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We propose a variational regularization approach based on a multiscale representation called cylindrical shearlets aimed at dynamic imaging problems, especially dynamic tomography. The intuitive idea of our approach is to integrate a…
We present a novel structure-preserving semi-implicit finite volume method on vertex-based staggered meshes for the compatible discretization of first order systems of time-dependent partial differential equations (PDEs). The method…
Symmetries have proven useful in machine learning models, improving generalisation and overall performance. At the same time, recent advancements in learning dynamical systems rely on modelling the underlying Hamiltonian to guarantee the…
In this paper we study a discrete variational optimal control problem for the rigid body. The cost to be minimized is the external torque applied to move the rigid body from an initial condition to a pre-specified terminal condition.…
We extend an implicit regularization scheme to be applicable in the $n$-dimensional space-time. Within this scheme divergences involving parity violating objects can be consistently treated without recoursing to dimensional continuation.…
A new method for the optimal solutions is proposed. Originating from the continuous-time dynamics stability theory in the control field, the optimal solution is anticipated to be obtained in an asymptotically evolving way. By introducing a…
We consider the continuous and discrete-time Hamilton's variational principle on phase space, and characterize the exact discrete Hamiltonian which provides an exact correspondence between discrete and continuous Hamiltonian mechanics. The…
Geometric flows, where an immersed manifold evolves in time according to its own geometry, exhibit important structural properties. For example, surface diffusion dissipates surface area while conserving volume; it is desirable to preserve…
This work presents an alternative view on the numerical simulation of diffusion processes applied to the heat and moisture transfer through porous building materials. Traditionally, by using the finite-difference approach, the…
Convergence of a full discretization of a second order stochastic evolution equation with nonlinear damping is shown and thus existence of a solution is established. The discretization scheme combines an implicit time stepping scheme with…
Dynamical black holes in the non-perturbative regime are not mathematically well understood. Studying approximate symmetries of spacetimes describing dynamical black holes gives an insight into their structure. Utilising the property that…
We present a method to linearize, without approximation, a specific class of eigenvalue problems with eigenvector nonlinearities (NEPv), where the nonlinearities are expressed by scalar functions that are defined by a quotient of linear…
We analyze, from the viewpoint of positivity preservation, certain discretizations of a fundamental partial differential equation, the one-dimensional advection equation with periodic boundary condition. The full discretization is obtained…
Variational integrators are well-suited for simulation of mechanical systems because they preserve mechanical quantities about a system such as momentum, or its change if external forcing is involved, and holonomic constraints. While they…
We present a collection of algorithms which utilize dimensional reduction to perform mesh refinement and study possibly singular solutions of time-dependent partial differential equations. The algorithms are inspired by constructions used…
We introduce a novel formulation for curvature regularization by penalizing normal curvatures from multiple directions. This total normal curvature regularization is capable of producing solutions with sharp edges and precise isotropic…
Obtainable computational efficiency is evaluated when using an Adaptive Mesh Refinement (AMR) strategy in time accurate simulations governed by sets of conservation laws. For a variety of 1D, 2D, and 3D hydro- and magnetohydrodynamic…
A class of abstract nonlinear time-periodic evolution problems is considered which arise in electrical engineering and other scientific disciplines. An efficient solver is proposed for the systems arising after discretization in time based…
We derive a formulation of the nonhydrostatic equations in spherical geometry with a Lorenz staggered vertical discretization. The combination conserves a discrete energy in exact time integration when coupled with a mimetic horizontal…
This work proposes a novel variational approximation of partial differential equations on moving geometries determined by explicit boundary representations. The benefits of the proposed formulation are the ability to handle large…