Mioara JoldesLAASCNRS7 Avenue du Colonel Roche, 31077 Toulouse, Cedex 4 France phone:(+33) (0)5 61 33 69 26 email: joldes AT laas DOT fr 
Rigorous computing (validated computing): use numerical computations, but provide mathematical
statements about the obtained result, such as sure, yet reasonably tight, error bounds. Build efficient
symbolicnumeric objects, algorithms and software tools with direct applications in control of dynamical
systems and in particular in the aerospace domain. Use and develop expertise and ideas from Computer Arithmetic and Computer Algebra.
A. Towards Fast and Certified Multiple Precision Arithmetic Libraries: develop a multiple precision arithmetic library tuned for Graphics Processing Units (GPUs). Targets highperformance computing problems which require multiple precision e.g., the study of strange attractors such as the Hénon attractor [2]. Use floatingpoint expansions that is, extended precision is represented as the unevaluated sum of standard machine floatingpoint numbers. Recent results: normalization, division and sqrt [8], [3]. Goals: formally proven algorithms, elementary functions implementation. Some talks: [ASAP14] [Henon13][SMC14] 

B. Rigorous Polynomial Approximation (RPA): a polynomial approximation together with rigorous error bounds. Use Taylor Models [1, 14, 15, 9] and Chebyshev polynomial interpolation/series Models [11], [19, Chap. 4]. Adapt to rigorous computations many numerical algorithms based on Chebyshev/Taylor Series for solving ordinary differential equations, quadrature, etc. Recent results: compute rigorous uniform approximations based on Chebyshev Series for Dfinite functions, i.e. solutions of ordinary differential equations with polynomial coefficients [5] e.g., erf, exp, sin, Bessel, Airy functions; efficient computations of supremum norms of approximation errors [1]; formally proven Taylor Models [9]. Objectives: efficient implementation of operations with Chebyshev Models, extensions to multivariate functions; efficient finite precision evaluation of power series. Some talks: [RPA] [TAMADI][SeaMac] 

