On nonlinear Feynman–Kac formulas for viscosity solutions of semilinear parabolic partial differential equations

The classical Feynman–Kac identity builds a bridge between stochastic analysis and partial differential equations (PDEs) by providing stochastic representations for classical solutions of linear Kolmogorov PDEs. This opens the door for the derivation of sampling based Monte Carlo approximation methods, which can be meshfree and thereby stand a chance to approximate solutions of PDEs without suffering from the curse of dimensionality. In this paper, we extend the classical Feynman–Kac formula to certain semilinear Kolmogorov PDEs. More specifically, we identify suitable solutions of stochastic fixed point equations (SFPEs), which arise when the classical Feynman–Kac identity is formally applied to semilinear Kolmorogov PDEs, as viscosity solutions of the corresponding PDEs. This justifies, in particular, employing full-history recursive multilevel Picard (MLP) approximation algorithms, which have recently been shown to overcome the curse of dimensionality in the numerical approximation of solutions of SFPEs, in the numerical approximation of semilinear Kolmogorov PDEs.

Score matching filters

<p>A widely popular group of data assimilation methods in meteorological and geophysical sciences is formed by filters based on Monte-Carlo approximation of the traditional Kalman filter, e.g. <span>E</span><span>nsemble Kalman filter </span><span>(EnKF)</span><span>, </span><span>E</span><span>nsemble </span><span>s</span><span>quare-root filter and others. Due to the computational cost, ensemble </span><span>size </span><span>is </span><span>usually </span><span>small </span><span>compar</span><span>ed</span><span> to the dimension of the </span><span>s</span><span>tate </span><span>vector. </span><span>Traditional </span> <span>EnKF implicitly uses the sample covariance which is</span><span> a poor estimate of the </span><span>background covariance matrix - singular and </span><span>contaminated by </span><span>spurious correlations. </span></p><p><span>W</span><span>e focus on modelling the </span><span>background </span><span>covariance matrix by means of </span><span>a linear model for its inverse. This is </span><span>particularly </span><span>useful</span> <span>in</span><span> Gauss-Markov random fields (GMRF), </span><span>where</span> <span>the inverse covariance matrix has </span><span>a banded </span><span>structure</span><span>. </span><span>The parameters of the model are estimated by the</span><span> score matching </span><span>method which </span><span>provides</span><span> estimators in a closed form</span><span>, cheap to compute</span><span>. The resulting estimate</span><span> is a key component of the </span><span>proposed </span><span>ensemble filtering algorithms. </span><span>Under the assumption that the state vector is a GMRF in every time-step, t</span><span>he Score matching filter with Gaussian resamplin</span><span>g (SMF-GR) </span><span>gives</span><span> in every time-step a consistent (in the large ensemble limit) estimator of mean and covariance matrix </span><span>of the forecast and analysis distribution</span><span>. Further, we propose a filtering method called Score matching ensemble filter (SMEF), based on regularization of the EnK</span><span>F</span><span>. Th</span><span>is</span><span> filter performs well even for non-Gaussian systems with non-linear dynamic</span><span>s</span><span>. </span><span>The performance of both filters is illustrated on a simple linear convection model and Lorenz-96.</span></p>

Monte carlo estimation of the solution of fractional partial differential equations

The paper is devoted to the numerical solutions of fractional PDEs based on its probabilistic interpretation, that is, we construct approximate solutions via certain Monte Carlo simulations. The main results represent the upper bound of errors between the exact solution and the Monte Carlo approximation, the estimate of the fluctuation via the appropriate central limit theorem (CLT) and the construction of confidence intervals. Moreover, we provide rates of convergence in the CLT via Berry-Esseen type bounds. Concrete numerical computations and illustrations are included.

Randomized exponential integrators for modulated nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation with dispersion modulated by a (formal) derivative of a time-dependent function with fractional Sobolev regularity of class $W^{\alpha ,2}$ for some $\alpha \in (0,1)$. Due to the loss of smoothness in the problem, classical numerical methods face severe order reduction. In this work, we develop and analyze a new randomized exponential integrator based on a stratified Monte Carlo approximation. The new discretization technique averages the high oscillations in the solution allowing for improved convergence rates of order $\alpha +1/2$. In addition, the new approach allows us to treat a far more general class of modulations than the available literature. Numerical results underline our theoretical findings and show the favorable error behavior of our new scheme compared to classical methods.