51 research outputs found
Models of Delay Differential Equations
This book gathers a number of selected contributions aimed at providing a balanced picture of the main research lines in the realm of delay differential equations and their applications to mathematical modelling. The contributions have been carefully selected so that they cover interesting theoretical and practical analysis performed in the deterministic and the stochastic settings. The reader will find a complete overview of recent advances in ordinary and partial delay differential equations with applications in other multidisciplinary areas such as Finance, Epidemiology or Engineerin
The Theory of Functional Connections: A Journey from Theory to Application
The Theory of Functional Connections (TFC) is a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The functionals derived from this method, called "constrained expressions," analytically satisfy the imposed constraints and can be leveraged to transform constrained optimization problems to unconstrained ones. By simplifying the optimization problem, this technique has been shown to produce a numerical scheme that is faster, more accurate, and robust to poor initialization. The content of this dissertation details the complete development of the Theory of Functional Connections. First, the seminal paper on the Theory of Functional Connections is discussed and motivates the discovery of a more general formulation of the constrained expressions. Leveraging this formulation, a rigorous structure of the constrained expression is produced with associated mathematical definitions, claims, and proofs. Furthermore, the second part of this dissertation explains how this technique can be used to solve ordinary differential equations providing a wide variety of examples compared to the state-of-the-art. The final part of this work focuses on unitizing the techniques and algorithms produced in the prior sections to explore the feasibility of using the Theory of Functional Connections to solve real-time optimal control problems, namely optimal landing problems
Performance of explicit and IMEX MRI multirate methods on complex reactive flow problems within modern parallel adaptive structured grid frameworks
Large-scale multiphysics simulations are computationally challenging due to
the coupling of multiple processes with widely disparate time scales. The
advent of exascale computing systems exacerbates these challenges, since these
enable ever increasing size and complexity. Recently, there has been renewed
interest in developing multirate methods as a means to handle the large range
of time scales, as these methods may afford greater accuracy and efficiency
than more traditional approaches of using IMEX and low-order operator splitting
schemes. However, there have been few performance studies that compare
different classes of multirate integrators on complex application problems. We
study the performance of several newly developed multirate infinitesimal (MRI)
methods, implemented in the SUNDIALS solver package, on two reacting flow model
problems built on structured mesh frameworks. The first model revisits the work
of Emmet et al. (2014) on a compressible reacting flow problem with complex
chemistry that is implemented using BoxLib but where we now include comparisons
between a new explicit MRI scheme with the multirate spectral deferred
correction (SDC) methods in the original paper. The second problem uses the
same complex chemistry as the first problem, combined with a simplified flow
model, but run at a large spatial scale where explicit methods become
infeasible due to stability constraints. Two recently developed
implicit-explicit MRI multirate methods are tested. These methods rely on
advanced features of the AMReX framework on which the model is built, such as
multilevel grids and multilevel preconditioners. The results from these two
problems show that MRI multirate methods can offer significant performance
benefits on complex multiphysics application problems and that these methods
may be combined with advanced spatial discretization to compound the advantages
of both
CutFEM and ghost stabilization techniques for higher order space-time discretizations of the Navier-Stokes equations
We propose and analyze computationally a new fictitious domain method, based
on higher order space-time finite element discretizations, for the simulation
of the nonstationary, incompressible Navier-Stokes equations on evolving
domains. The physical domain is embedded into a fixed computational mesh such
that arbitrary intersections of the moving domain's boundaries with the
background mesh occur. The potential of such cut finite element techniques for
higher order space-time finite element methods has rarely been studied in the
literature so far and deserves further elucidation. The key ingredients of the
approach are the weak formulation of Dirichlet boundary conditions by Nitsche's
method, the flexible and efficient integration over all types of intersections
of cells by moving boundaries and the spatial extension of the discrete
physical quantities to the entire computational background mesh including
fictitious (ghost) subdomains of fluid flow. Thereby, an expensive remeshing
and adaptation of the sparse matrix data structure are avoided and the
computations are accelerated. To prevent spurious oscillations caused by
irregular intersections of mesh cells, a penalization, defining also implicitly
the extension to ghost domains, is added. These techniques are embedded in an
arbitrary order, discontinuous Galerkin discretization of the time variable and
an inf-sup stable discretization of the spatial variables. The parallel
implementation of the matrix assembly is described. The optimal order
convergence properties of the algorithm are illustrated in a numerical
experiment for an evolving domain. The well-known 2d benchmark of flow around a
cylinder as well as flow around moving obstacles with arising cut cells and
fictitious domains are considered further
VPLanet: The Virtual Planet Simulator
We describe a software package called VPLanet that simulates fundamental
aspects of planetary system evolution over Gyr timescales, with a focus on
investigating habitable worlds. In this initial release, eleven physics modules
are included that model internal, atmospheric, rotational, orbital, stellar,
and galactic processes. Many of these modules can be coupled simultaneously to
simulate the evolution of terrestrial planets, gaseous planets, and stars. The
code is validated by reproducing a selection of observations and past results.