C. Applications to Optimal Control and Aerospace:


 
[1] Sylvain Chevillard, John Harrison, Mioara Joldeş, and Christoph Lauter. Efficient and accurate computation of upper bounds of approximation errors. Theoretical Computer Science, 16(412):1523–1543, 2011. Preliminary version.
[2] Mioara Joldes, Valentina Popescu, and Warwick Tucker. Searching for sinks for the Hénon map using a multipleprecision GPU arithmetic library. SIGARCH Comput. Archit. News, 42(4):63–68, December 2014. Preliminary version.
[3] Mioara Joldes, Olivier Marty, JeanMichel Muller, and Valentina Popescu. Arithmetic algorithms for extended precision using floatingpoint expansions. IEEE Transactions on Computers, 2015. In print, doi: 10.1109/TC.2015.2441714, Preliminary version.
[4] Romain Serra, Denis Arzelier, Mioara Joldes, JeanBernard Lasserre, Aude Rondepierre, and Bruno Salvy. Fast and Accurate Computation of Orbital Collision Probability for ShortTerm Encounters. Journal of Guidance, Control, and Dynamics, 2015. Preliminary version.
[5] Alexandre Benoit, Mioara Joldes, and Marc Mezzarobba. Rigorous uniform approximation of Dfinite functions using Chebyshev expansions. Accepted for publication to Mathematics of Computation, 2015. Preliminary version.
[6] Romain Serra, Denis Arzelier, Mioara Joldes, JeanBernard Lasserre, Aude Rondepierre, and Bruno Salvy. A new method to compute the probability of collision for shortterm space encounters. In Astrodynamics Specialist Conference, pages 1–7, Aug 2014.
[7] Romain Serra, Denis Arzelier, Mioara Joldes, and Aude Rondepierre. Probabilistic collision avoidance for longterm space encouters via risk selection. In 3rd CEAS European Aerospace Guidance, Navigation and Control (EuroGNC) Conference, pages –21, Dec 2014.
[8] Mioara Joldes, JeanMichel Muller, and Valentina Popescu. On the computation of the reciprocal of floating point expansions using an adapted NewtonRaphson iteration. In IEEE 25th International Conference on ApplicationSpecific Systems, Architectures and Processors, ASAP 2014, Zurich, Switzerland, June 1820, 2014, pages 63–67. IEEE, 2014. Preliminary version.
[9] Nicolas Brisebarre, Mioara Joldes, Érik MartinDorel, Micaela Mayero, JeanMichel Muller, Ioana Pasca, Laurence Rideau, and Laurent Théry. Rigorous polynomial approximation using Taylor Models in Coq. In Alwyn Goodloe and Suzette Person, editors, NASA Formal Methods  4th International Symposium, NFM 2012, Norfolk, VA, USA, April 35, 2012. Proceedings, volume 7226 of Lecture Notes in Computer Science, pages 85–99. Springer, 2012. Preliminary version.
[10] Nicolas Brisebarre, Mioara Joldes, Peter Kornerup, Érik MartinDorel, and JeanMichel Muller. Augmented precision square roots and 2d norms, and discussion on correctly rounding sqrt(x^2+y^2). In Elisardo Antelo, David Hough, and Paolo Ienne, editors, 20th IEEE Symposium on Computer Arithmetic, ARITH 2011, Tübingen, Germany, 2527 July 2011, pages 23–30. IEEE Computer Society, 2011. Preliminary version.
[11] Nicolas Brisebarre and Mioara Joldes. Chebyshev interpolation polynomialbased tools for rigorous computing. In Wolfram Koepf, editor, Symbolic and Algebraic Computation, International Symposium, ISSAC 2010, Munich, Germany, July 2528, 2010, Proceedings, pages 147–154. ACM, 2010. Preliminary version.
[12] Florent de Dinechin, Mioara Joldes, and Bogdan Pasca. Automatic generation of polynomialbased hardware architectures for function evaluation. In François Charot, Frank Hannig, Jürgen Teich, and Christophe Wolinski, editors, 21st IEEE International Conference on Applicationspecific Systems Architectures and Processors, ASAP 2010, Rennes, France, 79 July 2010, pages 216–222. IEEE, 2010. Preliminary version
[13] Florent de Dinechin, Mioara Joldes, Bogdan Pasca, and Guillaume Revy. Multiplicative square root algorithms for FPGAs. In International Conference on Field Programmable Logic and Applications, FPL 2010, August 31 2010  September 2, 2010, Milano, Italy, pages 574–577. IEEE, 2010.
[14] Sylvain Chevillard, Mioara Joldes, and Christoph Quirin Lauter. Sollya: An environment for the development of numerical codes. In Komei Fukuda, Joris van der Hoeven, Michael Joswig, and Nobuki Takayama, editors, Mathematical Software  ICMS 2010, Third International Congress on Mathematical Software, Kobe, Japan, September 1317, 2010. Proceedings, volume 6327 of Lecture Notes in Computer Science, pages 28–31. Springer, 2010.
[15] Sylvain Chevillard, Mioara Joldes, and Christoph Quirin Lauter. Certified and fast computation of supremum norms of approximation errors. In Javier D. Bruguera, Marius Cornea, Debjit Das Sarma, and John Harrison, editors, 19th IEEE Symposium on Computer Arithmetic, ARITH 2009, Portland, Oregon, USA, 910 June 2009, pages 169–176. IEEE Computer Society, 2009.
[16] Mioara Joldes, Valentina Popescu, and Warwick Tucker. Searching for sinks of Hénon map using a multipleprecision GPU arithmetic library. In Forum des Jeunes Mathématiciennes, pages –6, Nov 2013.
[17] Mioara Joldes. When a logarithm is a misspelled algorithm. In Proceedings of the Association Femmes et mathématiques, September 2010.
[18] Florent de Dinechin, Mioara Joldes, Bogdan Pasca, and Guillaume Revy. Racines carrées multiplicatives sur FPGA. In SYMPosium en Architectures nouvelles de machines (SYMPA), Toulouse, September 2009.
[19] Mioara Joldes. Rigorous Polynomial Approximations and Applications. Thesis, École Normale Supérieure de Lyon  ENS LYON, September 2011. https://tel.archivesouvertes.fr/tel00657843.
[20] Romain Serra, Denis Arzelier, Mioara Joldes, and Aude Rondepierre. Probability of collision between spherical space objects for shortterm space encounters. Technical report, LAASCNRS No. 14154, Mars 2014. Astrium Funding for PhD.
[21] Nicolas Brisebarre and Mioara Joldes. Rigorous polynomial approximations based on Chebyshev series expansions. In preparation, 2015.
[22] Denis Arzelier, Florent Bréhard, Norbert Deak, Mioara Joldes, Christophe Louembet, Aude Rondepierre and Romain Serra. Linearized Impulsive FixedTime FuelOptimal Space rendezvous: A New Numerical Approach. Submitted. Preliminary version