Reduced Order Optimal Control of the Convective FitzHugh-Nagumo Equation

In this paper, we compare three model order reduction methods: the proper orthogonal decomposition (POD), discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD) for the optimal control of the convective FitzHugh-Nagumo (FHN) equations. The convective FHN equations consists of the semi-linear activator and the linear inhibitor equations, modeling blood coagulation in moving excitable media. The semilinear activator equation leads to a non-convex optimal control problem (OCP). The most commonly used method in reduced optimal control is POD. We use DEIM and DMD to approximate efficiently the nonlinear terms in reduced order models. We compare the accuracy and computational times of three reduced-order optimal control solutions with the full order discontinuous Galerkin finite element solution of the convection dominated FHN equations with terminal controls. Numerical results show that POD is the most accurate whereas POD-DMD is the fastest

A rapidly converging domain decomposition method for the Helmholtz equation

A new domain decomposition method is introduced for the heterogeneous 2-D and 3-D Helmholtz equations. Transmission conditions based on the perfectly matched layer (PML) are derived that avoid artificial reflections and match incoming and outgoing waves at the subdomain interfaces. We focus on a subdivision of the rectangular domain into many thin subdomains along one of the axes, in combination with a certain ordering for solving the subdomain problems and a GMRES outer iteration. When combined with multifrontal methods, the solver has near-linear cost in examples, due to very small iteration numbers that are essentially independent of problem size and number of subdomains. It is to our knowledge only the second method with this property next to the moving PML sweeping method.Comment: 16 pages, 3 figures, 6 tables - v2 accepted for publication in the Journal of Computational Physic

A Level Set Approach to Eulerian-Lagrangian Coupling

We present a numerical method for coupling an Eulerian compressible flow solver with a Lagrangian solver for fast transient problems involving fluid-solid interactions. Such coupling needs arise when either specific solution methods or accuracy considerations necessitate that different and disjoint subdomains be treated with different (Eulerian or Lagrangian)schemes. The algorithm we propose employs standard integration of the Eulerian solution over a Cartesian mesh. To treat the irregular boundary cells that are generated by an arbitrary boundary on a structured grid, the Eulerian computational domain is augmented by a thin layer of Cartesian ghost cells. Boundary conditions at these cells are established by enforcing conservation of mass and continuity of the stress tensor in the direction normal to the boundary. The description and the kinematic constraints of the Eulerian boundary rely on the unstructured Lagrangian mesh. The Lagrangian mesh evolves concurrently, driven by the traction boundary conditions imposed by the Eulerian counterpart. Several numerical tests designed to measure the rate of convergence and accuracy of the coupling algorithm are presented as well. General problems in one and two dimensions are considered, including a test consisting of an isotropic elastic solid and a compressible fluid in a fully coupled setting where the exact solution is available

Pipeline Implementations of Neumann-Neumann and Dirichlet-Neumann Waveform Relaxation Methods

This paper is concerned with the reformulation of Neumann-Neumann Waveform Relaxation (NNWR) methods and Dirichlet-Neumann Waveform Relaxation (DNWR) methods, a family of parallel space-time approaches to solving time-dependent PDEs. By changing the order of the operations, pipeline-parallel computation of the waveform iterates are possible without changing the final solution. The parallel efficiency and the increased communication cost of the pipeline implementation is presented, along with weak scaling studies to show the effectiveness of the pipeline NNWR and DNWR algorithms.Comment: 20 pages, 8 figure

Digital waveguide modeling for wind instruments: building a state-space representation based on the Webster-Lokshin model

This paper deals with digital waveguide modeling of wind instruments. It presents the application of state-space representations for the refined acoustic model of Webster-Lokshin. This acoustic model describes the propagation of longitudinal waves in axisymmetric acoustic pipes with a varying cross-section, visco-thermal losses at the walls, and without assuming planar or spherical waves. Moreover, three types of discontinuities of the shape can be taken into account (radius, slope and curvature). The purpose of this work is to build low-cost digital simulations in the time domain based on the Webster-Lokshin model. First, decomposing a resonator into independent elementary parts and isolating delay operators lead to a Kelly-Lochbaum network of input/output systems and delays. Second, for a systematic assembling of elements, their state-space representations are derived in discrete time. Then, standard tools of automatic control are used to reduce the complexity of digital simulations in the time domain. The method is applied to a real trombone, and results of simulations are presented and compared with measurements. This method seems to be a promising approach in term of modularity, complexity of calculation and accuracy, for any acoustic resonators based on tubes

Characterising poroelastic materials in the ultrasonic range - A Bayesian approach

Acoustic fields scattered by poroelastic materials contain key information about the materials' pore structure and elastic properties. Therefore, such materials are often characterised with inverse methods that use acoustic measurements. However, it has been shown that results from many existing inverse characterisation methods agree poorly. One reason is that inverse methods are typically sensitive to even small uncertainties in a measurement setup, but these uncertainties are difficult to model and hence often neglected. In this paper, we study characterising poroelastic materials in the Bayesian framework, where measurement uncertainties can be taken into account, and which allows us to quantify uncertainty in the results. Using the finite element method, we simulate measurements where ultrasonic waves are incident on a water-saturated poroelastic material in normal and oblique angles. We consider uncertainties in the incidence angle and level of measurement noise, and then explore the solution of the Bayesian inverse problem, the posterior density, with an adaptive parallel tempering Markov chain Monte Carlo algorithm. Results show that both the elastic and pore structure parameters can be feasibly estimated from ultrasonic measurements.Comment: Published in JSV. https://doi.org/10.1016/j.jsv.2019.05.02