The need of accurate local predictions of environmental parameters has increased
significantly in recent years as a result of the large number of social and
commercial activities that are directly affected. The validity of such data is
particularly important in terms of renewable energy for the safe assessment of
available energy resources in wind farms and off shore platforms.
On European level, the previous requirements have led to the activation of
numerous research and operational centers that develop high quality scientific
tools able to provide reliable environmental predictions. In Greece, the
available forecasts, although satisfactorily describe the overall picture in
large or moderate scale, pose considerable problems to local weather
information. These difficulties are magnified in areas of complex orography and
coastline - characteristics of the Greek region.
To address these problems, the use of optimization techniques will be employed
based on a relatively new branch of mathematics the Information Geometry. The
latter implements techniques from the non-Euclidean Geometry in Statistics,
targeting to the optimization of the solution of nonlinear problems.
One of the key issues is the appropriate estimation of the "distance" between
two distributions or data sets.
The classical treatment
of such problems is usually based on regression techniques - least squares
methods. However, such an approach carries the assumption that the data
processed belongs to an Euclidian – finite dimensional space.
Within the framework of Information Geometry it is proved that this is not always the case since a family of distributions is a manifold – i.e. a generalization of an Euclidean space - where the geometric properties may be quite distant from the classic. Thus, determining the necessary tools leading to the safe estimation of the geometric properties requires the construction of a Riemannian structure. The distances in such a complex environment are closely connected with the curves of minimum length (geodesics). The evaluation of the latter is equivalent to solving differential equations of second order.
In classical Euclidean geometry these equations have always the form
which requires zero curvature and resolved in a segment. In the case of a non-Euclidean space M the corresponding requirement leads to a system of differential equations
with boundary conditions:
where
is the desired curve of minimum length.
The form of these equations and their solutions, depending heavily on the
functions that characterize the geometric properties of
M,
are connected with the distribution followed by the data in study: usually the
Gamma, the Normal or the Weibull distribution prove to be the optimal choices.
In most cases the determination of the analytic solutions of the differential
equations obtained is not possible and the need for using approximate-numerical
methods is imperative. The differential equations that should be solved are
second order boundary value problems (BVPs).
The new
techniques and methodologies proposed in this project will be applied in
selected areas of Greece where the use of renewable energy sources is of
increased interest. Hopefully, the obtained results will provide the necessary
information for the development of profitable renewable sources installations in
these areas.
A presentation of the project in Greek.
I.Th. Famelis, Georgios
Galanis, Matthias Ehrhardt and Dimitrios Triantafyllou,
Classical and Quasi-Newton methods for a Meteorological Parameters
Prediction Boundary Value Problem AMIS, v.8, No. 6
(2014), pp 2683-2693.
I.Th. Famelis,
«Runge-Kutta
solutions for an environmental parameter prediction boundary value
problem.», J. Coupled Syst. Multiscale Dyn.,
2(2) (2014), pp 62-69.
I.Th. Famelis, Ch. Tsitouras, Quadratic shooting solution for an environmental parameter prediction boundary value problem., Far East Journal of Applied Mathematics, v.91, No.2 (2015), pp.81-98.
George Galanis, Ioannis Famelis, George Kallos, Aristotelis Liakatas, A new Kalman Filter based on Information Geometry techniques for optimizing numerical environmental simulations, Submitted
On
the numerical solution of a boundary value problem which rises in the
prediction of meteorological parameters,
International Conference Of Numerical Analysis And Applied
Mathematics 2012, Kos, Greece, 19-25 September 2012. AIP Conf. Proc. 1479,
2118 (2012); 10.1063/1.4756609
Classical and Quasi-Newton methods on the numerical solution of a Boundary Value Problem which rises in the prediction of meteorological parameters using finite differences. , Seventh (7th) Workshop on Statistics Mathematics and Computations, 28 - 29 May 2013, Tomar, Portugal.
Τhe numerical solution of a BVP which rises in the prediction of
meteorological parameters.
International Conference on Scientific Computation And Differential
Equations (Scicade) 2013, Valladolid, Spain, 16-20 September, 2013.
Quadratic RK shooting solution for a environmental parameter
prediction boundary value problem, ICCMSE 2014 Athens, AIP Conf. Proc. 1618 , 839 (2014);
doi
10.1063/1.4897863
Optimization
of numerical weather/wave prediction models based on information geometry
and computational techniques
, ICCMSE 2014, AIP Conf. Proc. 1618 , 828 (2014) ;
doi:
10.1063/1.4897861
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