Continuous Stochastic Differential Equations Research Articles

High-Frequency Space Diffusion Model for Accelerated MRI.

Diffusion models with continuous stochastic differential equations (SDEs) have shown superior performances in image generation. It can serve as a deep generative prior to solving the inverse problem in magnetic resonance (MR) reconstruction. However, low-frequency regions of k -space data are typically fully sampled in fast MR imaging, while existing diffusion models are performed throughout the entire image or k -space, inevitably introducing uncertainty in the reconstruction of low-frequency regions. Additionally, existing diffusion models often demand substantial iterations to converge, resulting in time-consuming reconstructions. To address these challenges, we propose a novel SDE tailored specifically for MR reconstruction with the diffusion process in high-frequency space (referred to as HFS-SDE). This approach ensures determinism in the fully sampled low-frequency regions and accelerates the sampling procedure of reverse diffusion. Experiments conducted on the publicly available fastMRI dataset demonstrate that the proposed HFS-SDE method outperforms traditional parallel imaging methods, supervised deep learning, and existing diffusion models in terms of reconstruction accuracy and stability. The fast convergence properties are also confirmed through theoretical and experimental validation. Our code and weights are available at https://github.com/Aboriginer/HFS-SDE.

Estimation of the invariant density for discretely observed diffusion processes: impact of the sampling and of the asynchronicity

We aim at estimating in a non-parametric way the density π of the stationary distribution of a d-dimensional stochastic differential equation , for , from the discrete observations of a finite sample ,…, with . We propose a kernel density estimator and we study its convergence rates for the pointwise estimation of the invariant density under anisotropic Hölder smoothness constraints. First of all, we find some conditions on the discretization step that ensures it is possible to recover the same rates as if the continuous trajectory of the process was available. As proven in the recent work [Amorino C, Gloter A. Minimax rate of estimation for invariant densities associated to continuous stochastic differential equations over anisotropic Holder classes; 2021. arXiv preprint arXiv:2110.02774], such rates are optimal and new in the context of density estimator. Then we deal with the case where such a condition on the discretization step is not satisfied, which we refer to as the intermediate regime. In this new regime we identify the convergence rate for the estimation of the invariant density over anisotropic Hölder classes, which is the same convergence rate as for the estimation of a probability density belonging to an anisotropic Hölder class, associated to n iid random variables . After that we focus on the asynchronous case, in which each component can be observed at different time points. Even if the asynchronicity of the observations complexifies the computation of the variance of the estimator, we are able to find conditions ensuring that this variance is comparable to the one of the continuous case. We also exhibit that the non-synchronicity of the data introduces additional bias terms in the study of the estimator.

Rate of estimation for the stationary distribution of jump-processes over anisotropic Holder classes

We study the problem of the non-parametric estimation for the density of the stationary distribution of the multivariate stochastic differential equation with jumps (Xt) , when the dimension d is bigger than 3. From the continuous observation of the sampling path on [0, T ] we show that, under anisotropic Holder smoothness constraints, kernel based estimators can achieve fast convergence rates. In particular , they are as fast as the ones found by Dalalyan and Reiss [9] for the estimation of the invariant density in the case without jumps under isotropic Holder smoothness constraints. Moreover, they are faster than the ones found by Strauch [29] for the invariant density estimation of continuous stochastic differential equations, under anisotropic Holder smoothness constraints. Furthermore, we obtain a minimax lower bound on the L2-risk for pointwise estimation, with the same rate up to a log(T) term. It implies that, on a class of diffusions whose invariant density belongs to the anisotropic Holder class we are considering, it is impossible to find an estimator with a rate of estimation faster than the one we propose.

Drawdown and Drawup of Bi-Directional Grid Constrained Stochastic Processes

The Grid Trading Problem (GTP) of mathematical finance, used in portfolio loss minimization, generalized dynamic hedging and algorithmic trading, is researched by examining the impact of the drawdown and drawup of discrete random walks and of Ito diffusions on the Bi-Directional Grid Constrained (BGC) stochastic process for profit Pt and equity E_t over time. A comprehensive Discrete Difference Equation (DDE) and a continuous Stochastic Differential Equation (SDE) are derived and proved for the GTP. This allows fund managers and traders the ability to better stress test the impact of volatility to reduce risk and generate positive returns. These theorems are then simulated to complement the theoretical models with charts. Not only does this research extend a rich mathematical problem that can be further researched in its own right, but it also extends the applications into the above areas of finance.

Bessel-like birth–death process

We consider models of the population or opinion dynamics which result in the non-linear stochastic differential equations (SDEs) exhibiting the spurious long-range memory. In this context, the correspondence between the description of the birth–death processes as the continuous-time Markov chains and the continuous SDEs is of high importance for the alternatives of modeling. We propose and generalize the Bessel-like birth–death process having clear representation by the SDEs. The new process helps to integrate the alternatives of description and to derive the equations for the probability density function (PDF) of the burst and inter-burst duration of the proposed continuous time birth–death processes.

Error estimates on ergodic properties of discretized Feynman–Kac semigroups

We consider the numerical analysis of the time discretization of Feynman–Kac semigroups associated with diffusion processes. These semigroups naturally appear in several fields, such as large deviation theory, Diffusion Monte Carlo or non-linear filtering. We present error estimates à la Talay–Tubaro on their invariant measures when the underlying continuous stochastic differential equation is discretized; as well as on the leading eigenvalue of the generator of the dynamics, which corresponds to the rate of creation of probability. This provides criteria to construct efficient integration schemes of Feynman–Kac dynamics, as well as a mathematical justification of numerical results already observed in the Diffusion Monte Carlo community. Our analysis is illustrated by numerical simulations.

