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A further significant extension is presented of the infinitely large class of differential algebras of generalized functions which are the basic structures in the nonlinear algebraic theory listed under 46F30 in the AMS Mathematical Subject…
In this paper, we deal with an elliptic problem with the Dirichlet boundary condition. We operate in Sobolev spaces and the main analytic tool we use is the Lax-Milgram lemma. First, we present the variational approach of the problem which…
The spreadsheet application is among the most widely used computing tools in modern society. It provides excellent usability and usefulness, and it easily enables a non-programmer to perform programming-like tasks in a visual tabular "pen…
Goal-conditioned reinforcement learning (RL) is a promising direction for training agents that are capable of solving multiple tasks and reach a diverse set of objectives. How to \textit{specify} and \textit{ground} these goals in such a…
We study several numerical discretization techniques for the one-space plus one-time dimensional Dirac equation, including finite difference and space-time finite element methods. Two finite difference schemes and several space-time finite…
In this paper, we introduce the locally conservative enriched immersed finite element method (EIFEM) to tackle the elliptic problem with interface. The immersed finite element is useful for handling interface with mesh unfit with the…
Partially class-disjoint data (PCDD), a common yet under-explored data formation where each client contributes a part of classes (instead of all classes) of samples, severely challenges the performance of federated algorithms. Without full…
A finite element analysis of a Dirichlet boundary control problem governed by the linear parabolic equation is presented in this article. The Dirichlet control is considered in a closed and convex subset of the energy space $H^1(\Omega…
This review discusses progress in efficient solvers which have as their foundation a representation in real space, either through finite-difference or finite-element formulations. The relationship of real-space approaches to linear-scaling…
Remote Controlled laboratories is a teaching and learning tool that increasingly becomes fundamental in the teaching and learning processes at all the levels. A study of available systems highlights a series of limitations on the used…
We discuss computational and qualitative aspects of the fractional Plateau and the prescribed fractional mean curvature problems on bounded domains subject to exterior data being a subgraph. We recast these problems in terms of energy…
Quantum algorithms have begun to surpass classical ones in several computation fields, yet practical application remains challenging due to hardware and software limitations. Here, we introduce a quantum algorithm that quadratically…
The alignment of autonomous agents with human values is a pivotal challenge when deploying these agents within physical environments, where safety is an important concern. However, defining the agent's objective as a reward and/or cost…
Partial Differential Equations (PDEs) describe phenomena ranging from turbulence and epidemics to quantum mechanics and financial markets. Despite recent advances in computational science, solving such PDEs for real-world applications…
The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential…
This thesis presents the development of a novel finite element library in Rust based on the principles of Finite Element Exterior Calculus (FEEC). The library solves partial differential equations formulated using differential forms on…
We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations (PDEs). The approach is based on a finite element discretization and aims at representing the solution…
Reinforcement learning (RL) is attracting attention as an effective way to solve sequential optimization problems that involve high dimensional state/action space and stochastic uncertainties. Many such problems involve constraints…
Personalized federated learning (PFL) is a popular framework that allows clients to have different models to address application scenarios where clients' data are in different domains. The typical model of a client in PFL features a global…
We encounter a bottleneck when we try to borrow the strength of classical classifiers to classify functional data. The major issue is that functional data are intrinsically infinite dimensional, thus classical classifiers cannot be applied…