2,397 research outputs found
Innovative Approaches to the Numerical Approximation of PDEs
This workshop was about the numerical solution of PDEs for which classical
approaches,
such as the finite element method, are not well suited or need further
(theoretical) underpinnings.
A prominent example of PDEs for which classical methods are not well
suited are PDEs posed in high space dimensions.
New results on low rank tensor approximation for those problems were
presented.
Other presentations dealt with regularity of PDEs, the numerical solution
of PDEs on surfaces,
PDEs of fractional order, numerical solvers for PDEs that converge with
exponential rates, and
the application of deep neural networks for solving PDEs
Schnelle Löser für Partielle Differentialgleichungen
The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds
Optimization of mesh hierarchies in Multilevel Monte Carlo samplers
We perform a general optimization of the parameters in the Multilevel Monte
Carlo (MLMC) discretization hierarchy based on uniform discretization methods
with general approximation orders and computational costs. We optimize
hierarchies with geometric and non-geometric sequences of mesh sizes and show
that geometric hierarchies, when optimized, are nearly optimal and have the
same asymptotic computational complexity as non-geometric optimal hierarchies.
We discuss how enforcing constraints on parameters of MLMC hierarchies affects
the optimality of these hierarchies. These constraints include an upper and a
lower bound on the mesh size or enforcing that the number of samples and the
number of discretization elements are integers. We also discuss the optimal
tolerance splitting between the bias and the statistical error contributions
and its asymptotic behavior. To provide numerical grounds for our theoretical
results, we apply these optimized hierarchies together with the Continuation
MLMC Algorithm. The first example considers a three-dimensional elliptic
partial differential equation with random inputs. Its space discretization is
based on continuous piecewise trilinear finite elements and the corresponding
linear system is solved by either a direct or an iterative solver. The second
example considers a one-dimensional It\^o stochastic differential equation
discretized by a Milstein scheme
Topology optimization of multiple anisotropic materials, with application to self-assembling diblock copolymers
We propose a solution strategy for a multimaterial minimum compliance
topology optimization problem, which consists in finding the optimal allocation
of a finite number of candidate (possibly anisotropic) materials inside a
reference domain, with the aim of maximizing the stiffness of the body. As a
relevant and novel application we consider the optimization of self-assembled
structures obtained by means of diblock copolymers. Such polymers are a class
of self-assembling materials which spontaneously synthesize periodic
microstructures at the nanoscale, whose anisotropic features can be exploited
to build structures with optimal elastic response, resembling biological
tissues exhibiting microstructures, such as bones and wood. For this purpose we
present a new generalization of the classical Optimality Criteria algorithm to
encompass a wider class of problems, where multiple candidate materials are
considered, the orientation of the anisotropic materials is optimized, and the
elastic properties of the materials are assumed to depend on a scalar
parameter, which is optimized simultaneously to the material allocation and
orientation. Well-posedness of the optimization problem and well-definition of
the presented algorithm are narrowly treated and proved. The capabilities of
the proposed method are assessed through several numerical tests
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