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It is known that state-dependent, multi-step Lyapunov bounds lead to greatly simplified verification theorems for stability for large classes of Markov chain models. This is one component of the "fluid model" approach to stability of…
We consider a singularly-perturbed two-well problem in the context of planar geometrically linear elasticity to model a rectangular martensitic nucleus in an austenitic matrix. We derive the scaling regimes for the minimal energy in terms…
This paper contributes to the exploration of a recently introduced computational paradigm known as second-order flows, which are characterized by novel dissipative hyperbolic partial differential equations extending accelerated gradient…
We extend and analyze the energy-based discontinuous Galerkin method for second order wave equations on staggered and structured meshes. By combining spatial staggering with local time-stepping near boundaries, the method overcomes the…
The paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finitedimensional spaces. The main attention is paid to the…
In this paper, we study a novel second-order energy stable Backward Differentiation Formula (BDF) finite difference scheme for the epitaxial thin film equation with slope selection (SS). One major challenge for the higher oder in time…
We study topology optimization governed by the incompressible Navier-Stokes flows using a phase field model. Novel stabilized semi-implicit schemes for the gradient flows of Allen-Cahn and Cahn-Hilliard types are proposed for solving the…
Semi-discrete Galerkin formulations of transient wave equations, either with conforming or discontinuous Galerkin finite element discretizations, typically lead to large systems of ordinary differential equations. When explicit time…
We propose a mesh-free method to solve nonconvex energy minimization problems for martensitic phase transitions and twinning in crystals, using the deep learning approach. These problems pose multiple challenges to both analysis and…
Hyperbolic reaction-diffusion equations have recently attracted attention both for their application to a variety of biological and chemical phenomena, and for their distinct features in terms of propagation speed and novel instabilities…
In this paper, we propose second-order sufficient optimality conditions for a very general nonconvex constrained optimization problem, which covers many prominent mathematical programs.Unlike the existing results in the literature, our…
In this paper we consider a simplified two-dimensional scalar model for the formation of mesoscopic domain patterns in martensitic shape-memory alloys at the interface between a region occupied by the parent (austenite) phase and a region…
We propose a new numerical technique to deal with nonlinear terms in gradient flows. By introducing a scalar auxiliary variable (SAV), we construct efficient and robust energy stable schemes for a large class of gradient flows. The SAV…
When a crystalline solid such as iron is cooled across a structural transition, its final microstructure depends sensitively on the cooling rate. For instance, an adiabatic cooling across the transition results in an equilibrium `ferrite',…
We recapitulate recent developments of the functional renormalization group (FRG) approach to the steady state of systems out of thermal equilibrium. In particular, we discuss second-order truncation schemes which account for the…
Development of thermodynamic induction up to second order gives a dynamical bifurcation for thermodynamic variables and allows for the prediction and detailed explanation of nonequilibrium phase transitions with associated spontaneous…
Non-local systems of conservation laws play a crucial role in modeling flow mechanisms across various scenarios. The well-posedness of such problems is typically established by demonstrating the convergence of robust first-order schemes.…
An algorithm for a family of self-starting high-order implicit time integration schemes with controllable numerical dissipation is proposed for both linear and nonlinear transient problems. This work builds on the previous works of the…
We consider finite configurations of particles and holes sampled according to Bernoulli product measure and with a second class particle added to a random position. The stabilization time is the number of steps needed to reach an ordered…
Many differential equations with physical backgrounds are described as gradient systems, which are evolution equations driven by the gradient of some functionals, and such problems have energy conservation or dissipation properties. For…