181 research outputs found
A class of robust numerical methods for solving dynamical systems with multiple time scales
In this paper, we develop a class of robust numerical methods for solving dynamical systems with multiple time scales. We first represent the solution of a multiscale dynamical system as a transformation of a slowly varying solution. Then, under the scale separation assumption, we provide a systematic way to construct the transformation map and derive the dynamic equation for the slowly varying solution. We also provide the convergence analysis of the proposed method. Finally, we present several numerical examples, including ODE system with three and four separated time scales to demonstrate the accuracy and efficiency of the proposed method. Numerical results verify that our method is robust in solving ODE systems with multiple time scale, where the time step does not depend on the multiscale parameters
A Novel Stochastic Interacting Particle-Field Algorithm for 3D Parabolic-Parabolic Keller-Segel Chemotaxis System
We introduce an efficient stochastic interacting particle-field (SIPF)
algorithm with no history dependence for computing aggregation patterns and
near singular solutions of parabolic-parabolic Keller-Segel (KS) chemotaxis
system in three space dimensions (3D). The KS solutions are approximated as
empirical measures of particles coupled with a smoother field (concentration of
chemo-attractant) variable computed by the spectral method. Instead of using
heat kernels causing history dependence and high memory cost, we leverage the
implicit Euler discretization to derive a one-step recursion in time for
stochastic particle positions and the field variable based on the explicit
Green's function of an elliptic operator of the form Laplacian minus a positive
constant. In numerical experiments, we observe that the resulting SIPF
algorithm is convergent and self-adaptive to the high gradient part of
solutions. Despite the lack of analytical knowledge (e.g. a self-similar
ansatz) of the blowup, the SIPF algorithm provides a low-cost approach to study
the emergence of finite time blowup in 3D by only dozens of Fourier modes and
through varying the amount of initial mass and tracking the evolution of the
field variable. Notably, the algorithm can handle at ease multi-modal initial
data and the subsequent complex evolution involving the merging of particle
clusters and formation of a finite time singularity
Understanding the diffusion models by conditional expectations
This paper provide several mathematical analyses of the diffusion model in
machine learning. The drift term of the backwards sampling process is
represented as a conditional expectation involving the data distribution and
the forward diffusion. The training process aims to find such a drift function
by minimizing the mean-squared residue related to the conditional expectation.
Using small-time approximations of the Green's function of the forward
diffusion, we show that the analytical mean drift function in DDPM and the
score function in SGM asymptotically blow up in the final stages of the
sampling process for singular data distributions such as those concentrated on
lower-dimensional manifolds, and is therefore difficult to approximate by a
network. To overcome this difficulty, we derive a new target function and
associated loss, which remains bounded even for singular data distributions. We
illustrate the theoretical findings with several numerical examples
A class of robust numerical methods for solving dynamical systems with multiple time scales
In this paper, we develop a class of robust numerical methods for solving dynamical systems with multiple time scales. We first represent the solution of a multiscale dynamical system as a transformation of a slowly varying solution. Then, under the scale separation assumption, we provide a systematic way to construct the transformation map and derive the dynamic equation for the slowly varying solution. We also provide the convergence analysis of the proposed method. Finally, we present several numerical examples, including ODE system with three and four separated time scales to demonstrate the accuracy and efficiency of the proposed method. Numerical results verify that our method is robust in solving ODE systems with multiple time scale, where the time step does not depend on the multiscale parameters
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