preserving property and generalize the methods proposed in a recent article. We fill a gap in the analysis of these schemes and prove strong error estimates, which are uniform with respect to the time scale separation parameter.

We study a family of numerical schemes applied to a class of multiscale systems of stochastic differential equations. When the time scale separation parameter vanishes, a well-known Smoluchowski--Kramers diffusion approximation result states that the slow component of the considered system converges to the solution of a standard It\^o stochastic differential equation. We propose and analyse schemes for strong and weak effective approximation of the slow component. Such schemes satisfy an asymptotic preserving property and generalize the methods proposed in a recent article. We fill a gap in the analysis of these schemes and prove strong and weak error estimates, which are uniform with respect to the time scale separation parameter.

We design and study splitting integrators for the temporal discretization of the stochastic FitzHugh--Nagumo system. This system is a model for signal propagation in nerve cells where the voltage variable is solution of a one-dimensional parabolic PDE with a cubic nonlinearity driven by additive space-time white noise. We first show that the numerical solutions have finite moments. We then prove that the splitting schemes have, at least, the strong rate of convergence $1/4$. Finally, numerical experiments illustrating the performance of the splitting schemes are provided.

We study a class of fully-discrete schemes for the numerical approximation of solutions of stochastic Cahn--Hilliard equations with cubic nonlinearity and driven by additive noise. The spatial (resp. temporal) discretization is performed with a spectral Galerkin method (resp. a tamed exponential Euler method). We consider two situations: space-time white noise in dimension $d=1$ and trace-class noise in dimensions $d=1,2,3$. In both situations, we prove weak error estimates, where the weak order of convergence is twice the strong order of convergence with respect to the spatial and temporal discretization parameters. To prove these results, we show appropriate regularity estimates for solutions of the Kolmogorov equation associated with the stochastic Cahn--Hilliard equation, which have not been established previously and may be of interest in other contexts.

We study an asymptotic preserving scheme for the temporal discretization of a system of parabolic semilinear SPDEs with two time scales. Owing to the averaging principle, when the time scale separation $\epsilon$ vanishes, the slow component converges to the solution of a limiting evolution equation, which is captured when the time-step size $\Delta t$ vanishes by a limiting scheme. The objective of this work is to prove weak error estimates which are uniform with respect to $\epsilon$, in terms of $\Delta t$: the scheme satisfies a uniform accuracy property. This is a non trivial generalization of a recent article in an infinite dimensional framework. The fast component is discretized using the modified Euler scheme for SPDEs introduced in a recent work. Proving the weak error estimates requires delicate analysis of the regularity properties of solutions of infinite dimensional Kolmogorov equations.

We propose a modification of the standard linear implicit Euler integrator for the weak approximation of parabolic semilinear stochastic PDEs driven by additive space-time white noise. The new method can easily be combined with a finite difference method for the spatial discretization. The proposed method is shown to have improved qualitative properties compared with the standard method. First, for any time-step size, the spatial regularity of the solution is preserved, at all times. Second, the proposed method preserves the Gaussian invariant distribution of the infinite dimensional Ornstein--Uhlenbeck process obtained when the nonlinearity is absent, for any time-step size. The weak order of convergence of the proposed method is shown to be equal to $1/2$ in a general setting, like for the standard Euler scheme. A stronger weak approximation result is obtained when considering the approximation of a Gibbs invariant distribution, when the nonlinearity is a gradient: one obtains an approximation in total variation distance of order $1/2$, which does not hold for the standard method. This is the first result of this type in the literature. A key point in the analysis is the interpretation of the proposed modified Euler scheme as the accelerated exponential Euler scheme applied to a modified stochastic evolution equation. Finally, it is shown that the proposed method can be applied to design an asymptotic preserving scheme for a class of slow-fast multiscale systems, and to construct a Markov Chain Monte Carlo method which is well-defined in infinite dimension. We also revisit the analysis of the standard and the accelerated exponential Euler scheme, and we prove new results with approximation in the total variation distance, which serve to illustrate the behavior of the proposed modified Euler scheme.

