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Large deviation methods for stochastic reachability
Get PDFIn this paper, we propose to find upper/lower bounds for different measures that characterize the reachability problem defined in the context of stochastic hybrid systems, using the theory of large deviations. For stochastic hybrid processes, criteria for large deviation results are given using properties of their infinitesimal generators. This represents just the first step towards applying large deviation methods for stochastic hybrid systems for treating new topics like robust control, metastability, performance analysis
Long's vortex revisited
No full textWe reconsider exact solutions to the Navier--Stokes equations that describe a vortex in a viscous, incompressible fluid. This type of solution was first introduced by Long (1958) and is par ameterised by an inverse Reynolds number . Long's attention (and that of many subsequent investigators) was centred upon the `quasi-cylindrical' (QC) case corresponding to . We show that the limit is not straightforward, and that it reveals other solutions to this fundamental exact reduction of the Navier--Stokes system (which are not of QC form). Through careful numerical investigation, supported by asymptotic descriptions, we identify new solutions and describe the full parameter space that is spanned by and the pressure at the vortex core. Some erroneous results that exist in the literature are corrected
Bayesian inverse problems for functions and applications to fluid mechanics
No full textIn this paper we establish a mathematical framework for a range of inverse problems for functions, given a finite set of noisy observations. The problems are hence underdetermined and are often ill-posed. We study these problems from the viewpoint of Bayesian statistics, with the resulting posterior probability measure being defined on a space of functions. We develop an abstract framework for such problems which facilitates application of an infinite-dimensional version of Bayes theorem, leads to a well-posedness result for the posterior measure (continuity in a suitable probability metric with respect to changes in data), and also leads to a theory for the existence of maximizing the posterior probability (MAP) estimators for such Bayesian inverse problems on function space. A central idea underlying these results is that continuity properties and bounds on the forward model guide the choice of the prior measure for the inverse problem , leading to the desired results on well-posedness and MAP estimators; the PDE analysis and probability theory required are thus clearly dileneated, allowing a straightforward derivation of results. We show that the abstract theory applies to some concrete applications of interest by studying problems arising from data assimilation in fluid mechanics. The objective is to make inference about the underlying velocity field, on the basis of either Eulerian or Lagrangian observations. We study problems without model error, in which case the inference is on the initial condition, and problems with model error in which case the inference is on the initial condition and on the driving noise process or, equivalently, on the entire time-dependent velocity field. In order to undertake a relatively uncluttered mathematical analysis we consider the two-dimensional Navier???Stokes equation on a torus. The case of Eulerian observations???direct observations of the velocity field itself???is then a model for weather forecasting. The case of Lagrangian observations???observations of passive tracers advected by the flow???is then a model for data arising in oceanography. The methodology which we describe herein may be applied to many other inverse problems in which it is of interest to find, given observations, an infinite-dimensional object, such as the initial condition for a PDE. A similar approach might be adopted, for example, to determine an appropriate mathematical setting for the inverse problem of determining an unknown tensor arising in a constitutive law for a PDE, given observations of the solution. The paper is structured so that the abstract theory can be read independently of the particular problems in fluid mechanics which are subsequently studied by application of the theory
Efficient Solvers for a Linear Stochastic Galerkin Mixed Formulation of Diffusion Problems with Random Data
Get PDFWe introduce a stochastic Galerkin mixed formulation of the
steady-state diffusion equation and focus on the efficient iterative solution of the saddle-point systems obtained by combining standard finite element discretisations with two distinct types of stochastic basis functions. So-called mean-based preconditioners, based on fast solvers for scalar diffusion problems, are introduced for use
with the minimum residual method. We derive eigenvalue bounds for the preconditioned system matrices and report
on the efficiency of the chosen preconditioning schemes with respect to all the discretisation parameter
Toric genera
Get PDFOur primary aim is to develop a theory of equivariant genera for stably complex manifolds equipped with compatible actions of a torus T^k. In the case of omnioriented quasitoric manifolds, we present computations that depend only on their defining combinatorial data;
these draw inspiration from analogous calculations in toric
geometry, which seek to express arithmetic, elliptic, and associated genera of toric varieties in terms only of their fans. Our theory focuses on the universal toric genus \varPhi, which was introduced independently by Krichever and L\"offler in 1974, albeit from radically different viewpoints. In fact \varPhi is a version of tom Dieck's bundling transformation of 1970, defined on T^k-equivariant complex cobordism classes and taking values in the
complex cobordism algebra \varOmega^*_U(BT^k_+) of the classifying space. We proceed by combining the analytic, the formal group theoretic, and the homotopical approaches to genera, and refer to the index theoretic approach as a recurring source of insight and motivation. The resultant flexibility allows us to identify several distinct genera within our framework, and to introduce parametrised
versions that apply to bundles equipped with a stably complex structure on the tangents along their fibres. In the presence of isolated fixed points, we obtain universal localisation formulae, whose applications include the identification of Krichever's generalised elliptic genus as universal amongst genera that are rigid on SU-manifolds. We follow the traditions of toric geometry by working with a variety of illustrative examples wherever possible. For background and prerequisites we attempt to reconcile the literature of east and west, which developed independently for several decades after the 1960s
On the dimension of iterated sumsets
Get PDFLet A be a subset of the real line. We study the fractal dimensions of the k-fold iterated sumsets kA, defined as kA = A+...+A (k times).
