Computing first passage times for Markov-modulated fluid models using numerical PDE problem solvers

Efficient numerical approximation of a non-regular Fokker--Planck equation associated with first-passage time distributions

In neuroscience, the distribution of a decision time is modelled by means of a one-dimensional Fokker--Planck equation with time-dependent boundaries and space-time-dependent drift. Efficient approximation of the solution to this equation is required, e.g., for model evaluation and parameter fitting. However, the prescribed boundary conditions lead to a strong singularity and thus to slow convergence of numerical approximations. In this article we demonstrate that the solution can be related to the solution of a parabolic PDE on a rectangular space-time domain with homogeneous initial and boun…

First-Passage Time Statistics on Surfaces of General Shape: Surface PDE Solvers using Generalized Moving Least Squares (GMLS)

We develop numerical methods for computing statistics of stochastic processes on surfaces of general shape with drift-diffusion dynamics $d\mathbf{X}_t = a(\mathbf{X}_t)dt + \mathbf{b}(\mathbf{X}_t)d\mathbf{W}_t$. We formulate descriptions of Brownian motion and general drift-diffusion processes on surfaces. We consider statistics of the form $u(\mathbf{x}) = \mathbb{E}^{\mathbf{x}}\left[\int_0^τg(\mathbf{X}_t)dt \right] + \mathbb{E}^{\mathbf{x}}\left[f(\mathbf{X}_τ)\right]$ for a domain $Ω$ and the exit stopping time $τ= \inf_t \{t > 0 \; |\; \mathbf{X}_t \not\in Ω\}$, where $f,g$ are general…

Parallel-in-Time Simulation of Power Converters Using Multirate PDEs

This paper presents an efficient numerical algorithm for the simulation of pulse-width modulated power converters via parallelization in time domain. The method applies the multirate partial differential equation approach on the coarse grid of the (two-grid) parallel-in-time algorithm Parareal. Performance of the proposed approach is illustrated via its application to a DC-DC converter.

线性偏微分方程解的多频高度振荡势的渐近展开

Asymptotic expansions for the solution of a linear PDE with a multifrequency highly oscillatory potential

本文针对多频高度振荡势的线性PDE，解析推导调制傅里叶展开，给出误差公式与系数确定方法，无需数值求解，并用实验验证。

积分微分方程描述的电路的多速率瞬态仿真技术

A Multirate Transient Simulation Technique for Circuits described by Integro-Differential Equations

本文针对射频电路仿真中多速率PDE技术的局限，提出用于积分微分方程描述电路的多速率瞬态仿真新方法，以应对混合器件和脉冲波形挑战。

Bianchi $VI_{-1/9}^{^{*}}$ 中的 18-循环：联合线性局部过渡与数值模拟

The 18-cycle in Bianchi $VI_{-1/9}^{^{*}}$: Combined Linear Local Passage and Numerical Simulation

本文在 Bianchi $VI_{-1/9}^{^{*}}$ 中发现了一个允许 Takens 线性化的 18-循环周期异宿链，证明其联合线性局部过渡具有收缩性，使其成为证明该模型首个严格收敛定理的候选，并利用 VSVO 求解器进行了数值模拟。