To solve variational indefinite problems, a celebrated tool is the Banach–Nečas– Babuška theory, which relies on the inf-sup condition. Here, we choose an alternate theory, T-coercivity. This theory relies on explicit inf-sup operators, both at the continuous and discrete levels. It is applied to solve Helmholtz-like problems in acoustics and electromagnetics. We provide simple proofs to solve ...

For an anisotropic discrete nonlinear problem with variable exponent, we demonstrate both the existence and multiplicity of nontrivial solutions in this study. The variational principle critical point theory are key techniques employed here.

Nowadays color image processing is an essential issue in computer vision. Variational formulations provide a framework for color image restoration, smoothing and segmentation problems. The solutions of variational models can be obtained by minimizing appropriate energy functions and this minimization is usually performed by continuous partial differential equations (PDEs). The problem is usuall...

Determining the ground state of multi-orbital Hubbard models is critical for understanding strongly correlated electron materials, yet existing methods struggle to simultaneously reach zero temperature and infinite system size. The \textit{de facto} standard approximate a finite dimension model with $d=\infty$ version, which can then be formally solved via dynamical mean-field theory (DMFT), th...

My research has focused on developing the mathematical foundations of discrete geometry and mechanics to enable the systematic construction of geometric structure-preserving numerical schemes based on the approach of geometric mechanics, with a view towards obtaining more robust and accurate numerical implementations of feedback and optimal control laws arising from geometric control theory. Th...

In this short review, we introduce the mathematical framework of the principle of the fermionic projector and set up a variational principle in discrete space-time. The connection to the continuum theory is outlined. Recent results and open problems are discussed. The principle of the fermionic projector [1] provides a new model of space-time together with the mathematical framework for the for...

We discuss the principles to be used in the construction of discrete time classical and quantum mechanics as applied to point particle systems. In the classical theory this includes the concept of virtual path and the construction of system functions from classical Lagrangians, Cadzow's variational principle applied to the action sum, Maeda-Noether and Logan invariants of the motion, elliptic a...

A discrete regularization framework on graphs is proposed and studied for color image filtering purposes when images are represented by grid graphs. Image filtering is considered as a variational problem which consists in minimizing an appropriate energy function. In this paper, we propose a general discrete regularization framework defined on weighted graphs which can be seen as a discrete ana...