arXiv subjects

Torbjørn Ringholm

Publications and source records attributed to Torbjørn Ringholm.

7 recordsLinked to original sources

arXiv2024

A geometric integration approach to smooth optimisation: Foundations of the discrete gradient method

Discrete gradient methods are geometric integration techniques that can preserve the dissipative structure of gradient flows. Due to the monotonic decay of the function values, they are well suited for general convex and nonconvex optimisation problems. Both zero- and first-order algorithms can be derived from the discrete gradient method by selecting different discrete gradients. In this paper, we present a thorough analysis of the discrete gradient method for optimisation which provides a solid theoretical foundation. We show that the discrete gradient method is well-posed by proving the existence of iterates for any positive time step, as well as uniqueness in some cases, and propose an efficient method for solving the associated discrete gradient equation. Moreover, we establish an $O(1/k)$ convergence rate for convex objectives and prove linear convergence if instead the Polyak-Lojasiewicz inequality is satisfied. The analysis is carried out for three discrete gradients-the Gonzalez discrete gradient, the mean value discrete gradient, and the Itoh-Abe discrete gradient, as well as for a randomised Itoh-Abe method. Our theoretical results are illustrated with a variety of numerical experiments, and we furthermore demonstrate that the methods are robust with respect to stiffness.

math.OC↗

arXiv2019

Boundary determination for hybrid imaging from a single measurement

We recover the conductivity $σ$ at the boundary of a domain from a combination of interior and boundary data, with a single quite arbitrary measurement, in AET or CDII. The argument is elementary and local. More generally, we consider the variable exponent $p(\cdot)$-Laplacian as a forward model with the interior data $σ|\nabla u|^q$, and find out that single measurement specifies the boundary conductivity when $p-q \ge 1$, and otherwise the measurement specifies two alternatives. We present heuristics for selecting between these alternatives. Both $p$ and $q$ may depend on the spatial variable $x$, but they are assumed to be a priori known. We illustrate the practical situations with numerical examples.

math.NA↗

arXiv2018

Dissipative numerical schemes on Riemannian manifolds with applications to gradient flows

This paper concerns an extension of discrete gradient methods to finite-dimensional Riemannian manifolds termed discrete Riemannian gradients, and their application to dissipative ordinary differential equations. This includes Riemannian gradient flow systems which occur naturally in optimization problems. The Itoh--Abe discrete gradient is formulated and applied to gradient systems, yielding a derivative-free optimization algorithm. The algorithm is tested on two eigenvalue problems and two problems from manifold valued imaging: InSAR denoising and DTI denoising.

math.NA↗

arXiv2018

Variational image regularization with Euler's elastica using a discrete gradient scheme

This paper concerns an optimization algorithm for unconstrained non-convex problems where the objective function has sparse connections between the unknowns. The algorithm is based on applying a dissipation preserving numerical integrator, the Itoh--Abe discrete gradient scheme, to the gradient flow of an objective function, guaranteeing energy decrease regardless of step size. We introduce the algorithm, prove a convergence rate estimate for non-convex problems with Lipschitz continuous gradients, and show an improved convergence rate if the objective function has sparse connections between unknowns. The algorithm is presented in serial and parallel versions. Numerical tests show its use in Euler's elastica regularized imaging problems and its convergence rate and compare the execution time of the method to that of the iPiano algorithm and the gradient descent and Heavy-ball algorithms.

math.OC↗

arXiv2018

Adaptive Energy Preserving Methods for Partial Differential Equations

A method for constructing first integral preserving numerical schemes for time-dependent partial differential equations on non-uniform grids is presented. The method can be used with both finite difference and partition of unity approaches, thereby also including finite element approaches. The schemes are then extended to accommodate $r$-, $h$- and $p$-adaptivity. The method is applied to the Korteweg-de Vries equation and the Sine-Gordon equation and results from numerical experiments are presented.

math.NA↗

arXiv2018

Energy preserving methods on Riemannian manifolds

The energy preserving discrete gradient methods are generalized to finite-dimensional Riemannian manifolds by definition of a discrete approximation to the Riemannian gradient, a retraction, and a coordinate center function. The resulting schemes are intrinsic and do not depend on a particular choice of coordinates, nor on embedding of the manifold in a Euclidean space. Generalizations of well-known discrete gradient methods, such as the average vector field method and the Itoh--Abe method are obtained. It is shown how methods of higher order can be constructed via a collocation-like approach. Local and global error bounds are derived in terms of the Riemannian distance function and the Levi-Civita connection. Some numerical results on spin system problems are presented.

math.NA↗

arXiv2017

Energy preserving moving mesh methods applied to the BBM equation

Energy preserving numerical methods for a certain class of PDEs are derived, applying the partition of unity method. The methods are extended to also be applicable in combination with moving mesh methods by the rezoning approach. These energy preserving moving mesh methods are then applied to the Benjamin--Bona--Mahony equation, resulting in schemes that exactly preserve an approximation to one of the Hamiltonians of the system. Numerical experiments that demonstrate the advantages of the methods are presented.

math.NA↗