## Research Themes Projects Teaching Talks |
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**Current research/collaborations and future plans**

**KAUST**

I started a collaboration with KAUST Extreme Computing Research Center. Here are a few ideas: a) extend/improve parallel implementation (R. Kriemann 2005, 2014) of hierarchical matrices [Hackbusch et al, 98] to the new hardware architecture, b) to research how such factors as admissibility condition (i.e. block partition), the minimal leaf size (nmin) and the H-matrix rank influence on the efficiency of the parallel Hierarchical matrix implementation. A practical application of both ideas above is reduce storage cost and computational times of large covariance matrices, which appears in statistical weather forecast (Europa, Saudi Arabia, Red Sea etc). c) The last idea is a parallel implementation of low-rank tensor formats for solving high-dimensional problems.

I started a cooperation with Prof. Genton and his group (Spatio-Temporal Statistics and Data Analysis). The subject of this cooperation is “Hierarchical matrix approximation of large Matern covariance matrices”. The class of Matern covariance functions becomes very popular in spatial statistics and especially in geostatistics. We reduce the storage and computational cost from quadratic to a log-linear. Application: to reduce the cost of the Kriging and of the Kalman filter/Bayesian update.

Furthermore, together with other colleagues from Tempone’s group, I started a cooperation with the group of Prof. Hakan Bagci. The subject of this cooperation is: “Efficient Computation of Electromagnetic Fields Scattered From Dielectric Objects of Uncertain Shapes Using a Multilevel Monte Carlo Scheme”. Simulators capable of computing scattered fields from objects of uncertain shapes are highly useful in electromagnetics and photonics, where device designs are typically subject to fabrication tolerances. Knowledge of statistical variations in scattered fields is useful in ensuring error-free functioning of devices. Application: e.g. shape control of very small particles.

I also collaborate with Prof. Victor Solovyev from the Computational Bioscience Research Center at KAUST. Together, we wrote a joint (CRG3 KAUST) proposal “Efficient technique of computing gene expression levels on noisy RNA-Seq data”. The goal is to develop robust bioinformatics tools for efficient identification of alternatively spliced transcripts and accurate estimation of their expression levels. We plan to develop a new approach to gene isoforms discovery by combining transcript assembling with alignment of RNA-Seq reads and transcripts to the genomic sequence. My task is to reduce the computing times from days to hours by using low-rank and sparse data approximation techniques.

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**Collaboration outside of KAUST**

During 2015, I plan to do the following research and publish the following works:

[R1]Mike Espig, Wolfgang Hackbusch, A. Litvinenko and Hermann G. Matthies,

An Efficient Method for the Computation of the Stochastic Galerkin Projection by Means of Tensor Format Representations

This joint work [R1] with colleagues from (RWTH Aachen, MPI Leipzig and Technical University Braunschweig, Germany) is about the low-rank tensor solution of the multi-dimensional stochastic elliptic equation. This paper will be the third in a series. In [9] we assumed that the stochastic solution has a low-rank representation and we proposed some effective algorithms for post-processing. The second paper [5, 21] was devoted to the analysis of the tensor rank of stochastic Galerkin matrices. We studied how this tensor rank depends on the number of KLE terms, on the PCE basis, on the number of quadrature points, on type of the covariance function, and on the covariance lengths. Now, in the third paper, we develop an iterative method which solves the stochastic linear system efficiently in the low-rank tensor format.

Together with H. Matthies (TU Braunschweig) we also research a non-linear Bayesian update method:

[R2]Alexander Litvinenko and Hermann G. Matthies,

Inverse problems and uncertainty quantification, arXiv:1312.5048, 2014, submitted.

The update formula is derived and the first successful numerical experiments are performed. The future work is to compare our results with the results obtained by the classical Monte Carlo Markov Chain method (cooperation with another member of SRI UQ Kody Law).

In

[R3]Uncertainty quantification and treatment in aircraft design - comparison of approaches, Volker Schulz (University of Trier, Germany), Dishi Liu (the national aeronautics and space research centre DLR) and Claudia Schillings (ETH Zürich), submitted to AIAA in 2013 and is currently in the second stage of the review process.

we develop response surfaces for optimal design of airfoil profile.

