Research

My research is focused on the following three topics:

- Low rank approximation of discrete matrices and tensors, and multidimensional functions

Representation of 3D functions in Chebfun3 involving three quasimatrices and a discrete core tensor. See [H. & Trefethen, SISC, 2017] for details.

- Verified numerical algorithms

Histograms for correct digits in 2500 entries of the square root of a 50 x 50 matrix. Left: Number of correct digits in the floating point approximation. Middle: Number of correct digits in computed enclosing intervals. Right: Additional correct digits obtained through enclosing intervals. For details see [Frommer & H., SIMAX, 2009].

Interpolation of the wiggly function at 120 intervals containing Chebyshev points. Rigorous lower and upper bounds on the interpolant are depicted using verified evaluation at 1000 equispaced points. This is work in progress.

- Matrix theory over ordered semirings

Comparison of the connections between vertices in the graphs corresponding with the Kronecker sums and products over semirings. See [H., Tavakolipour & Shirazi, LAA, 2014] to find out how we do it!

SELECTED JOURNAL PUBLICATIONS:

B. Hashemi and Y. Nakatsukasa, RTSMS: Randomized Tucker with single-mode sketching, under review, arXiv preprint, link to codes in GitHub
B. Hashemi, Y. Nakatsukasa and L. N, Trefethen, Rectangular eigenvalue problems, Advances in Computational Mathematics 48 (2022) Article No. 80. Link to full-text
B. Hashemi and Y. Nakatsukasa, Least-squares spectral methods for ODE eigenvalue problems, SIAM Journal on Scientific Computing 44 (2022) A3244-A3264. PDF file, link to supplementary material and codes
B. Hashemi, Sufficient conditions for the solvability of a Sylvester-like absolute value matrix equation, Applied Mathematics Letters 112 (2021) 106818. Link to full-text (valid until Dec. 4 2020)
A. Paganini, B. Hashemi, E. Alsina Ballester and L. Belluzzi, Fast and accurate approximation of the angle-averaged redistribution function for polarized radiation, Astronomy & Astrophysics (A&A), accepted, Sep. 2020. arXiv preprint
A. Frommer and B. Hashemi, Computing enclosures for the matrix exponential, SIAM Journal on Matrix Analysis and Applications 41 (2020) 1674–1703. PDF file
J. L. Aurentz and B. Hashemi, The Laurent–Horner method for validated evaluation of Chebyshev expansions, Applied Mathematics Letters 102 (2020) 106113.
B. Hashemi, Enclosing Chebyshev expansions in linear time, ACM Transactions on Mathematical Software 45 (2019) 1-33. PDF file
B. Hashemi and Y. Nakatsukasa, On the spectral problem for trivariate functions, BIT Numerical Mathematics 58 (2018) 981-1008. PDF file
B. Hashemi and L. N. Trefethen, Chebfun in three dimensions, SIAM Journal on Scientific Computing 39 (2017) C341-C363. PDF file
B. Hashemi, H. Tavakolipour and M. N. Shirazi, Explicit formulae for the greatest least-squares and minimum norm g-inverses and the unique group inverse of matrices over commutative residuated dioids, Semigroup Forum 92 (2016) 541-550. PDF file
B. Hashemi, M. M. Khalilabadi and H. Tavakolipour, A cubic time algorithm for finding the principal solution to Sylvester matrix equations over (max; +), Linear and Multilinear Algebra 63 (2015) 283-295.
B. Hashemi, M. M. Khalilabadi and H. Tavakolipour, On the fixed-point type Sylvester matrix equations over complete commutative dioids, Electronic Journal of Linear Algebra 27 (2014) 882-891. PDF file
A. Frommer, B. Hashemi and T. Sablik, Computing enclosures for the inverse square root and the sign function of a matrix, Linear Algebra and its Applications 456 (2014) 199-213. PDF file
B. Hashemi, H. Tavakolipour and M. N. Shirazi, Comparison of the quasi-inverses of the Kronecker sum and product of matrices over complete commutative dioids with applications, Linear Algebra and its Applications 448 (2014) 22-36. PDF file
B. Hashemi and H. Tavakolipour, A non-induced interval matrix norm, Reliable Computing 18 (2013) 144-146. PDF file
A. Frommer and B. Hashemi, Verified stability analysis using the Lyapunov matrix equation, Electronic Transactions on Numerical Analysis 40 (2013) 187-203. PDF file
A. Frommer and B. Hashemi, Verified error bounds for solutions of Sylvester matrix equations, Linear Algebra and its Applications 436 (2012) 405-420. PDF file
B. Hashemi and M. Dehghan, The interval Lyapunov matrix equation: Analytical results and an efficient numerical technique for outer estimation of the united solution set, Mathematical and Computer Modelling 55 (2012) 622-633. PDF file
B. Hashemi and M. Dehghan, Results concerning interval linear systems with multiple right-hand sides and the interval matrix equation AX = B, Journal of Computational and Applied Mathematics 235 (2011) 2969-2978. PDF file
B. Hashemi and M. Dehghan, Efficient computation of enclosures for the exact solvents of a quadratic matrix equation, Electronic Journal of Linear Algebra 20 (2010) 519-536. PDF file
A. Frommer and B. Hashemi, Verified computation of square roots of a matrix, SIAM Journal on Matrix Analysis and Applications 31 (2009) 1279-1302. PDF file
M. Dehghan and B. Hashemi, Determination of the degrees of P-property and nonnegative invertibility for a fuzzy matrix, International Journal of Approximate Reasoning 46 (2007) 98-108. PDF file

