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

Spectrally accurate space-time solution of Hamiltonian PDEs

Recently, the numerical solution of multi-frequency, highly-oscillatory Hamiltonian problems has been attacked by using Hamiltonian Boundary Value Methods (HBVMs) as spectral methods in time. When the problem derives from the space semi- discretization of (possibly Hamiltonian) partial differential equations (PDEs), the resulting problem may be stiffly-oscillatory, rather than highly-oscillatory. In such a case, a different implementation of the methods is needed, in order to gain the maximum efficiency.Comment: 17 pages, 3 figure

Shape Calculus for Shape Energies in Image Processing

Many image processing problems are naturally expressed as energy minimization or shape optimization problems, in which the free variable is a shape, such as a curve in 2d or a surface in 3d. Examples are image segmentation, multiview stereo reconstruction, geometric interpolation from data point clouds. To obtain the solution of such a problem, one usually resorts to an iterative approach, a gradient descent algorithm, which updates a candidate shape gradually deforming it into the optimal shape. Computing the gradient descent updates requires the knowledge of the first variation of the shape energy, or rather the first shape derivative. In addition to the first shape derivative, one can also utilize the second shape derivative and develop a Newton-type method with faster convergence. Unfortunately, the knowledge of shape derivatives for shape energies in image processing is patchy. The second shape derivatives are known for only two of the energies in the image processing literature and many results for the first shape derivative are limiting, in the sense that they are either for curves on planes, or developed for a specific representation of the shape or for a very specific functional form in the shape energy. In this work, these limitations are overcome and the first and second shape derivatives are computed for large classes of shape energies that are representative of the energies found in image processing. Many of the formulas we obtain are new and some generalize previous existing results. These results are valid for general surfaces in any number of dimensions. This work is intended to serve as a cookbook for researchers who deal with shape energies for various applications in image processing and need to develop algorithms to compute the shapes minimizing these energies