And in

[R4]Sergey Dolgov, Boris N. Khoromskij, Alexander Litvinenko and Hermann G. Matthies,

Approximation of Response Surface in Tensor Train data format, submitted in 2015 to SIAM JUQ, in the second stage of the review process.

the new method for approximation of the solution of high-dimensional (multi-parametric) PDEs is offered. To avoid the curse of dimensionality as well as to reduce the number of runs of the deterministic code, a low-rank surrogate model is used. The question, which we addressed in this article, is: how to approximate the high-dimensional response surface in a low-rank tensor train format. The procedure is demonstrated in the solution of the elliptic partial differential coefficient with uncertain coefficient.

In

[R5]Alexander Litvinenko, Youssef Marzouk, Hermann G. Matthies,

Bayesian update of PCE coefficients and distance between two updates, in preparation

we develop a fast low-rank tensor based method for computing the probability density function of a random vector from its characteristic function. The low-rank properties of the characteristic function are combined with the multi-dimensional Fourier Transformation as well as with the polynomial Chaos expansion. The whole plan is to approximate the random vector in generalized polynomial chaos expansion, from the characteristic function sample probability density function, and after that to perform the Bayesian update. All computations will be done in a tensor data format.

Last but not least, in

[R6]Alexander Litvinenko, Habib Najm, Hermann G. Matthies,

Data free inference of uncertain parameters in chemical models by the (non)-linear Bayesian Update and MCMC,in preparation

we compare our non-linear Bayesian update with the new version of the classical Monte Carlo Marcov Chain offered by H. Najm et. al. (International Journal for Uncertainty Quantification, 06/2012; 4(2).)

