相关论文: Variational Multiscale Closures for Finite Element…
We develop rigorous estimates and provably convergent approximations for the memory integral in the Mori-Zwanzig (MZ) formulation. The new theory is built upon rigorous mathematical foundations and is presented for both state-space and…
A new technique to derive delay models from systems of partial differential equations, based on the Mori-Zwanzig formalism, is used to derive a delay difference equation model for the Atlantic Multidecadal Oscillation. The Mori-Zwanzig…
A typical problem with the conventional Galerkin approach for the construction of finite-mode models is to keep structural properties unaffected in the process of discretization. We present two examples of finite-mode approximations that in…
We describe a compatible finite element discretisation for the shallow water equations on the rotating sphere, concentrating on integrating consistent upwind stabilisation into the framework. Although the prognostic variables are velocity…
A first-principle multiscale modeling approach is presented, which is derived from the solution of the Ornstein-Zernike equation for the coarse-grained representation of polymer liquids. The approach is analytical, and for this reason is…
Capturing the correct dynamics at the Coarse-Grained (CG) scale remains a central challenge in the advancement of systematic CG models for soft matter simulations. The Generalized Langevin Equation (GLE), rooted in the Mori-Zwanzig…
We develop a thorough mathematical analysis of the effective Mori-Zwanzig (EMZ) equation governing the dynamics of noise-averaged observables in stochastic differential equations driven by multiplicative Gaussian white noise. Building upon…
Many unsteady flows exhibiting complex dynamics are nevertheless characterized by emergent large-scale coherence in space and time. Reduced-order models based on Galerkin projection of the governing equations onto an orthogonal modal basis…
We introduce a variational multiscale closure modeling strategy for the numerical stabilization of proper orthogonal decomposition reduced-order models of convection-dominated equations. As a first step, the new model is analyzed and tested…
In this paper we consider the problem of deriving approximate autonomous dynamics for a number of variables of a dynamical system, which are weakly coupled to the remaining variables. In a previous paper we have used the Ruelle response…
Algebraic multigrid (AMG) methods derive their optimal efficiency from the interplay between a relaxation process and a corresponding coarse grid correction. In many standard formulations, relaxation and coarse-graining are analyzed and…
Coarse-grained (CG) models are simplified representations of soft matter systems that are commonly employed to overcome size and time limitations in computational studies. Many approaches have been developed to construct and parametrise…
We introduce and analyze a discontinuous Galerkin method for the numerical modelling of the equations of Multiple-Network Poroelastic Theory (MPET) in the dynamic formulation. The MPET model can comprehensively describe functional changes…
We introduce a combinatorial version Mori-Zwanzig theory and develop from it a family of self-consistent evolution equations for the correlation function or Green's function of interactive many-body systems. The core idea is to use an…
In this work, we illustrate the underlying mathematical structure of mixed-dimensional models arising from the composition of graphs and continuous domains. Such models are becoming popular in applications, in particular, to model the human…
We examine a variational multiscale method in which the unresolved fine-scales are approximated element-wise using a discontinuous Galerkin method. We establish stability and convergence results for the methodology as applied to the scalar…
The Mori-Zwanzig projection formalism is widely used in studying systems with many degrees of freedom. We used a system-bath Hamiltonian system to show that the Mori's and Zwanzig's projection procedures are mutual limiting cases of each…
We present a Ritz-Galerkin discretization on sparse grids using pre-wavelets, which allows to solve elliptic differential equations with variable coefficients for dimension $d=2,3$ and higher dimensions $d>3$. The method applies multilinear…
Recent advances in learning dynamical systems from data have shown significant promise. However, many existing methods assume access to the full state of the system -- an assumption that is rarely satisfied in practice, where systems are…
In many applications of practical interest, solutions of partial differential equation models arise as critical points of an underlying (energy) functional. If such solutions are saddle points, rather than being maxima or minima, then the…