602 research outputs found
Robust isogeometric preconditioners for the Stokes system based on the Fast Diagonalization method
In this paper we propose a new class of preconditioners for the isogeometric
discretization of the Stokes system. Their application involves the solution of
a Sylvester-like equation, which can be done efficiently thanks to the Fast
Diagonalization method. These preconditioners are robust with respect to both
the spline degree and mesh size. By incorporating information on the geometry
parametrization and equation coefficients, we maintain efficiency on
non-trivial computational domains and for variable kinematic viscosity. In our
numerical tests we compare to a standard approach, showing that the overall
iterative solver based on our preconditioners is significantly faster.Comment: 31 pages, 4 figure
Model reduction of controlled Fokker--Planck and Liouville-von Neumann equations
Model reduction methods for bilinear control systems are compared by means of
practical examples of Liouville-von Neumann and Fokker--Planck type. Methods
based on balancing generalized system Gramians and on minimizing an H2-type
cost functional are considered. The focus is on the numerical implementation
and a thorough comparison of the methods. Structure and stability preservation
are investigated, and the competitiveness of the approaches is shown for
practically relevant, large-scale examples
Consistent Dynamic Mode Decomposition
We propose a new method for computing Dynamic Mode Decomposition (DMD)
evolution matrices, which we use to analyze dynamical systems. Unlike the
majority of existing methods, our approach is based on a variational
formulation consisting of data alignment penalty terms and constitutive
orthogonality constraints. Our method does not make any assumptions on the
structure of the data or their size, and thus it is applicable to a wide range
of problems including non-linear scenarios or extremely small observation sets.
In addition, our technique is robust to noise that is independent of the
dynamics and it does not require input data to be sequential. Our key idea is
to introduce a regularization term for the forward and backward dynamics. The
obtained minimization problem is solved efficiently using the Alternating
Method of Multipliers (ADMM) which requires two Sylvester equation solves per
iteration. Our numerical scheme converges empirically and is similar to a
provably convergent ADMM scheme. We compare our approach to various
state-of-the-art methods on several benchmark dynamical systems
Exploring multimodal data fusion through joint decompositions with flexible couplings
A Bayesian framework is proposed to define flexible coupling models for joint
tensor decompositions of multiple data sets. Under this framework, a natural
formulation of the data fusion problem is to cast it in terms of a joint
maximum a posteriori (MAP) estimator. Data driven scenarios of joint posterior
distributions are provided, including general Gaussian priors and non Gaussian
coupling priors. We present and discuss implementation issues of algorithms
used to obtain the joint MAP estimator. We also show how this framework can be
adapted to tackle the problem of joint decompositions of large datasets. In the
case of a conditional Gaussian coupling with a linear transformation, we give
theoretical bounds on the data fusion performance using the Bayesian Cramer-Rao
bound. Simulations are reported for hybrid coupling models ranging from simple
additive Gaussian models, to Gamma-type models with positive variables and to
the coupling of data sets which are inherently of different size due to
different resolution of the measurement devices.Comment: 15 pages, 7 figures, revised versio
Structure-Preserving Model Reduction of Physical Network Systems
This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p
Numerical Analysis of Three-dimensional Acoustic Cloaks and Carpets
We start by a review of the chronology of mathematical results on the
Dirichlet-to-Neumann map which paved the way towards the physics of
transformational acoustics. We then rederive the expression for the
(anisotropic) density and bulk modulus appearing in the pressure wave equation
written in the transformed coordinates. A spherical acoustic cloak consisting
of an alternation of homogeneous isotropic concentric layers is further
proposed based on the effective medium theory. This cloak is characterised by a
low reflection and good efficiency over a large bandwidth for both near and far
fields, which approximates the ideal cloak with a inhomogeneous and anisotropic
distribution of material parameters. The latter suffers from singular material
parameters on its inner surface. This singularity depends upon the sharpness of
corners, if the cloak has an irregular boundary, e.g. a polyhedron cloak
becomes more and more singular when the number of vertices increases if it is
star shaped. We thus analyse the acoustic response of a non-singular spherical
cloak designed by blowing up a small ball instead of a point, as proposed in
[Kohn, Shen, Vogelius, Weinstein, Inverse Problems 24, 015016, 2008]. The
multilayered approximation of this cloak requires less extreme densities
(especially for the lowest bound). Finally, we investigate another type of
non-singular cloaks, known as invisibility carpets [Li and Pendry, Phys. Rev.
Lett. 101, 203901, 2008], which mimic the reflection by a flat ground.Comment: Latex, 21 pages, 7 Figures, last version submitted to Wave Motion.
OCIS Codes: (000.3860) Mathematical methods in physics; (260.2110)
Electromagnetic theory; (160.3918) Metamaterials; (160.1190) Anisotropic
optical materials; (350.7420) Waves; (230.1040) Acousto-optical devices;
(160.1050) Acousto-optical materials; (290.5839) Scattering,invisibility;
(230.3205) Invisibility cloak
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