We show that for any non-decreasing sequence {a_k} taking values in [0,1], there exists a compact set A such that kA has Hausdorff dimension a_k for all k. We also show how to control various kinds of dimension simultaneously for families of iterated sumsets.
These results are in stark contrast to the Plunnecke-Rusza inequalities in additive combinatorics. However, for lower box-counting dimension, the analogue of the Plunnecke-Rusza inequalities does hold
The Hausdorff dimension of the projections of self-affine carpets
Get PDFWe study the orthogonal projections of a large class of self-affine carpets, which contains the carpets of Bedford and McMullen as special cases. Our main result is that if is such a carpet, and certain natural irrationality conditions hold, then every orthogonal projection of in a non-principal direction has Hausdorff dimension , where is the Hausdorff dimension of . This generalizes a recent result of Peres and Shmerkin on sums of Cantor sets
Base change and K-theory for GL(n,R)
Get PDFWe investigate base change at the level of -theory for the real general linear group . In the course of this study, we compute in detail the -algebra -theory of this disconnected group. We investigate the interaction of base change with the Baum-Connes correspondence for and . This article is the archimedean companion of our previous article in J. Noncommutative Geometry 1 (2007) 311-331
Adams operations on the Green ring of a cyclic group of prime-power order
Get PDFWe consider the Green ring for a cyclic -group over a field of prime characteristic and determine the Adams operations in the case where is not divisible by . This gives information on the decomposition into indecomposables of exterior powers and symmetric powers of -modules
Workshop on Formal Methods for Aerospace (FMA)
Get PDFThe coexistence of multiple disciplinary perspectives on the same class of critical applications (aerospace) and investigation (formal) methods leads naturally to the opportunity to define multidisciplinary approaches. Thus, work in this area will likely underline the importance of
some research problems from aerospace to the formal methods community, and promote new formal techniques combining the principles of artificial intelligence and control engineering.
The source of new problems for formal methods comes from the great diversity of aerospace systems. These can be satellites, unmanned aerial vehicles (UAVs), terrestrial or other kinds of flying robots. These systems can be involved in complex activities such as space exploration,
telecommunication support, fire detection, geo-mapping, weather prognoses, geo-rectification, search and rescue, naval traffic surveillance, tracking high value targets. From these applications, new research problems appear: autonomy, collective behaviour, information fusion, cognitive skills, coordination, flocking, etc. In addition, new concepts must be formalised: digital pheromones, swarms, system of systems of robots, sensing, physical actuation, and so on.
Aerospace systems are not only safety critical, but also mission critical and have very high performance requirements. For example, there is no safety issue regarding a planetary rover, but the system performance must justify the great cost of deploying it. Consequently,
aerospace enriches traditional formal methods topics with new (or, at least, rarely investigated) research issues.
Formal methods could greatly benefit from integration with approaches from other disciplines, and many such opportunities are now appearing. A good example is the problem of coordination for platoons of UAVs or satellites, which have been successfully, tackled using various
techniques from control engineering and numerical tools from dynamic programming. In addition, there exist an abundance of examples artificial intelligence techniques in aerospace (target tracking, rover planning, multi-agent technologies and so on). The implementation of these methods could benefit from formal development. From the cross-fertilization of related multidisciplinary approaches, we expect more robust, safe and mechanizable development and verification methods for aerospace systems