92 research outputs found
PRECONDITIONERS AND TENSOR PRODUCT SOLVERS FOR OPTIMAL CONTROL PROBLEMS FROM CHEMOTAXIS
In this paper, we consider the fast numerical solution of an optimal control
formulation of the Keller--Segel model for bacterial chemotaxis. Upon
discretization, this problem requires the solution of huge-scale saddle point
systems to guarantee accurate solutions. We consider the derivation of
effective preconditioners for these matrix systems, which may be embedded
within suitable iterative methods to accelerate their convergence. We also
construct low-rank tensor-train techniques which enable us to present efficient
and feasible algorithms for problems that are finely discretized in the space
and time variables. Numerical results demonstrate that the number of
preconditioned GMRES iterations depends mildly on the model parameters.
Moreover, the low-rank solver makes the computing time and memory costs
sublinear in the original problem size.Comment: 23 page
pde2path - A Matlab package for continuation and bifurcation in 2D elliptic systems
pde2path is a free and easy to use Matlab continuation/bifurcation package
for elliptic systems of PDEs with arbitrary many components, on general two
dimensional domains, and with rather general boundary conditions. The package
is based on the FEM of the Matlab pdetoolbox, and is explained by a number of
examples, including Bratu's problem, the Schnakenberg model, Rayleigh-Benard
convection, and von Karman plate equations. These serve as templates to study
new problems, for which the user has to provide, via Matlab function files, a
description of the geometry, the boundary conditions, the coefficients of the
PDE, and a rough initial guess of a solution. The basic algorithm is a one
parameter arclength continuation with optional bifurcation detection and
branch-switching. Stability calculations, error control and mesh-handling, and
some elementary time-integration for the associated parabolic problem are also
supported. The continuation, branch-switching, plotting etc are performed via
Matlab command-line function calls guided by the AUTO style. The software can
be downloaded from www.staff.uni-oldenburg.de/hannes.uecker/pde2path, where
also an online documentation of the software is provided such that in this
paper we focus more on the mathematics and the example systems
Analysis of the discontinuous Galerkin method for elliptic problems on surfaces
We extend the discontinuous Galerkin (DG) framework to a linear second-order
elliptic problem on a compact smooth connected and oriented surface. An
interior penalty (IP) method is introduced on a discrete surface and we derive
a-priori error estimates by relating the latter to the original surface via the
lift introduced in Dziuk (1988). The estimates suggest that the geometric error
terms arising from the surface discretisation do not affect the overall
convergence rate of the IP method when using linear ansatz functions. This is
then verified numerically for a number of test problems. An intricate issue is
the approximation of the surface conormal required in the IP formulation,
choices of which are investigated numerically. Furthermore, we present a
generic implementation of test problems on surfaces.Comment: 21 pages, 4 figures. IMA Journal of Numerical Analysis 2013, Link to
publication: http://imajna.oxfordjournals.org/cgi/content/abstract/drs033?
ijkey=45b23qZl5oJslZQ&keytype=re
Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), 10-11 July 2017, Nottingham Conference Centre, Nottingham Trent University
This book contains the abstracts and papers presented at the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), held at Nottingham Trent University in July 2017. The work presented at the conference, and published in this volume, demonstrates the wide range of work that is being carried out in the UK, as well as from further afield
Data-driven modelling of biological multi-scale processes
Biological processes involve a variety of spatial and temporal scales. A
holistic understanding of many biological processes therefore requires
multi-scale models which capture the relevant properties on all these scales.
In this manuscript we review mathematical modelling approaches used to describe
the individual spatial scales and how they are integrated into holistic models.
We discuss the relation between spatial and temporal scales and the implication
of that on multi-scale modelling. Based upon this overview over
state-of-the-art modelling approaches, we formulate key challenges in
mathematical and computational modelling of biological multi-scale and
multi-physics processes. In particular, we considered the availability of
analysis tools for multi-scale models and model-based multi-scale data
integration. We provide a compact review of methods for model-based data
integration and model-based hypothesis testing. Furthermore, novel approaches
and recent trends are discussed, including computation time reduction using
reduced order and surrogate models, which contribute to the solution of
inference problems. We conclude the manuscript by providing a few ideas for the
development of tailored multi-scale inference methods.Comment: This manuscript will appear in the Journal of Coupled Systems and
Multiscale Dynamics (American Scientific Publishers
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