Related papers: A deterministic particle approximation for a fourt…
We introduce deterministic perturbation schemes for the recently proposed random directions stochastic approximation (RDSA) [17], and propose new first-order and second-order algorithms. In the latter case, these are the first second-order…
This manuscript introduces a regression-type formulation for approximating the Perron-Frobenius Operator by relying on distributional snapshots of data. These snapshots may represent densities of particles. The Wasserstein metric is…
The crucial step in designing a particle filter for a particular application is the choice of importance density. The optimal scheme is to use the conditional posterior density of the state, but this cannot be sampled or calculated…
The question of optimally approximating an arbitrary probability measure in the Wasserstein distance by a discrete one with uniform weights is considered. Estimates are obtained for the optimal approximation distance, with an explicit rate…
We consider a linear partial integro-differential equation that arises in the modeling of various physical and biological processes. We study the problem in a spatial periodic domain. We analyze numerical stability and numerical convergence…
In many applications, a state-space model depends on a parameter which needs to be inferred from a data set. Quite often, it is necessary to perform the parameter inference online. In the maximum likelihood approach, this can be done using…
Gradient flow in the 2-Wasserstein space is widely used to optimize functionals over probability distributions and is typically implemented using an interacting particle system with $n$ particles. Analyzing these algorithms requires showing…
In many recent applications when new materials and technologies are developed it is important to describe and simulate new nonlinear and nonlocal diffusion transport processes. A general class of such models deals with nonlocal fractional…
When classical particle filtering algorithms are used for maximum likelihood parameter estimation in nonlinear state-space models, a key challenge is that estimates of the likelihood function and its derivatives are inherently noisy. The…
The author discusses particular solutions of a second order equation designated by source equation. This equation is special because the metric of the space where it is written is influenced by the solution, rendering the equation…
The well-posedness of a non-local advection-selection-mutation problem deriving from adaptive dynamics models is shown for a wide family of initial data. A particle method is then developed, in order to approximate the solution of such…
Particle-based variational inference offers a flexible way of approximating complex posterior distributions with a set of particles. In this paper we introduce a new particle-based variational inference method based on the theory of…
We propose a deterministic particle method for a one-dimensional nonlocal equation with interactions through the repulsive Morse potential. We show that the particle method converges as the number of particles goes to infinity towards weak…
In this contribution, we address the numerical solutions of high-order asymptotic equivalent partial differential equations with the results of a lattice Boltzmann scheme for an inhomogeneous advection problem in one spatial dimension. We…
Numerical solutions to the Vlasov-Poisson system of equations have important applications to both plasma physics and cosmology. In this paper, we present a new Particle-in-Cell (PIC) method for solving this system that is 4th-order accurate…
Many important physical problems, such as fluid structure interaction or conjugate heat transfer, require numerical methods that compute boundary derivatives or fluxes to high accuracy. This paper proposes a novel alternative to calculating…
We give a {\em deterministic} algorithm for approximately computing the fraction of Boolean assignments that satisfy a degree-$2$ polynomial threshold function. Given a degree-2 input polynomial $p(x_1,\dots,x_n)$ and a parameter $\eps >…
A probabilistic approach is developed for the exact solution $u$ to a determinist partial differential equation as well as for its associated approximation $u^{(k)}_{h}$ performed by $P_k$ Lagrange finite element. Two limitations motivated…
We develop a one step matrix method in order to obtain approximate solutions of first order systems and non-linear ordinary differential equations, reducible to first order systems. We find a sequence of such solutions that converge to the…
A nonlinear diffusion equation, interpreted as a Wasserstein gradient flow, is numerically solved in one space dimension using a higher-order minimizing movement scheme based on the BDF (backward differentiation formula) discretization. In…