A hybrid stochastic method with adaptive time step control for reaction–diffusion systems

Randomness often plays an important role in the spatial and temporal dynamics of biological systems. General stochastic simulation methods may lead to excessive computational cost for a system in which a large number of molecules involved. Therefore, multi-scale hybrid simulation methods become important for stochastic simulations. Here we build a spatially hybrid method which couples two approaches: discrete stochastic simulation and continuous stochastic differential equations. In our method, the locations of the interfaces between the two approaches are changing according to the distribution of molecules in a one-dimensional domain. To balance the accuracy and efficiency, the time step of the numerical method for the continuous stochastic differential equations is adapted to the dynamics of the molecules near the adaptive interfaces. The simulation results for a linear system and two nonlinear biological systems in different one-dimensional domains demonstrate the effectiveness and advantage of our new hybrid method with the adaptive time step control.

Jump-Diffusion Approximation of Stochastic Reaction Dynamics: Error Bounds and Algorithms

Biochemical processes in living cells are comprised of reactions with vastly varying speeds and molecular counts of the reactant species. Classical deterministic or stochastic approaches to modeling often fail to exploit this multiscale nature of the reaction systems. In this paper, we propose a jump-diffusion approximation to these types of multiscale systems that couples the two traditional modeling approaches. An error bound of the proposed approximation is derived and used to partition the reactions into two sets, where one set is modeled by continuous stochastic differential equations and the other by discrete state processes. The methodology leads to a very efficient dynamic partitioning algorithm which has been implemented for several multiscale reaction systems. The gain in computational efficiency is evident in all these examples, which include a realistically sized model of a signal transduction cascade coupled to a gene expression dynamics.

An efficient particle Fokker–Planck algorithm for rarefied gas flows

This paper is devoted to the algorithmic improvement and careful analysis of the Fokker–Planck kinetic model derived by Jenny et al. [1] and Gorji et al. [2]. The motivation behind the Fokker–Planck based particle methods is to gain efficiency in low Knudsen rarefied gas flow simulations, where conventional direct simulation Monte Carlo (DSMC) becomes expensive. This can be achieved due to the fact that the resulting model equations are continuous stochastic differential equations in velocity space. Accordingly, the computational particles evolve along independent stochastic paths and thus no collision needs to be calculated. Therefore the computational cost of the solution algorithm becomes independent of the Knudsen number. In the present study, different computational improvements were persuaded in order to augment the method, including an accurate time integration scheme, local time stepping and noise reduction. For assessment of the performance, gas flow around a cylinder and lid driven cavity flow were studied. Convergence rates, accuracy and computational costs were compared with respect to DSMC for a range of Knudsen numbers (from hydrodynamic regime up to above one). In all the considered cases, the model together with the proposed scheme give rise to very efficient yet accurate solution algorithms.

Discrete, continuous, and stochastic models of protein sorting in the Golgi apparatus

The Golgi apparatus plays a central role in processing and sorting proteins and lipids in eukaryotic cells. Golgi compartments constantly exchange material with each other and with other cellular components, allowing them to maintain and reform distinct identities despite dramatic changes in structure and size during cell division, development, and osmotic stress. We have developed three minimal models of membrane and protein exchange in the Golgi-a discrete, stochastic model, a continuous ordinary differential equation model, and a continuous stochastic differential equation model-each based on two fundamental mechanisms: vesicle-coat-mediated selective concentration of cargoes and soluble N-ethylmaleimide-sensitive factor attachment protein receptor (SNARE) proteins during vesicle formation and SNARE-mediated selective fusion of vesicles. By exploring where the models differ, we hope to discover whether the discrete, stochastic nature of vesicle-mediated transport is likely to have appreciable functional consequences for the Golgi. All three models show similar ability to restore and maintain distinct identities over broad parameter ranges. They diverge, however, in conditions corresponding to collapse and reassembly of the Golgi. The results suggest that a continuum model provides a good description of Golgi maintenance but that considering the discrete nature of vesicle-based traffic is important to understanding assembly and disassembly of the Golgi. Experimental analysis validates a prediction of the models that altering guanine nucleotide exchange factor expression levels will modulate Golgi size.

Energy and Momentum Conservation for Diffusion - A Stochastic Mechanics Approximation - Part I

This paper models the classical diffusion of a main particle through a heatbath by means of a pre-limit microscopic representation of its drifted momentum and energy transfers at collision times. The collision point linear interpolated path can be approximated by the solution to the inscribed continuous stochastic differential equation using the same drift function. Employing results from stochastic mechanics it is then shown that the combined main particle/heatbath system does not exchange or radiate energy if the probability distribution for the position of the main particle is derived from Schroedinger's equation. Furthermore it is shown that the main particle distance traveled between collisions and the mean inter-collision time must satisfy a type of Minkowski invariant. Hence if there is a correlation between the pre- and post-collision velocities of the main particle through a collision point then the mean distance traveled can be related to the mean inter-particle collision times via a Lorentz transformation. The last Section shows that this approach can be applied to all elastic main particle/heatbath particle collisions either via direct calculation involving modeling the collision scattering or by altering the properties of the heatbath.

Dispersion de particules solides en mouvement de saltation dans un écoulement turbulent

The solid particle dispersion in saltating motion is studied in an homogeneous turbulence and in a turbulent boundary layer. The fluid velocity along the particle trajectory is estimated using a continuous stochastic differential equation in which the correlation integral time takes into account gravity and inertia effects. As far as the boundary layer is concerned, the aerodynamic entrainment of particles and the rebound are modelised as random variables with Gaussian probability density functions. Compared with experimental results, the numerical results show good agreement for dispersion, although velocity fluctuations are slightly under evaluated. To cite this article: C. Aguirre et al., C. R. Mecanique 332 (2004).