Krylov projection methods for linear Hamiltonian systems

We study geometric properties of Krylov projection methods for large and sparse linear Hamiltonian systems. We consider in particular energy preservation. We discuss the connection to structure preserving model reduction. We illustrate the performance of the methods by applying them to Hamiltonian PDEs.Comment: 16 pages, 17 figure

Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems

Reduced basis methods are popular for approximately solving large and complex systems of differential equations. However, conventional reduced basis methods do not generally preserve conservation laws and symmetries of the full order model. Here, we present an approach for reduced model construction, that preserves the symplectic symmetry of dissipative Hamiltonian systems. The method constructs a closed reduced Hamiltonian system by coupling the full model with a canonical heat bath. This allows the reduced system to be integrated with a symplectic integrator, resulting in a correct dissipation of energy, preservation of the total energy and, ultimately, in the stability of the solution. Accuracy and stability of the method are illustrated through the numerical simulation of the dissipative wave equation and a port-Hamiltonian model of an electric circuit

Weakly Supervised Audio Source Separation via Spectrum Energy Preserved Wasserstein Learning

Separating audio mixtures into individual instrument tracks has been a long standing challenging task. We introduce a novel weakly supervised audio source separation approach based on deep adversarial learning. Specifically, our loss function adopts the Wasserstein distance which directly measures the distribution distance between the separated sources and the real sources for each individual source. Moreover, a global regularization term is added to fulfill the spectrum energy preservation property regardless separation. Unlike state-of-the-art weakly supervised models which often involve deliberately devised constraints or careful model selection, our approach need little prior model specification on the data, and can be straightforwardly learned in an end-to-end fashion. We show that the proposed method performs competitively on public benchmark against state-of-the-art weakly supervised methods

Performance and error analysis of structure-preserving time-integration procedures for incompressible-flow simulations

The effects of kinetic-energy preservation errors due to Runge-Kutta (RK) temporal integrators have been analyzed for the case of large-eddy simulations of incompressible turbulent channel flow. Simulations have been run using the open-source solver Xcompact3D with an implicit spectral vanishing viscosity model and a variety of temporal Runge-Kutta integrators. Explicit pseudo-symplectic schemes, with improved energy preservation properties, have been compared to standard RK methods. The results show a marked decrease in the temporal error for higher-order pseudo-symplectic methods, and suggest that these schemes could be used to attain results comparable to traditional methods at a reduced computational cost.Postprint (published version