T.E.I. of Athens

Research Program EPEAEK - Archimedes III

New numerical and computational methods for the solution of differential equations with applications in environmental issues

financed by The European Union & The Ministry of Education, Lifelong Learning and Religious Affairs of Greece

(1/3/2012 - 30/11/2014)

Project goals

The proposed project has as main target the development and use of new models and computational methods for the optimization of the prediction of meteorological parameters. More precisely, an integrated approach will be adopted aiming at the development of:
  1. A new methodology for the estimation of the "distance" between data sets or distributions emerging in meteorological and renewable energy forecasting procedures.
  2. New computational methods for the solution of the problems/equations that arise.
  3. Novel statistical methods for the optimization of the prediction of meteorological parameters with emphasis in the local adaptation of the results.
These goals-objectives will be pursuit based on advanced scientific tools combined with the development of new techniques promoting the cooperation between the research members of our scientific team and proposing new, more effective and widely applicable techniques.

State of the art and the new perspective

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 numerical methods solving these equations must take into consideration the nature of the problem. In this respect, new optimization technique proposed will be based on the minimization of the computational cost which will take into account the geometry of the distributions space.

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.


WorkPackages in Progress


Papers in Scientific Journals

  1. 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.

  2. 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.

  3.  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.

  4.  George Galanis, Ioannis Famelis, George Kallos, Aristotelis Liakatas, A new Kalman Filter based on Information Geometry techniques for optimizing numerical environmental simulations, Submitted


Presentations in Conferences

  1. 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

  2.  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.

  3. Numerical and geometric optimization techniques for environmental prediction systems,  SIAM Conference on Mathematical and Computational Issues in the Geosciences Padova, Italy, 17-20 June, 2013.

  4. 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.

  5.  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

  6.  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

  7.  Information Geometry and applications to optimization techniques for numerical environmental models,  12th Pan-Hellenic  Geometry Conference with international participation, Thessaloniki, 29-31.05.2015

  8. New Efficient Optimizing Techniques for Kalman Filters and Numerical Weather Prediction Models, International Conference Of Numerical Analysis And Applied Mathematics 2012, Rhodes, Greece, 23-29 September 2015


Other Webpages related to this project:


Main Research Team:

  • Ioannis Th. Famelis, Coordinator, Department of Mathematics, School of Technological Applications, TEI of Athens, Greece
  • Georgios Galanis, Atmospheric Modeling and Weather Forecasting Group, Department of Physics, University of Athens, Greece
  • Georgios Kallos, Atmospheric Modeling and Weather Forecasting Group, Department of Physics, University of Athens, Greece
  • Georgios Smyrlis, Department of Mathematics, TEI of Athens, Greece
  • Charalambos Tsitouras, Department of Applied Sciences, TEI of Chalkis, Greece
  • Matthias Ehrhardt, Applied Mathematics and Numerical Analysis, University of Wuppertal, Germany

External Collaborators:

  • Dimitrios Triandafyllou, Greece
  • Georgios Emmanouil,  Hellenic National Meteorological Service, Greece
  • Alexander Adam, Atmospheric Modeling and Weather Forecasting Group, Department of Physics, University of Athens, Greece
  • Kalogeri Christina, Atmospheric Modeling and Weather Forecasting Group, Department of Physics, University of Athens, Greece
  • Michael Günther, Applied Mathematics and Numerical Analysis, University of Wuppertal, Germany