We study stochastic Poisson integrators for a class of stochastic Poisson systems driven by Stratonovich noise. Such geometric integrators preserve Casimir functions and the Poisson map property. For this purpose, we propose explicit stochastic Poisson integrators based on a splitting strategy, and analyse their qualitative and quantitative properties: preservation of Casimir functions, existence of almost sure or moment bounds, asymptotic preserving property, and strong and weak rates of convergence. The construction of the schemes and the theoretical results are illustrated through extensive numerical experiments for three examples of stochastic Lie--Poisson systems, namely: stochastically perturbed Maxwell--Bloch, rigid body and sine--Euler equations.

We consider a class of stochastic kinetic equations, depending on two time scale separation parameters $\epsilon$ and $\delta$: the evolution equation contains singular terms with respect to $\epsilon$, and is driven by a fast ergodic process which evolves at the time scale $t/\delta^2$. We prove that when $(\epsilon,\delta)\to (0,0)$ the density converges to the solution of a linear diffusion PDE. This is a mixture of diffusion approximation in the PDE sense (with respect to the parameter $\epsilon$) and of averaging in the probabilistic sense (with respect to the parameter $\delta$). The proof employs stopping times arguments and a suitable perturbed test functions approach which is adapted to consider the general regime $\epsilon\neq \delta$.

We design numerical schemes for a class of slow-fast systems of stochastic differential equations, where the fast component is an Ornstein-Uhlenbeck process and the slow component is driven by a fractional Brownian motion with Hurst index $H>1/2$. We establish the asymptotic preserving property of the proposed scheme: when the time-scale parameter goes to $0$, a limiting scheme which is consistent with the averaged equation is obtained. With this numerical analysis point of view, we thus illustrate the recently proved averaging result for the considered SDE systems and the main differences with the standard Wiener case., Comment: Comments are welcome

We consider a one-dimensional stochastic differential equation driven by a Wiener process, where the diffusion coefficient depends on an ergodic fast process. The averaging principle is satisfied: it is well-known that the slow component converges in distribution to the solution of an averaged equation, with generator determined by averaging the square of the diffusion coefficient. We propose a version of the averaging principle, where the solution is interpreted as the sum of two terms: one depending on the average of the diffusion coefficient, the other giving fluctuations around that average. Both the average and fluctuation terms contribute to the limit, which illustrates why it is required to average the square of the diffusion coefficient to find the limit behavior., Comment: Comments are welcome

Explicit stabilized integrators are an efficient alternative to implicit or semi-implicit methods to avoid the severe timestep restriction faced by standard explicit integrators applied to stiff diffusion problems. In this paper, we provide a fully discrete strong convergence analysis of a family of explicit stabilized methods coupled with finite element methods for a class of parabolic semilinear deterministic and stochastic partial differential equations. Numerical experiments including the semilinear stochastic heat equation with space-time white noise confirm the theoretical findings., Comment: Assyr Abdulle passed away on September 1st, 2021 prior to the revision of this paper. To appear in IMA Journal of Numerical Analysis

We introduce and study a notion of Asymptotic Preserving schemes, related to convergence in distribution, for a class of slow-fast Stochastic Differential Equations. In some examples, crude schemes fail to capture the correct limiting equation resulting from averaging and diffusion approximation procedures. We propose examples of Asymptotic Preserving schemes: when the time-scale separation vanishes, one obtains a limiting scheme, which is shown to be consistent in distribution with the limiting Stochastic Differential Equation. Numerical experiments illustrate the importance of the proposed Asymptotic Preserving schemes for several examples. In addition, in the averaging regime, error estimates are obtained and the proposed scheme is proved to be uniformly accurate.

