212 research outputs found
Mini-Workshop: Applied Koopmanism
Koopman and Perron–Frobenius operators are linear operators that encapsulate dynamics of nonlinear dynamical systems without loss of information. This is accomplished by embedding the dynamics into a larger infinite-dimensional space where the focus of study is shifted from trajectory curves to measurement functions evaluated along trajectories and densities of trajectories evolving in time. Operator-theoretic approach to dynamics shares many features with an optimization technique: the Lasserre moment–sums-of-squares (SOS) hierarchies, which was developed for numerically solving non-convex optimization problems with semialgebraic data. This technique embeds the optimization problem into a larger primal semidefinite programming (SDP) problem consisting of measure optimization over the set of globally optimal solutions, where measures are manipulated through their truncated moment sequences. The dual SDP problem uses SOS representations to certify bounds on the global optimum. This workshop highlighted the common threads between the operator-theoretic dynamical systems and moment–SOS hierarchies in optimization and explored the future directions where the synergy of the two techniques could yield results in fluid dynamics, control theory, optimization, and spectral theory
Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals
Among the fastest methods for solving stiff PDE are exponential integrators, which require the evaluation of , where is a negative definite matrix and is the exponential function or one of the related `` functions'' such as . Building on previous work by Trefethen and Gutknecht, Gonchar and Rakhmanov, and Lu, we propose two methods for the fast evaluation of that are especially useful when shifted systems can be solved efficiently, e.g. by a sparse direct solver. The first method method is based on best rational approximations to on the negative real axis computed via the Carathéodory-Fejér procedure, and we conjecture that the accuracy scales as , where is the number of complex matrix solves. In particular, three matrix solves suffice to evaluate to approximately six digits of accuracy. The second method is an application of the trapezoid rule on a Talbot-type contour
Some approximation schemes in polynomial optimization
Cette thèse est dédiée à l'étude de la hiérarchie moments-sommes-de-carrés, une famille de problèmes de programmation semi-définie en optimisation polynomiale, couramment appelée hiérarchie de Lasserre. Nous examinons différents aspects de ses propriétés et applications. Comme application de la hiérarchie, nous approchons certains objets potentiellement compliqués, comme l'abscisse polynomiale et les plans d'expérience optimaux sur des domaines semi-algébriques. L'application de la hiérarchie de Lasserre produit des approximations par des polynômes de degré fixé et donc de complexité bornée. En ce qui concerne la complexité de la hiérarchie elle-même, nous en construisons une modification pour laquelle un taux de convergence amélioré peut être prouvé. Un concept essentiel de la hiérarchie est l'utilisation des modules quadratiques et de leurs duaux pour appréhender de manière flexible le cône des polynômes positifs et le cône des moments. Nous poursuivons cette idée pour construire des approximations étroites d'ensembles semi-algébriques à l'aide de séparateurs polynomiaux.This thesis is dedicated to investigations of the moment-sums-of-squares hierarchy, a family of semidefinite programming problems in polynomial optimization, commonly called the Lasserre hierarchy. We examine different aspects of its properties and purposes. As applications of the hierarchy, we approximate some potentially complicated objects, namely the polynomial abscissa and optimal designs on semialgebraic domains. Applying the Lasserre hierarchy results in approximations by polynomials of fixed degree and hence bounded complexity. With regard to the complexity of the hierarchy itself, we construct a modification of it for which an improved convergence rate can be proved. An essential concept of the hierarchy is to use quadratic modules and their duals as a tractable characterization of the cone of positive polynomials and the moment cone, respectively. We exploit further this idea to construct tight approximations of semialgebraic sets with polynomial separators
Nonnormality in Lyapunov Equations
The singular values of the solution to a Lyapunov equation determine the potential accuracy of the low-rank approximations constructed by iterative methods. Low- rank solutions are more accurate if most of the singular values are small, so a priori bounds that describe coefficient matrix properties that correspond to rapid singular value decay are valuable. Previous bounds take similar forms, all of which weaken (quadratically) as the coefficient matrix departs from normality. Such bounds suggest that the more nonnormal the coefficient matrix becomes, the slower the singular values of the solution will decay. However, simple examples typically exhibit an eventual acceleration of decay if the coefficient becomes very nonnormal. We will show that this principle is universal: decay always improves as departure from normality increases beyond a given threshold, specifically as the numerical range of the coefficient matrix extends farther into the right half-plane. We also give examples showing that similar behavior can occur for general Sylvester equations, though the right-hand side plays a more important role
Optimization and Applications
Proceedings of a workshop devoted to optimization problems, their theory and resolution, and above all applications of them. The topics covered existence and stability of solutions; design, analysis, development and implementation of algorithms; applications in mechanics, telecommunications, medicine, operations research
Spectral triples for the Sierpinski Gasket
We construct a family of spectral triples for the Sierpinski Gasket . For
suitable values of the parameters, we determine the dimensional spectrum and
recover the Hausdorff measure of in terms of the residue of the volume
functional tr at its abscissa of convergence , which
coincides with the Hausdorff dimension of the fractal. We determine the
associated Connes' distance showing that it is bi-Lipschitz equivalent to the
distance on induced by the Euclidean metric of the plane, and show that the
pairing of the associated Fredholm module with (odd) -theory is non-trivial.
When the parameters belong to a suitable range, the abscissa of convergence
of the energy functional
tr takes the value
, which we call energy dimension, and the
corresponding residue gives the standard Dirichlet form on .Comment: 48 pages, 9 figures. Final version, to appear in J.Funct.Ana
High-order DG solvers for under-resolved turbulent incompressible flows: A comparison of and (div) methods
The accurate numerical simulation of turbulent incompressible flows is a
challenging topic in computational fluid dynamics. For discretisation methods
to be robust in the under-resolved regime, mass conservation as well as energy
stability are key ingredients to obtain robust and accurate discretisations.
Recently, two approaches have been proposed in the context of high-order
discontinuous Galerkin (DG) discretisations that address these aspects
differently. On the one hand, standard -based DG discretisations enforce
mass conservation and energy stability weakly by the use of additional
stabilisation terms. On the other hand, pointwise divergence-free
-conforming approaches ensure exact mass conservation
and energy stability by the use of tailored finite element function spaces. The
present work raises the question whether and to which extent these two
approaches are equivalent when applied to under-resolved turbulent flows. This
comparative study highlights similarities and differences of these two
approaches. The numerical results emphasise that both discretisation strategies
are promising for under-resolved simulations of turbulent flows due to their
inherent dissipation mechanisms.Comment: 24 pages, 13 figure
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