VPLanet is written in C and designed so that the user can choose the physics
modules to apply to an individual object at runtime without recompiling, i.e.,
a single executable can simulate the diverse phenomena that are relevant to a
wide range of planetary and stellar systems. This feature is enabled by
matrices and vectors of function pointers that are dynamically allocated and
populated based on user input. The speed and modularity of VPLanet enables
large parameter sweeps and the versatility to add/remove physical phenomena to
assess their importance. VPLanet is publicly available from a repository that
contains extensive documentation, numerous examples, Python scripts for
plotting and data management, and infrastructure for community input and future
development.Comment: 75 pages, 34 figures, 10 tables, accepted to the Proceedings of the
Astronomical Society of the Pacific. Source code, documentation, and examples
available at https://github.com/VirtualPlanetaryLaboratory/vplane
Numerical scalar curvature deformation and a gluing construction
In this work a new numerical technique to prepare Cauchy data for the initial value problem (IVP) formulation of Einstein's field equations (EFE) is presented. Our method is directly inspired by the exterior asymptotic gluing (EAG) result of Corvino (2000). The argument assumes a moment in time symmetry and allows for a composite, initial data set to be assembled from (a finite subdomain of) a known asymptotically Euclidean initial data set which is glued (in a controlled manner) over a compact spatial region to an exterior Schwarzschildean representative. We demonstrate how (Corvino, 2000) may be directly adapted to a numerical scheme and under the assumption of axisymmetry construct composite Hamiltonian constraint satisfying initial data featuring internal binary black holes (BBH) glued to exterior Schwarzschild initial data in isotropic form. The generality of the method is shown in a comparison of properties of EAG composite initial data sets featuring internal BBHs as modelled by Brill-Lindquist and Misner data.
The underlying geometric analysis character of gluing methods requires work within suitably weighted function spaces, which, together with a technical impediment preventing (Corvino, 2000) from being fully constructive, is the principal difficulty in devising a numerical technique. Thus the single previous attempt by Giulini and Holzegel (2005) (recently implemented by Doulis and Rinne (2016)) sought to avoid this by embedding the result within the well known Lichnerowicz-York conformal framework which required ad-hoc assumptions on solution form and a formal perturbative argument to show that EAG may proceed. In (Giulini and Holzegel, 2005) it was further claimed that judicious engineering of EAG can serve to reduce the presence of spurious gravitational radiation - unfortunately, in line with the general conclusion of (Doulis and Rinne, 2016) our numerical investigation does not appear to indicate that this is the case.
Concretising the sought initial data to be specified with respect to a spatial manifold with underlying topology R×S² our method exploits a variety of pseudo-spectral (PS) techniques. A combination of the eth-formalism and spin-weighted spherical harmonics together with a novel complex-analytic based numerical approach is utilised. This is enabled by our Python 3 based numerical toolkit allowing for unified just-in-time compiled, distributed calculations with seamless extension to arbitrary precision for problems involving generic, geometric partial differential equations (PDE) as specified by tensorial expressions. Additional features include a layer of abstraction that allows for automatic reduction of indicial (i.e., tensorial) expressions together with grid remapping based on chart specification - hence straight-forward implementation of IVP formulations of the EFE such as ADM-York or ADM-York-NOR is possible. Code-base verification is performed by evolving the polarised Gowdy T³ space-time with the above formulations utilising high order, explicit time-integrators in the method of lines approach as combined with PS techniques.
As the initial data we prepare has a precise (Schwarzschild) exterior this may be of interest to global evolution schemes that incorporate information from spatial-infinity. Furthermore, our approach may shed light on how more general gluing techniques could potentially be adapted for numerical work. The code-base we have developed may also be of interest in application to other problems involving geometric PDEs
PFASST-ER: Combining the Parallel Full Approximation Scheme in Space and Time with parallelization across the method
To extend prevailing scaling limits when solving time-dependent partial
differential equations, the parallel full approximation scheme in space and
time (PFASST) has been shown to be a promising parallel-in-time integrator.
Similar to a space-time multigrid, PFASST is able to compute multiple
time-steps simultaneously and is therefore in particular suitable for
large-scale applications on high performance computing systems. In this work we
couple PFASST with a parallel spectral deferred correction (SDC) method,
forming an unprecedented doubly time-parallel integrator. While PFASST provides
global, large-scale "parallelization across the step", the inner parallel SDC
method allows to integrate each individual time-step "parallel across the
method" using a diagonalized local Quasi-Newton solver. This new method, which
we call "PFASST with Enhanced concuRrency" (PFASST-ER), therefore exposes even
more temporal parallelism. For two challenging nonlinear reaction-diffusion
problems, we show that PFASST-ER works more efficiently than the classical
variants of PFASST and can be used to run parallel-in-time beyond the number of
time-steps.Comment: 12 pages, 12 figures, CVS PinT Workshop Proceeding
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