1,942 research outputs found
Controlling overestimation of error covariance in ensemble Kalman filters with sparse observations: A variance limiting Kalman filter
We consider the problem of an ensemble Kalman filter when only partial
observations are available. In particular we consider the situation where the
observational space consists of variables which are directly observable with
known observational error, and of variables of which only their climatic
variance and mean are given. To limit the variance of the latter poorly
resolved variables we derive a variance limiting Kalman filter (VLKF) in a
variational setting. We analyze the variance limiting Kalman filter for a
simple linear toy model and determine its range of optimal performance. We
explore the variance limiting Kalman filter in an ensemble transform setting
for the Lorenz-96 system, and show that incorporating the information of the
variance of some un-observable variables can improve the skill and also
increase the stability of the data assimilation procedure.Comment: 32 pages, 11 figure
Towards a General Theory of Extremes for Observables of Chaotic Dynamical Systems
In this paper we provide a connection between the geometrical properties of a
chaotic dynamical system and the distribution of extreme values. We show that
the extremes of so-called physical observables are distributed according to the
classical generalised Pareto distribution and derive explicit expressions for
the scaling and the shape parameter. In particular, we derive that the shape
parameter does not depend on the chosen observables, but only on the partial
dimensions of the invariant measure on the stable, unstable, and neutral
manifolds. The shape parameter is negative and is close to zero when
high-dimensional systems are considered. This result agrees with what was
derived recently using the generalized extreme value approach. Combining the
results obtained using such physical observables and the properties of the
extremes of distance observables, it is possible to derive estimates of the
partial dimensions of the attractor along the stable and the unstable
directions of the flow. Moreover, by writing the shape parameter in terms of
moments of the extremes of the considered observable and by using linear
response theory, we relate the sensitivity to perturbations of the shape
parameter to the sensitivity of the moments, of the partial dimensions, and of
the Kaplan-Yorke dimension of the attractor. Preliminary numerical
investigations provide encouraging results on the applicability of the theory
presented here. The results presented here do not apply for all combinations of
Axiom A systems and observables, but the breakdown seems to be related to very
special geometrical configurations.Comment: 16 pages, 3 Figure
Shepherding Hordes of Markov Chains
This paper considers large families of Markov chains (MCs) that are defined
over a set of parameters with finite discrete domains. Such families occur in
software product lines, planning under partial observability, and sketching of
probabilistic programs. Simple questions, like `does at least one family member
satisfy a property?', are NP-hard. We tackle two problems: distinguish family
members that satisfy a given quantitative property from those that do not, and
determine a family member that satisfies the property optimally, i.e., with the
highest probability or reward. We show that combining two well-known
techniques, MDP model checking and abstraction refinement, mitigates the
computational complexity. Experiments on a broad set of benchmarks show that in
many situations, our approach is able to handle families of millions of MCs,
providing superior scalability compared to existing solutions.Comment: Full version of TACAS'19 submissio
A Novel Stochastic Interacting Particle-Field Algorithm for 3D Parabolic-Parabolic Keller-Segel Chemotaxis System
We introduce an efficient stochastic interacting particle-field (SIPF)
algorithm with no history dependence for computing aggregation patterns and
near singular solutions of parabolic-parabolic Keller-Segel (KS) chemotaxis
system in three space dimensions (3D). The KS solutions are approximated as
empirical measures of particles coupled with a smoother field (concentration of
chemo-attractant) variable computed by the spectral method. Instead of using
heat kernels causing history dependence and high memory cost, we leverage the
implicit Euler discretization to derive a one-step recursion in time for
stochastic particle positions and the field variable based on the explicit
Green's function of an elliptic operator of the form Laplacian minus a positive
constant. In numerical experiments, we observe that the resulting SIPF
algorithm is convergent and self-adaptive to the high gradient part of
solutions. Despite the lack of analytical knowledge (e.g. a self-similar
ansatz) of the blowup, the SIPF algorithm provides a low-cost approach to study
the emergence of finite time blowup in 3D by only dozens of Fourier modes and
through varying the amount of initial mass and tracking the evolution of the
field variable. Notably, the algorithm can handle at ease multi-modal initial
data and the subsequent complex evolution involving the merging of particle
clusters and formation of a finite time singularity
Linear statistics and pushed Coulomb gas at the edge of beta random matrices: four paths to large deviations
The Airy point process, , describes
the eigenvalues at the edge of the Gaussian ensembles of
random matrices for large matrix size . We study the probability
distribution function (PDF) of linear statistics for large parameter . We show the large deviation
forms and for the cumulant
generating function and the PDF. We obtain the exact rate function
using four apparently different methods (i) the
