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In this paper, a weak Local Linearization scheme for Stochastic Differential Equations (SDEs) with multiplicative noise is introduced. First, for a time discretization, the solution of the SDE is locally approximated by the solution of the…
The paper introduces a very simple and fast computation method for high-dimensional integrals to solve high-dimensional Kolmogorov partial differential equations (PDEs). The new machine learning-based method is obtained by solving a…
In the theory and practice of inverse problems for partial differential equations (PDEs) much attention is paid to the problem of the identification of coefficients from some additional information. This work deals with the problem of…
This work proposes and analyzes a generalized acceleration technique for decreasing the computational complexity of using stochastic collocation (SC) methods to solve partial differential equations (PDEs) with random input data. The SC…
The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…
We will introduce Euler-Maruyama approximations given by an orthogonal system in $L^{2}[0,1]$ for high dimensional SDEs, which could be finite dimensional approximations of SPDEs. In general, the higher the dimension is, the more one needs…
A new approximation format for solutions of partial differential equations depending on infinitely many parameters is introduced. By combining low-rank tensor approximation in a selected subset of variables with a sparse polynomial…
This paper introduces a new approximation scheme for solving high-dimensional semilinear partial differential equations (PDEs) and backward stochastic differential equations (BSDEs). First, we decompose a target semilinear PDE (BSDE) into…
We consider state-dependent delay equations (SDDE) obtained by adding delays to a planar ordinary differential equation with a limit cycle. These situations appear in models of several physical processes, where small delay effects are…
Stochastic differential equations (SDEs) are increasingly used in longitudinal data analysis, compartmental models, growth modelling, and other applications in a number of disciplines. Parameter estimation, however, currently requires…
Previous studies on stochastic primal-dual algorithms for solving min-max problems with faster convergence heavily rely on the bilinear structure of the problem, which restricts their applicability to a narrowed range of problems. The main…
High-dimensional partial-differential equations (PDEs) arise in a number of fields of science and engineering, where they are used to describe the evolution of joint probability functions. Their examples include the Boltzmann and…
In this paper, we present a deep learning-based numerical method for approximating high dimensional stochastic partial differential equations (SPDEs). At each time step, our method relies on a predictor-corrector procedure. More precisely,…
To study the nonlinear properties of complex natural phenomena, the evolution of the quantity of interest can be often represented by systems of coupled nonlinear stochastic differential equations (SDEs). These SDEs typically contain…
We present a novel deep learning method for estimating time-dependent parameters in Markov processes through discrete sampling. Departing from conventional machine learning, our approach reframes parameter approximation as an optimization…
Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex, possibly nonsmooth part that promotes regularization. Most analyses of iteration…
Asynchronous iterations arise naturally in parallel computing if one wants to solve large problems with a minimization of the idle times. This paper presents an original model of asynchronous iterations for a time-domain decomposition…
The challenging problem of conducting fully Bayesian inference for the reaction rate constants governing stochastic kinetic models (SKMs) is considered. Given the challenges underlying this problem, the Markov jump process representation is…
In this paper, a nonlinear system of fractional ordinary differential equations with multiple scales in time is investigated. We are interested in the effective long-term computation of the solution. The main challenge is how to obtain the…
In this paper, we propose a model order reduction based adaptive parareal method for time-dependent partial differential equations. By using the data obtained by the fine propagator in each iteration of the plain parareal method together…