BOOK CHAPTER:

B. Hashemi and L. N. Trefethen, Chebfun3, Chapter 18 in Chebfun Guide, T. A. Driscoll, N. Hale, and L. N. Trefethen, editors, Pafnuty Publications, Oxford, 2014.

BOOK TRANSLATION:

M. L. Overton, Numerical Computing with IEEE Floating Point Arithmetic, SIAM, Philadelphia, 2001, Translation to Persian, to be published by Sharif University Press.

REFEREED CONFERENCE PROCEEDINGS:

B. Hashemi and L. N. Trefethen, Slice-Tucker decomposition for Chebfun in 3D, Workshop on Tensor Decompositions and Applications, Leuven, Belgium, Jan. 18-22, 2016.
Verified computation of symmetric solutions to continuous-time algebraic Riccati matrix equations, SCAN 2012: 15th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Verified Numerical Computations, Institute of Computational Technologies, Siberian Department of the Russian Academy of Sciences, Novosibirsk, Russia, Sep. 23-29, 2012.
A. Frommer and B. Hashemi, Computational proofs of the stability of Lyapunov equations, Householder Symposium XVIII on Numerical Linear Algebra, Tahoe City, CA, USA, Jun. 12-17, 2011.
A. Frommer and B. Hashemi, Computing guaranteed error bounds for solutions of matrix equations, SCAN 2010: 14th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, ENS Lyon, France, Sep. 27-30, 2010.
A. Frommer and B. Hashemi, Error bounds for solutions of Sylvester equations and tests for stability, 81st Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), University of Karlsruhe, Germany, Mar. 22-26, 2010.
A. Frommer and B. Hashemi, Computing enclosures for the matrix square root, SIAM Conference on Applied Linear Algebra, Monterey Bay-Seaside, CA, USA, Oct. 26-29, 2009.
A. Frommer and B. Hashemi, Computing enclosures for the matrix square root, International Workshop on Verified Computations and Related Topics, University of Karlsruhe, Germany, Mar. 7-10, 2009. Published in COE Lecture Notes, Vol. 15, Kyushu University, ISSN 1881-4042, pp. 25-30.
M. Dehghan and B. Hashemi, How to compute the degree of positive definiteness for a fuzzy matrix of fuzzy numbers?, 37th Annual Iranian Mathematics Conference, Azarbaijan University of Tarbiat Moallem, Iran, Sep. 2-5, 2006.

SELECTED INVITED TALKS:

B. Hashemi, Spectral transformation of Clenshaw recurrence for enclosing Chebyshev expansions, Institute of Mathematical Sciences (ICMAT), Madrid, Spain, Sep. 10, 2018.
B. Hashemi, Tensor based methods for approximation of 3D functions, Scientific Computing Group, University of Wuppertal, Germany, Apr. 29, 2016.
B. Hashemi, On the norms of interval matrices, 7th CREST-SBM international conference, Japan SIAM, Kyoto University, Japan, Mar. 19-20, 2014.
B. Hashemi, Verified computation of inverse square root and the sign functions of a matrix, International Workshop on Numerical Verification and its Applications 2014, Waseda University, Tokyo, Japan, Mar. 15-17, 2014.

SELECTED TALKS IN CONFERENCES & SEMINARS:

B. Hashemi, On the spectral problem for trivariate functions, Numerical Analysis Group Internal Seminar, Mathematical Institute, University of Oxford, Jan. 24, 2017.
B. Hashemi, Chebfun in three dimensions, Numerical Analysis Group Internal Seminar, Mathematical Institute, University of Oxford, Mar. 8, 2016.
B. Hashemi, Numerical computing with 3D functions, British Applied Mathematics Colloquium (BAMC), University of Oxford, Apr. 5-8, 2016.
B. Hashemi, Toward Chebfun in 3D, 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015), Beijing, Aug. 10-14, 2015.
B. Hashemi, Chebfun: From 2D to 3D, 26th Biennial Numerical Analysis Conference, University of Strathclyde, Glasgow, Jun. 23-26, 2015.
B. Hashemi, Rigorous computational proof of Hurwitz stability for a matrix by Lyapunov equation, Numerical Analysis Group Internal Seminar, Mathematical Institute, University of Oxford, Feb. 3, 2015