Related papers: Geometric Methods for Adjoint Systems
Gradient-based techniques are becoming increasingly critical in quantitative fields, notably in statistics and computer science. The utility of these techniques, however, ultimately depends on how efficiently we can evaluate the derivatives…
In this paper, we present and study discontinuous Galerkin (DG) methods for one-dimensional multi-symplectic Hamiltonian partial differential equations. We particularly focus on semi-discrete schemes with spatial discretization only, and…
An interesting family of geometric integrators for Lagrangian systems can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators. In this…
We find the Lie point symmetries of the Novikov equation and demonstrate that it is strictly self-adjoint. Using the self-adjointness and the recent technique for constructing conserved vectors associated with symmetries of differential…
Computation methods for solving entropy-regularized reward optimization -- a class of problems widely used for fine-tuning generative models -- have advanced rapidly. Among those, Adjoint Matching (AM, Domingo-Enrich et al., 2025) has…
We formulate a geometric measurement theory of dynamical classical systems possessing both continuous and discrete degrees of freedom. The approach is covariant with respect to choices of clocks and canonically incorporates laboratories.…
Several recent works have explored stochastic gradient methods for variational inference that exploit the geometry of the variational-parameter space. However, the theoretical properties of these methods are not well-understood and these…
Asynchronous stochastic gradient descent (ASGD) is a popular parallel optimization algorithm in machine learning. Most theoretical analysis on ASGD take a discrete view and prove upper bounds for their convergence rates. However, the…
Variational integrators for Lagrangian dynamical systems provide a systematic way to derive geometric numerical methods. These methods preserve a discrete multisymplectic form as well as momenta associated to symmetries of the Lagrangian…
We explore the connection between two problems that have arisen independently in the signal processing and related fields: the estimation of the geometric mean of a set of symmetric positive definite (SPD) matrices and their approximate…
A new proof for adjoint systems of linear equations is presented. The argument is built on the principles of Algorithmic Differentiation. Application to scalar multiplication sets the base line. Generalization yields adjoint inner vector,…
To study discrete dynamical systems of different types --- deterministic, statistical and quantum --- we develop various approaches. We introduce the concept of a system of discrete relations on an abstract simplicial complex and develop…
This work investigates an elliptic optimal control problem defined on uncertain domains and discretized by a fictitious domain finite element method and cut elements. Key ingredients of the study are to manage cases considering the usually…
In this paper we show how almost cosymplectic structures are a natural framework to study thermodynamical systems. Indeed, we are able to obtain the same evolution equations obtained previously by Gay-Balmaz and Yoshimura (see Entropy,…
This paper applies the gradient discretisation method (GDM) for fourth order elliptic variational inequalities. The GDM provides a new formulation of error estimates and a complete convergence analysis of several numerical methods. We show…
Algorithmic differentiation (AD) has become increasingly capable and straightforward to use. However, AD is inefficient when applied directly to solvers, a feature of most engineering analyses. We can leverage implicit differentiation to…
A new method is proposed for integrating the equations of motion of an elastic filament. In the standard finite-difference and finite-element formulations the continuum equations of motion are discretized in space and time, but it is then…
Discrete conjugate systems are quadrilateral nets with all planar faces. Discrete orthogonal systems are defined by the additional property of all faces being concircular. Their geometric properties allow one to consider them as proper…
This work extends the Ibragimov's conservation theorem for partial differential equations [{\it J. Math. Anal. Appl. 333 (2007 311-328}] to under determined systems of differential equations. The concepts of adjoint equation and formal…
This paper presents a geometric variational discretization of compressible fluid dynamics. The numerical scheme is obtained by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups…