Related papers: The Heterogeneous Multi-Scale Method
Definite integrals with parameters of holonomic functions satisfy holonomic systems of linear partial differential equations. When we restrict parameters to a one dimensional curve, the system becomes a linear ordinary differential equation…
We develop a multiscale hybrid scheme for simulations of soft condensed matter systems, which allows one to treat the system at the particle level in selected regions of space, and at the continuum level elsewhere. It is derived…
Structure-preserving algorithms and algorithms with uniform error bound have constituted two interesting classes of numerical methods. In this paper, we blend these two kinds of methods for solving nonlinear Hamiltonian systems with highly…
We present a Lagrangian Heterogeneous Multiscale Method (LHMM) for simulating the non-Newtonian rheology of polymer melts in complex two-dimensional flows. The method couples Dissipative Particle Dynamics (DPD) at the microscale with a…
The two-scale computational homogenization method is proposed for modelling of locally periodic fluid-saturated media subjected a to large deformation induced by quasistatic loading. The periodic heterogeneities are relevant to the…
Recent advances in operator learning have produced two distinct approaches for solving partial differential equations (PDEs): attention-based methods offering point-level adaptability but lacking spectral constraints, and spectral-based…
We analyze the asymptotic behavior of a multiscale problem given by a sequence of integral functionals subject to differential constraints conveyed by a constant-rank operator with two characteristic length scales, namely the film thickness…
The method of harmonic balance (HB) is a spectrally accurate method used to obtain periodic steady state solutions to dynamical systems subjected to periodic perturbations. We adapt HB to solve for the stress response of the Giesekus model…
We propose the segmented iHMM (siHMM), a hierarchical infinite hidden Markov model (iHMM) that supports a simple, efficient inference scheme. The siHMM is well suited to segmentation problems, where the goal is to identify points at which a…
The problem of Hybrid Linear Modeling (HLM) is to model and segment data using a mixture of affine subspaces. Different strategies have been proposed to solve this problem, however, rigorous analysis justifying their performance is missing.…
In practice, optimization tasks have some structure that allows developing new algorithms for every problem with faster convergence rates. Using the structure of optimization tasks, we can propose algorithms with more optimistic convergence…
What is a systematic way to efficiently apply a wide spectrum of advanced ML programs to industrial scale problems, using Big Models (up to 100s of billions of parameters) on Big Data (up to terabytes or petabytes)? Modern parallelization…
Inspired by multigrid methods for linear systems of equations, multilevel optimization methods have been proposed to solve structured optimization problems. Multilevel methods make more assumptions regarding the structure of the…
In this paper a model for partial melts is constructed using two-scale homogenization theory. While this technique is well known to the mathematics and materials communities, it is relatively novel to problems in the solid Earth. This…
In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950's, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on…
Many biological and physical systems exhibit behaviour at multiple spatial, temporal or population scales. Multiscale processes provide challenges when they are to be simulated using numerical techniques. While coarser methods such as…
Simulation of stochastic spatially-extended systems is a challenging problem. The fundamental quantities in these models are individual entities such as molecules, cells, or animals, which move and react in a random manner. In big systems,…
We consider the problem of solving a large-scale system of linear equations in a distributed or federated manner by a taskmaster and a set of machines, each possessing a subset of the equations. We provide a comprehensive comparison of two…
Common techniques for the spatial discretisation of PDEs on a macroscale grid include finite difference, finite elements and finite volume methods. Such methods typically impose assumed microscale structures on the subgrid fields, so…
Model merging is a technique that combines multiple large pretrained models into a single model with enhanced performance and broader task adaptability. It has gained popularity in large pretrained model development due to its ability to…