We analyse a splitting integrator for the time discretization of the Schr\"odinger equation with nonlocal interaction cubic nonlinearity and white noise dispersion. We prove that this time integrator has order of convergence one in the $p$-th mean sense, for any $p\geq1$ in some Sobolev spaces. We prove that the splitting schemes preserves the $L^2$-norm, which is a crucial property for the proof of the strong convergence result. Finally, numerical experiments illustrate the performance of the proposed numerical scheme.

We consider the long-time behavior of an explicit tamed Euler scheme applied to a class of stochastic differential equations driven by additive noise, under a one-sided Lipschitz continuity condition. The setting encompasses drift nonlinearities with polynomial growth. First, we prove that moment bounds for the numerical scheme hold, with at most polynomial dependence with respect to the time horizon. Second, we apply this result to obtain error estimates, in the weak sense, in terms of the time-step size and of the time horizon, to quantify the error to approximate averages with respect to the invariant distribution of the continuous-time process. We justify the efficiency of using the explicit tamed Euler scheme to approximate the invariant distribution, since the computational cost does not suffer from the at most polynomial growth of the moment bounds. To the best of our knowledge, this is the first result in the literature concerning the approximation of the invariant distribution for SDEs with non-globally Lipschitz coefficients using an explicit tamed scheme.

We consider the long-time behavior of an explicit tamed exponential Euler scheme applied to a class of parabolic semilinear stochastic partial differential equations driven by additive noise, under a one-sided Lipschitz continuity condition. The setting encompasses nonlinearities with polynomial growth. First, we prove that moment bounds for the numerical scheme hold, with at most polynomial dependence with respect to the time horizon. Second, we apply this result to obtain error estimates, in the weak sense, in terms of the time-step size and of the time horizon, to quantify the error to approximate averages with respect to the invariant distribution of the continuous-time process. We justify the efficiency of using the explicit tamed exponential Euler scheme to approximate the invariant distribution, since the computational cost does not suffer from the at most polynomial growth of the moment bounds. To the best of our knowledge, this is the first result in the literature concerning the approximation of the invariant distribution for SPDEs with non-globally Lipschitz coefficients using an explicit tamed scheme.

We analyze the qualitative properties and the order of convergence of a splitting scheme for a class of nonlinear stochastic Schr\"odinger equations driven by additive It\^o noise. The class of nonlinearities of interest includes nonlocal interaction cubic nonlinearities. We show that the numerical solution is symplectic and preserves the expected mass for all times. On top of that, for the convergence analysis, some exponential moment bounds for the exact and numerical solutions are proved. This enables us to provide strong orders of convergence as well as orders of convergence in probability and almost surely. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.

Instruments achieve sharper and finer observations of micron-in-size dust grains in the top layers of young stellar discs. To provide accurate models, we revisit the theory of dust settling for small grains, when gas stratification, dust inertia and finite correlation times for the turbulence should be handled simultaneously. We start from a balance of forces and derive distributions at steady-state. Asymptotic expansions require caution since limits do not commute. In particular, non-physical bumpy distributions appear when turbulence is purely diffusive. This excludes very short correlation times for real discs, as predicted by numerical simulations., Comment: 13 pages, 4 figures, accepted for publication in MNRAS

This article fills a gap in the mathematical analysis of Adaptive Biasing algorithms, which are extensively used in molecular dynamics computations. Given a reaction coordinate, ideally, the bias in the overdamped Langevin dynamics would be given by the gradient of the associated free energy function, which is unknown. We consider an adaptive biased version of the overdamped dynamics, where the bias depends on the past of the trajectory and is designed to approximate the free energy. The main result of this article is the consistency and efficiency of this approach. More precisely we prove the almost sure convergence of the bias as time goes to infinity, and that the limit is close to the ideal bias, as an auxiliary parameter of the algorithm goes to $0$. The proof is based on interpreting the process as a self-interacting dynamics, and on the study of a non-trivial fixed point problem for the limiting flow obtained using the ODE method.

