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.Comment: 27 pages; some changes to notation and figure

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

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.Comment: Post-revision version. To appear in SIAM Journal on Scientific Computin

Target Depth Estimation Using Hull Mounted Active Sonar

High false alarm rates are a problem in anti-submarine warfare in littoral waters using active broadband sonar. Automatic classification algorithms may help combat this problem by filtering out detections due to non-threatening targets. An important feature for classification purposes is knowledge of the target's depth. Using active sonar with vertical beamforming capabilities, the received signal from a target can be used to find an estimate of the target's depth given an initial guess of the target's horizontal distance from the ship, the bottom profile and the sound speed profile. The estimation is done by an optimization algorithm. The algorithm varies relevant parameters and models signals based on these parameters, comparing the modelled signals with the received signal until parameters providing an optimal fit are found. The modelling is based on using a ray tracing procedure to find eigenrays for a candidate target depth, finding vertical arrival angles and arrival times by use of these eigenrays, and synthesizing a signal based on the arrival angles and arrival times. The ray tracing procedure is done numerically using LYBIN, a platform developed by the Norwegian Defence Logistics Organization (NDLO). Three candidate objective functions for comparing recorded signals to modelled signals are presented. The validity of the eigenray finding procedure is confirmed, and results from testing the optimization procedure on synthetic data when applying the different objective functions are presented. The results show that the method produces target depth estimates which are suitable for classification purposes

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

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.submittedVersio