Related papers: Adjoint and Its roles in Sciences, Engineering, an…
As more and more multiphysics effects are entering the field of CFD simulations, this raises the question how they can be accurately captured in gradient computations for shape optimization. The latter has been successfully enriched over…
Symmetries and adjoint-symmetries are two fundamental (coordinate-free) structures of PDE systems. Recent work has developed several new algebraic aspects of adjoint-symmetries: three fundamental actions of symmetries on adjoint-symmetries;…
Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define - perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which…
This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits…
The theory of abstract convexity, also known as convexity without linearity, is an extension of the classical convex analysis. There are a number of remarkable results, mostly concerning duality, and some numerical methods, however, this…
This note is an invitation to the theory of geometric functions. The foundation techniques and some of the developments in the field are explained with the mindset that the audience is principally young researchers wishing to understand…
We present a new software system PETSc TSAdjoint for first-order and second-order adjoint sensitivity analysis of time-dependent nonlinear differential equations. The derivative calculation in PETSc TSAdjoint is essentially a high-level…
Embedders play a central role in machine learning, projecting any object into numerical representations that can, in turn, be leveraged to perform various downstream tasks. The evaluation of embedding models typically depends on…
We survey some recent applications of machine learning to problems in geometry and theoretical physics. Pure mathematical data has been compiled over the last few decades by the community and experiments in supervised, semi-supervised and…
We study selfadjoint functors acting on categories of finite dimensional modules over finite dimensional algebras with an emphasis on functors satisfying some polynomial relations. Selfadjoint functors satisfying several easy relations, in…
Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications. They have been used in many areas, including combinatorial optimization, machine learning, and economics. In…
Derivative-based algorithms are ubiquitous in statistics, machine learning, and applied mathematics. Automatic differentiation offers an algorithmic way to efficiently evaluate these derivatives from computer programs that execute relevant…
Joint inversion refers to the simultaneous inference of multiple parameter fields from observations of systems governed by single or multiple forward models. In many cases these parameter fields reflect different attributes of a single…
Network embedding methods aim at learning low-dimensional latent representation of nodes in a network. These representations can be used as features for a wide range of tasks on graphs such as classification, clustering, link prediction,…
We construct self-adjoint operators in the direct sum of a complex Hilbert space $H$ and a finite dimensional complex inner product space $W$. The operator theory developed in this paper for the Hilbert space $H\oplus W$ is originally…
A recent paper by Boughammoura (2023) describes the back-propagation algorithm in terms of an alternative formulation called the F-adjoint method. In particular, by the F-adjoint algorithm the computation of the loss gradient, with respect…
A densely-defined symmetric linear map from/to a real Hilbert space extends to a self-adjoint map. Extension is expressed via Riesz representation. For a case including Friedrichs extension of a strongly monotone map, self-adjoint extension…
We present a logical and algebraic description of right adjoint functors between generalized quasi-varieties, inspired by the work of McKenzie on category equivalence. This result is achieved by developing a correspondence between the…
Optimization problems, generalized equations, and the multitude of other variational problems invariably lead to the analysis of sets and set-valued mappings as well as their approximations. We review the central concept of set-convergence…
Adjoint methods have gained popularity in recent years for driving adaptation procedures which aim to reduce error in solution functionals. While adjoint methods have been proven effective for functional-based adaptation, the practical…