Parareal algorithms are studied for semilinear parabolic stochastic partial differential equations. These algorithms proceed as two-level integrators, with fine and coarse schemes, and have been designed to achieve a `parallel in real time' implementation. In this work, the fine integrator is given by the exponential Euler scheme. Two choices for the coarse integrator are considered: the linear implicit Euler scheme, and the exponential Euler scheme. The influence on the performance of the parareal algorithm, of the choice of the coarse integrator, of the regularity of the noise, and of the number of parareal iterations, is investigated, with theoretical analysis results and with extensive numerical experiments.

We investigate the application of the Adaptive Multilevel Splitting algorithm for the estimation of tail probabilities of solutions of Stochastic Differential Equations evaluated at a given time, and of associated temporal averages. We introduce a new, very general and effective family of score functions which is designed for these problems. We illustrate its behavior on a series of numerical experiments. In particular, we demonstrate how it can be used to estimate large deviation rate functionals for the longtime limit of temporal averages.

This article is devoted to the analysis of semilinear, parabolic, Stochastic Partial Differential Equations, with slow and fast time scales. Asymptotically, an averaging principle holds: the slow component converges to the solution of another semilinear, parabolic, SPDE, where the nonlinearity is averaged with respect to the invariant distribution of the fast process. We exhibit orders of convergence, in both strong and weak senses, in two relevant situations, depending on the spatial regularity of the fast process and on the covariance of the Wiener noise in the slow equation. In a very regular case, strong and weak orders are equal to $\frac12$ and $1$. In a less regular case, the weak order is also twice the strong order. This study extends previous results concerning weak rates of convergence, where either no stochastic forcing term was included in the slow equation, or the covariance of the noise was extremely regular. An efficient numerical scheme, based on Heterogeneous Multiscale Methods, is briefly discussed.

This article is devoted to the analysis of the weak rates of convergence of schemes introduced by the authors in a recent work, for the temporal discretization of the stochastic Allen-Cahn equation driven by space-time white noise. The schemes are based on splitting strategies and are explicit. We prove that they have a weak rate of convergence equal to $\frac12$, like in the more standard case of SPDEs with globally Lipschitz continuous nonlinearity. To deal with the polynomial growth of the nonlinearity, several new estimates and techniques are used. In particular, new regularity results for solutions of related infinite dimensional Kolmogorov equations are established. Our contribution is the first one in the literature concerning weak convergence rates for parabolic semilinear SPDEs with non globally Lipschitz nonlinearities.

This article analyzes an explicit temporal splitting numerical scheme for the stochastic Allen-Cahn equation driven by additive noise, in a bounded spatial domain with smooth boundary in dimension $d\le 3$. The splitting strategy is combined with an exponential Euler scheme of an auxiliary problem. When $d=1$ and the driving noise is a space-time white noise, we first show some a priori estimates of this splitting scheme. Using the monotonicity of the drift nonlinearity, we then prove that under very mild assumptions on the initial data, this scheme achieves the optimal strong convergence rate $\OO(\delta t^{\frac 14})$. When $d\le 3$ and the driving noise possesses some regularity in space, we study exponential integrability properties of the exact and numerical solutions. Finally, in dimension $d=1$, these properties are used to prove that the splitting scheme has a strong convergence rate $\OO(\delta t)$., Comment: 33 pages

We introduce and analyze an explicit time discretization scheme for the one-dimensional stochastic Allen-Cahn, driven by space-time white noise. The scheme is based on a splitting strategy, and uses the exact solution for the nonlinear term contribution. We first prove boundedness of moments of the numerical solution. We then prove strong convergence results: first, L^2 ($\Omega$)-convergence of order almost 1/4, localized on an event of arbitrarily large probability, then convergence in probability of order almost 1/4. The theoretical analysis is supported by numerical experiments, concerning strong and weak orders of convergence.