electrostatics of a Coulomb gas (ii) a random Schr\"odinger problem, i.e. the
stochastic Airy operator (iii) a cumulant expansion (iv) a non-local non-linear
differential Painlev\'e type equation. Each method was independently introduced
to obtain the lower tail of the KPZ equation. Here we show their equivalence in
a more general framework. Our results are obtained for a class of functions
, the monotonous soft walls, containing the monomials
and the exponential and
equivalently describe the response of a Coulomb gas pushed at its edge. The
small behavior of the excess energy exhibits a change at
between a non-perturbative hard wall like regime for
(third order free-to-pushed transition) and a perturbative deformation of the
edge for (higher order transition). Applications are given, among
them: (i) truncated linear statistics such as , leading
to a formula for the PDF of the ground state energy of
noninteracting fermions in a linear plus random potential (ii)
interacting spinless fermions in a trap at the edge of a Fermi gas (iii) traces
of large powers of random matrices.Comment: Main text : 8 pages. Supp mat : 49 page
Optimization of Spinning Reserve in Stand-alone Wind-Diesel Power Systems
Spinning reserve carried on synchronized units is the most effective resource available to the system operators for managing unforeseen power unbalances, such as demand fluctuations and the sudden loss of generation equipment. The amount of reserve and the speed that it can effectively be deployed determine the supply reliability that the generation system can achieve. Carrying more spinning reserve reduces the probability that the generation system become unable to preserve the momentary power balance and costly remedial actions, such as involuntary load shedding, turns unavoidable to prevent a system collapse. Nevertheless, providing spinning reserve on a continuous basis is expensive. Indeed, the provision of spinning reserve entails incurring in startup costs to commit generating units in excess of the forecasted load, which consequently have to be dispatched at less efficient operating points. The problem of keeping the power balance is still more difficult in stand-alone wind-diesel power systems, since these systems are additionally subjected to random power fluctuations originated in the uncertain and intermittent nature of the wind resource. Furthermore, autonomous power system cannot rely on power imported from interconnections for preserving the power balance. The inherent characteristics of these systems require scheduling more reserve on synchronized units for ensuring adequate security and reliability levels. The higher reserve requirements may substantially deteriorate the economy of these supply systems. The costs of keeping spinning reserve must be compared with the benefits that it provides in terms of lower expected costs of interruptions. In essence, the optimal reserve level can be set so that the marginal cost of carrying an additional MW equals the marginal reduction of the expected load curtailment costs. Despite the apparent simplicity of this optimality condition, determining the optimal amount of spinning reserve in a practical setting presents substantial modelling complexities and computational challenges. Given the random nature of the disturbances and contingencies that may face a generation system, assessing the benefits of carrying a certain amount of spinning reserve involves quantifying the occurrence probability, Wind Power 2 duration, extent and costs of load loss events. Such evaluation entails modelling the stochastic behaviour of system operation by considering the random failure of system components and the stochastic fluctuations of load and wind generation. The problem is probabilistic in its very nature and thus it may be appropriately treated by applying stochastic modelling techniques. Only with the advent of more powerful computing hardware, the problem of optimizing the spinning reserve has attracted the interest of researchers and its solution is currently deemed practicable. This work proposes a novel method for determining the optimal amount of spinning reserve that should be carried in autonomous hybrid wind-diesel generation systems. The optimal spinning reserve is determined by comparing the cost of its provision with the economic benefits it delivers in terms of supply reliability. The proposed approach is still general and can be applied in straightforward manner to establish the optimal reserve level in large interconnected systems. The presented methodology considers with accuracy the probabilistic features of the load and the wind generation, as well as the random outages of the conventional generating units. By applying high-resolution chronological simulation techniques, the stochastic features of the integrated operation of the diesel units and the wind turbine can be detailed replicated. The mathematical model appropriately considers all relevant characteristics and operational constraints of the generating units, e.g. non-linear heat rate curve, maximum and minimum output, startup and synchronization time, minimum down and uptime, ramping, etc. Massive stochastic simulation methods allow assessing the system reliability and valuing the economic costs of loss load events. Global search methods like particle swarm optimization (PSO) are proposed for finding the optimal scheduling policy and spinning reserve requirement that minimizes the sum of the expected operation costs and the expected costs of the energy not served.Fil: Olsina, Fernando Gabriel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - San Juan; Argentina. Universidad Nacional de San Juan. Facultad de Ingeniería. Instituto de Energía Eléctrica; ArgentinaFil: Larisson, Carlos Hugo. No especifíca
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