** ****References**

- W. Nowak, A. Litvinenko, Kriging accelerated by orders of magnitude: combining low-rank covariance approximations with FFT-techniques, accepted for publication in Mathematical Geosciences, 2013
- A. Litvinenko and H. G. Matthies and T. A. El-Moselhy, Sampling and Low-Rank Tensor Approximation of the Response Surface, accepted for publication in MCQMC Proceedings, Editors: J. Dick, F. Y. Kuo, G. W. Peters, I. H. Sloan, 2013
- A. Litvinenko and H. G. Matthies, Numerical Methods for Uncertainty Quantiﬁcation and Bayesian update in Aerodynamics, chapter in the book Management and Minimisation of Uncertainties and Errors in Numerical Aerodynamics – Results of the German collaborative project MUNA, pp. 267-283, Editors: B. Eisfeld, H. Barnewitz, W. Fritz, F. Thiele, Springer, 2013
- H. G. Matthies, A. Litvinenko, O. Pajonk, B. V. Rosić, E. Zander, Parametric and Uncertainty Computations with Tensor Product Representations, Book “Uncertainty Quantification in Scientific Computing”, IFIP Advances in Information and Communication Technology Vol. 377, 2012, pp 139-150,doi:10.1007/978-3-642-32677-6.
- M. Espig, W. Hackbusch, A. Litvinenko, H. G. Matthies, Ph. Wähnert, Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats, Computers & Mathematics with Applications, (2012), ISSN 0898-1221, 10.1016/j.camwa.2012.10.008.
- B. V. Rosic, A. Kucerová, J. Sykora, O. Pajonk, A. Litvinenko, H. G. Matthies: Parameter Identification in a Probabilistic Setting, Engineering Structures (2013), Vol. 50, pp 179–196, doi:10.1016/j.engstruct.2012.12.029.
- B. Rosic, A. Litvinenko, O. Pajonk and H. G. Matthies, Sampling-free linear Bayesian update of polynomial chaos representations, J. Comp. Physics, 231(2012), pp 5761-5787
- O. Pajonk, B. Rosic, A. Litvinenko, and H. G. Matthies, A Deterministic Filter for non Gaussian Bayesian Estimation, Physica D: Nonlinear Phenomena, 241(2012), pp 775-788.
- M. Espig, W. Hackbusch, A. Litvinenko, H. G. Matthies and E. Zander, Efficien Analysis of High Dimensional Data in Tensor Formats, Springer Lecture Note series for Computational Science and Engineering ''Sparse Grids and Applications'', vol. 88, pp 31-56, Garcke, Jochen; Griebel, Michael (Eds.) (2012) http://dx.doi.org/10.1007/978-3-642-31703-3_2.
- B. Rosic; H. G. Matthies, A. Litvinenko, O. Pajonk, A. Kucerova, and J. Sykora, Bayesian Updating of uncertainties in the description of heat and moisture transport in heterogenous materials, International Conference on Adaptive Modeling and Simulation ADMOS 2011, D. Aubry and P. Diez (Eds), pp. 415-423, 2011.
- A. Litvinenko and H. G. Matthies, Low-Rank Data Format for Uncertainty Quantification, International Conference on Stochastic Modeling Techniques and Data Analysis Proceedings, pp. 477-484, Chania, Greece, (2010). Editor: Christos H. Skiadas.
- B. N. Khoromskij, A. Litvinenko and H. G. Matthies, Application of hierarchical matrices for computing the Karhunen-Loeve expansion, Computing, 84 (2009), pp.49-67.
- B. N. Khoromskij, A. Litvinenko, H-matrix based preconditioner for the skin problem Lecture Notes in Computational Science and Engineering, Series Editors: Barth, T.J., Griebel, M., Keyes, D.E., Nieminen, R.M., Roose, D., Schlick, T., 2007
- A. Litvinenko, Application of hierarchical matrices for solving multiscale problems, Dissertation, University of Leipzig, Germany, (2006).
- V. B. Berikov, A. G. Litvinenko, The Influence of Prior Knowledge on the Expected Performance of a Classifier. Pattern Recognition Letters, Vol. 24, Issue 15, 2003, pp 2537–2548.
- V. B. Berikov, A. G. Litvinenko, Decision trees knowledge on the expected performance of a classifier, Pattern Recognition Letters, Vol. 24/15, 2003. pp 2537-2548.
- A. A. Cherepanov , A.G.Litvinenko, Book: Global problems of humanity and searching of ways of solving. Material for discussions. Moscow 2000, 197 pages.
- A. Litvinenko, H. G. Matthies and B. Khoromskij, Data sparse approximation of the Karhunen-Loeve expansion , Oberwolfach Report to Miniworkshop Numerical Upscaling for Media with Deterministic and Stochastic Heterogeneity, 10-16 Feb. 2013;
- A. Litvinenko and H. G. Matthies, Sampling and Low-Rank Tensor Approximations , Oberwolfach Report to Workshop Numerical Methods for PDE Constrained Optimization with Uncertain Data, 27 Jan.- 2 Feb. 2013;
- H. G. Matthies, A. Litvinenko, B. V. Rosic, A. Kucerova, J. Sykora, O. Pajonk, Stochastic Setting for Inverse Identification Problems , Oberwolfach Report to Workshop Numerical Methods for PDE Constrained Optimization with Uncertain Data , 27 Jan.- 2 Feb. 2013;
- P. Wähnert, A. Litvinenko, M. Espig, H. G. Matthies and W. Hackbusch, Approximation of the stochastic Galerkin matrix in the low-rank canonical tensor format , PAMM Proc. Appl. Math. Mech. 12, 785 – 788 (2012) / DOI 10.1002/pamm.201210380
- A. Litvinenko and H. G. Matthies Uncertainty Quantification in numerical Aerodynamic via low-rank Response Surface, PAMM Proc. Appl. Math. Mech. 12, 781 – 784 (2012) / DOI 10.1002/pamm.201210379
- B. N. Khoromskij and A. Litvinenko, Data Sparse Computation of the Karhunen-Loeve Expansion. AIP Conference Proceedings, volume 1048, Num. 1, pp 311-314, 2008. Editors: Theodore E. Simos, George Psihoyios and Ch. Tsitouras.
- A. Litvinenko and H. G. Matthies Uncertainties Quantification and Data Compression in numerical Aerodynamics, PAMM Proc. Appl. Math. Mech. 11, 877-878 / DOI 10.1002/pamm.201110425. Special Issue: 82nd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Graz 2011; Editors: G. Brenn, G.A. Holzapfel, M. Schanz and O. Steinbach.
- Berikov, V.B., Lbov G.S. Litvinenko A.G. Evaluation of recognition performance for discrete classifier, The 6-th German-Russian Workshop Pattern Recognition and Image Understanding ORGW2003. Katun, Altai Region, Russia, pp 34-37.
- Berikov V.B., Litvinenko A.G. Estimation of reliability of classification in discrete pattern recognition problem // International Scientific Conference IOI2002). (June 17-21, 2002), Ukraina.
- Berikov V.B., Litvinenko A.G. On the evaluation of discrete classifiers, Computer Data Analysis and Modeling. Robustness and Computer Intensive Methods. Proceedings of the Sixth International Conference (September 10-14), Minsk, Belarus 2001, pp 10-15.