The average time between two occurrences of the same event, referred to as its return time (or return period), is a useful statistical concept for practical applications. For instance insurances or public agency may be interested by the return time of a 10m flood of the Seine river in Paris. However, due to their scarcity, reliably estimating return times for rare events is very difficult using either observational data or direct numerical simulations. For rare events, an estimator for return times can be built from the extrema of the observable on trajectory blocks. Here, we show that this estimator can be improved to remain accurate for return times of the order of the block size. More importantly, we show that this approach can be generalised to estimate return times from numerical algorithms specifically designed to sample rare events. So far those algorithms often compute probabilities, rather than return times. The approach we propose provides a computationally extremely efficient way to estimate numerically the return times of rare events for a dynamical system, gaining several orders of magnitude of computational costs. We illustrate the method on two kinds of observables, instantaneous and time-averaged, using two different rare event algorithms, for a simple stochastic process, the Ornstein-Uhlenbeck process. As an example of realistic applications to complex systems, we finally discuss extreme values of the drag on an object in a turbulent flow.

This article investigates the role of the regularity of the test function when considering the weak error for standard discretizations of SPDEs of the form $dX(t)=AX(t)dt+F(X(t))dt+dW(t)$, driven by space-time white noise. In previous results, test functions are assumed (at least) of class $\mathcal{C}^2$ with bounded derivatives, and the weak order is twice the strong order. We prove, in the case $F=0$, that to quantify the speed of convergence, it is crucial to control some derivatives of the test functions, even when the noise is non-degenerate. First, the supremum of the weak error over all bounded continuous functions, which are bounded by $1$, does not converge to $0$ as the discretization parameter vanishes. Second, when considering bounded Lipschitz test functions, the weak order of convergence is divided by $2$, i.e. it is not better than the strong order. This is in contrast with the finite dimensional case, where the Euler-Maruyama discretization of elliptic SDEs $dY(t)=f(Y(t))dt+dB_t$ has weak order of convergence $1$ even for bounded continuous functions.

This article is concerned with the mathematical analysis of a family of adaptive importance sampling algorithms applied to diffusion processes. These methods, referred to as Adaptive Biasing Potential methods, are designed to efficiently sample the invariant distribution of the diffusion process, thanks to the approximation of the associated free energy function (relative to a reaction coordinate). The bias which is introduced in the dynamics is computed adaptively; it depends on the past of the trajectory of the process through some time-averages. We give a detailed and general construction of such methods. We prove the consistency of the approach (almost sure convergence of well-chosen weighted empirical probability distribution). We justify the efficiency thanks to several qualitative and quantitative additional arguments. To prove these results , we revisit and extend tools from stochastic approximation applied to self-interacting diffusions, in an original context.

We provide new regularity results for the solutions of the Kolmogorov equation associated to a SPDE with nonlinear diffusion coefficients and a Burgers type nonlinearity. This generalizes previous results in the simpler cases of additive or affine noise. The basic tool is a discrete version of a two sided stochastic integral which allows a new formulation for the derivatives of these solutions. We show that this can be used to generalize the weak order analysis performed in [16]. The tools we develop are very general and can be used to study many other examples of applications.

We prove the consistency of an adaptive importance sampling strategy based on biasing the potential energy function $V$ of a diffusion process $dX_t^0=-\nabla V(X_t^0)dt+dW_t$; for the sake of simplicity, periodic boundary conditions are assumed, so that $X_t^0$ lives on the flat $d$-dimensional torus. The goal is to sample its invariant distribution $\mu=Z^{-1}\exp\bigl(-V(x)\bigr)\,dx$. The bias $V_t-V$, where $V_t$ is the new (random and time-dependent) potential function, acts only on some coordinates of the system, and is designed to flatten the corresponding empirical occupation measure of the diffusion $X$ in the large time regime. The diffusion process writes $dX_t=-\nabla V_t(X_t)dt+dW_t$, where the bias $V_t-V$ is function of the key quantity $\overline{\mu}_t$: a probability occupation measure which depends on the past of the process, {\it i.e.} on $(X_s)_{s\in [0,t]}$. We are thus dealing with a self-interacting diffusion. In this note, we prove that when $t$ goes to infinity, $\overline{\mu}_t$ almost surely converges to $\mu$. Moreover, the approach is justified by the convergence of the bias to a limit which has an intepretation in terms of a free energy. The main argument is a change of variables, which formally validates the consistency of the approach. The convergence is then rigorously proven adapting the ODE method from stochastic approximation.

