Related papers: A new discretization of the Euler equation via the…
We introduce a novel integrability-preserving discretization for a broad class of differential equations with variable coefficients, encompassing both linear and nonlinear cases. The construction is achieved via a categorical approach that…
Euler's elastica model has a wide range of applications in Image Processing and Computer Vision. However, the non-convexity, the non-smoothness and the nonlinearity of the associated energy functional make its minimization a challenging…
A categorical theory for the discretization of a large class of dynamical systems with variable coefficients is proposed. It is based on the existence of covariant functors between the Rota category of Galois differential algebras and…
We construct a continuous domain for temporal discretization of differential equations. By using this domain, and the domain of Lipschitz maps, we formulate a generalization of the Euler operator, which exhibits second-order convergence. We…
We develop in this paper a new framework for discrete calculus of variations when the actions have densities involving an arbitrary discretization operator. We deduce the discrete Euler-Lagrange equations for piecewise continuous critical…
The derivation of second-order ordinary differential equations (ODEs) as continuous-time limits of optimization algorithms has been shown to be an effective tool for the analysis of these algorithms. Additionally, discretizing…
In this paper we present a new Eulerian finite element method for the discretization of scalar partial differential equations on evolving surfaces. In this method we use the restriction of standard space-time finite element spaces on a…
For a differential operator $L$ of order $n$ over $C(z)$ with a finite (differential) Galois group $G\subset {\rm GL}(C^n)$, there is an algorithm, by M. van Hoeij and J.-A.~Weil, which computes the associated evaluation of the invariants…
An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a…
We investigate an approach for the numerical solution of differential equations which is based on the perfect discretization of actions. Such perfect discretizations show up at the fixed points of renormalization group transformations. This…
A certain representation for the Heisenberg algebra in finite-difference operators is established. The Lie-algebraic procedure of discretization of differential equations with isospectral property is proposed. Using $sl_2$-algebra based…
Viewing optimization methods as numerical integrators for ordinary differential equations (ODEs) provides a thought-provoking modern framework for studying accelerated first-order optimizers. In this literature, acceleration is often…
A method, recently advanced as the conformable Euler method, a general method for the finite difference discretization of fractional initial value problems for fractions in (0, 1], is shown to be valid only for the integer derivative. The…
We consider a numerical approximation of a linear quadratic control problem constrained by the stochastic heat equation with non-homogeneous Neumann boundary conditions. This involves a combination of distributed and boundary control, as…
The numerical solution of differential equations using machine learning-based approaches has gained significant popularity. Neural network-based discretization has emerged as a powerful tool for solving differential equations by…
An operator-splitting finite element scheme for the time-dependent, high-dimensional radiative transfer equation is presented in this paper. The streamline upwind Petrov-Galerkin finite element method and discontinuous Galerkin finite…
Discrete optimisation problems arise in many different areas and are studied under many different names. In many such problems the quantity to be optimised can be expressed as a sum of functions of a restricted form. Here we present a…
We introduce calculus-based formulas for the continuous Euler and homotopy operators. The 1D continuous homotopy operator automates integration by parts on the jet space. Its 3D generalization allows one to invert the total divergence…
The problem of discretization of Darboux integrable equations is considered. Given a Darboux integrable continuous equation, one can obtain a Darboux integrable differential-discrete equation, using the integrals of the continuous equation.…
In [1] new discretizations of the Euler top have been found. They can be discribed with a Lax pair with a spectral parameter on an elliptic curve. This is used in this paper to perform a finite gap integration.