We consider stochastic semi-linear evolution equations which are driven by additive, spatially correlated, Wiener noise, and in particular consider problems of heat equation (analytic semigroup) and damped-driven wave equations (bounded semigroup) type. We discretize these equations by means of a spectral Galerkin projection, and we study the approximation of the probability distribution of the trajectories: test functions are regular, but depend on the values of the process on the interval $[0,T]$. We introduce a new approach in the context of quantative weak error analysis for discretization of SPDEs. The weak error is formulated using a deterministic function (It\^o map) of the stochastic convolution found when the nonlinear term is dropped. The regularity properties of the It\^o map are exploited, and in particular second-order Taylor expansions employed, to transfer the error from spectral approximation of the stochastic convolution into the weak error of interest. We prove that the weak rate of convergence is twice the strong rate of convergence in two situations. First, we assume that the covariance operator commutes with the generator of the semigroup: the first order term in the weak error expansion cancels out thanks to an independence property. Second, we remove the commuting assumption, and extend the previous result, thanks to the analysis of a new error term depending on a commutator.

We introduce a time-integrator to sample with high order of accuracy the invariant distribution for a class of semilinear SPDEs driven by an additive space-time noise. Combined with a postprocessor, the new method is a modification with negligible overhead of the standard linearized implicit Euler-Maruyama method. We first provide an analysis of the integrator when applied for SDEs (finite dimension), where we prove that the method has order $2$ for the approximation of the invariant distribution, instead of $1$. We then perform a stability analysis of the integrator in the semilinear SPDE context, and we prove in a linear case that a higher order of convergence is achieved. Numerical experiments, including the semilinear heat equation driven by space-time white noise, confirm the theoretical findings and illustrate the efficiency of the approach., Comment: 25 pages

The Adaptive Multilevel Splitting (AMS) algorithm is a powerful and versatile method for the simulation of rare events. It is based on an interacting (via a mutation-selection procedure) system of replicas, and depends on two integer parameters: n $\in$ N * the size of the system and the number k $\in$ {1, . . . , n -- 1} of the replicas that are eliminated and resampled at each iteration. In an idealized setting, we analyze the performance of this algorithm in terms of a Large Deviations Principle when n goes to infinity, for the estimation of the (small) probability P(X \textgreater{} a) where a is a given threshold and X is real-valued random variable. The proof uses the technique introduced in [BLR15]: in order to study the log-Laplace transform, we rely on an auxiliary functional equation. Such Large Deviations Principle results are potentially useful to study the algorithm beyond the idealized setting, in particular to compute rare transitions probabilities for complex high-dimensional stochastic processes.

The Adaptive Multilevel Splitting algorithm is a very powerful and versatile iterative method to estimate the probability of rare events, based on an interacting particle systems. In an other article, in a so-called idealized setting, the authors prove that some associated estimators are unbiased, for each value of the size n of the systems of replicas and of resampling number k. Here we go beyond and prove these estimator's asymptotic normality when h goes to infinity, for any fixed value of k. The main ingredient is the asymptotic analysis of a functional equation on an appropriate characteristic function. Some numerical simulations illustrate the convergence to rely on Gaussian confidence intervals.

The Adaptive Multilevel Splitting algorithm is a very powerful and versatile method to estimate rare events probabilities. It is an iterative procedure on an interacting particle system, where at each step, the $k$ less well-adapted particles among $n$ are killed while $k$ new better adapted particles are resampled according to a conditional law. We analyze the algorithm in the idealized setting of an exact resampling and prove that the estimator of the rare event probability is unbiased whatever $k$. We also obtain a precise asymptotic expansion for the variance of the estimator and the cost of the algorithm in the large $n$ limit, for a fixed $k$.