People

Dr Dmitry Savostyanov

Lecturer
Department of Mathematical Sciences
Dr Dmitry Savostyanov

Profile

Biography

I develop efficient algorithms for high-dimensional problems using low-rank tensor product approximations, linear and multilinear algebra, matrix analysis, and other methods of modern numerical mathematics.

Qualifications

  • PhD (Cand. Sci.) in Computational Mathematics Institute of Numerical Mathematics of Russian Academy of Sciences, (2006)

  • MSc in Applied Physics and Mathematics Moscow Institute of Physics and Technology, (2003)

  • BSc in Applied Physics and Mathematics Institute of Physics and Technology, (2001)

  • PGCert in Learning and Teaching in Higher Education University of Brighton, (2016)

Appointments

Other academic

  • Senior Lecturer, Computing, Engineering and Mathematics, University of Brighton (20/10/2014 - 31/1/2020)

  • Senior Research Fellow, Chemistry, University of Southampton (1/10/2012 - 30/9/2014)

  • Leverhulme Trust Visiting Research Fellow, Mathematical and Physical Sciences, University of Chester (1/10/2011 - 30/9/2012)

  • Researcher, Institute of Numerical Mathematics of Russian Academy of Sciences (1/1/2007 - 30/9/2011)

Research and professional activities

Research interests

High-dimensional problems

High-dimensional problems appear in a variety of applications, including quantum physics and chemistry, probability and statistics, stochastic differential equations, problems with uncertain parameters or noise. Typically, these problems are too complex to be solved using pen and paper only --- numerical methods and high-performance calculations are essential. However, the numerical complexity (computer memory and time required to crunch the numbers) grow exponentially with the dimension and standard off-the-shelf algorithms struggle. To break the curse of dimensionality, we need to develop a new family of effective algorithms, capable to meet the main computational challenge of the 21st century.

Key words: curse of dimensionality
Open to supervise

Tensor product decompositions

To break the curse of dimensionality and make high-dimensional problems possible to solve in our computational practice, we need to develop a new family of effective algorithms. One of the most promising approaches is based on the idea of tensor product decomposition. Similar to a low-rank decomposition of a matrix, tensor product decomposition applied to a high-dimensional array (tensor) or a multivariate function separates the variables (indices) and treats each dimension (mode) separately. In many applications this leads to exponential savings of computational costs, dramatically reducing the time required to solve the problem and improving the accuracy of the results. Often the problems which were considered impossible to solve become possible with the help of tensor product decompositions.

Key words: effective algorithms
Open to supervise

Teaching and supervision

Current teaching responsibilities

  • Numerical Methods (MA209)

Publications

Journal articles (24)

Dolgov, SV. and Savostyanov, DV., One-site density matrix renormalization group and alternating minimum energy algorithm. in Lecture Notes in Computational Science and Engineering.. 103, 335-343

Dolgov, SV. and Savostyanov, DV., Alternating minimal energy methods for linear systems in higher dimensions. Part I: SPD systems. SIAM J. Sci. Comput. 36(5): A2248-A2271, 2014

Dolgov, S. and Savostyanov, D., (2020). Parallel cross interpolation for high-precision calculation of high-dimensional integrals. Computer Physics Communications. 246, 106869-106869

Dolgov, S., Pearson, JW., Savostyanov, DV. and Stoll, M., (2016). Fast tensor product solvers for optimization problems with fractional differential equations as constraints. Applied Mathematics and Computation. 273, 604-623

Savostyanov, DV., Dolgov, SV., Werner, JM. and Kuprov, I., (2014). Exact NMR simulation of protein-size spin systems using tensor train formalism. Physical Review B. 90 (8), 085139-

Edwards, LJ., Savostyanov, DV., Welderufael, ZT., Lee, D. and Kuprov, I., (2014). Quantum mechanical NMR simulation algorithm for protein-size spin systems. Journal of Magnetic Resonance. 243, 107-113

Ford, NJ., Savostyanov, DV. and Zamarashkin, NL., (2014). On the Decay of the Elements of Inverse Triangular Toeplitz Matrices. SIAM Journal on Matrix Analysis and Applications. 35 (4), 1288-1302

Savostyanov, DV., (2014). Quasioptimality of maximum-volume cross interpolation of tensors. Linear Algebra and its Applications. 458, 217-244

Dolgov, SV., Khoromskij, BN., Oseledets, IV. and Savostyanov, DV., (2014). Computation of extreme eigenvalues in higher dimensions using block tensor train format. Computer Physics Communications. 185 (4), 1207-1216

Dolgov, SV. and Savostyanov, DV., (2014). Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions. SIAM Journal on Scientific Computing. 36 (5), A2248-A2271

Roberts, JA., Savostyanov, DV. and Tyrtyshnikov, EE., (2014). Superfast solution of linear convolutional Volterra equations using QTT approximation. Journal of Computational and Applied Mathematics. 260, 434-448

Edwards, LJ., Savostyanov, DV., Nevzorov, AA., Concistrè, M., Pileio, G. and Kuprov, I., (2013). Grid-free powder averages: On the applications of the Fokker–Planck equation to solid state NMR. Journal of Magnetic Resonance. 235, 121-129

Goreinov, SA., Oseledets, IV. and Savostyanov, DV., (2012). Wedderburn Rank Reduction and Krylov Subspace Method for Tensor Approximation. Part 1: Tucker Case. SIAM Journal on Scientific Computing. 34 (1), A1-A27

Savostyanov, DV., Tyrtyshnikov, EE. and Zamarashkin, NL., (2012). Fast truncation of mode ranks for bilinear tensor operations. Numerical Linear Algebra with Applications. 19 (1), 103-111

Savostyanov, D., (2012). QTT-rank-one vectors with QTT-rank-one and full-rank Fourier images. Linear Algebra and its Applications. 436 (9), 3215-3224

Dolgov, S., Khoromskij, B. and Savostyanov, D., (2012). Superfast Fourier Transform Using QTT Approximation. Journal of Fourier Analysis and Applications. 18 (5), 915-953

Oseledets, IV., Savostyanov, DV. and Tyrtyshnikov, EE., (2010). Cross approximation in tensor electron density computations. Numerical Linear Algebra with Applications. 17 (6), 935-952

Savostyanov, DV., (2010). Tensor algorithms of blind separation of electromagnetic signals. Russian Journal of Numerical Analysis and Mathematical Modelling. 25 (4), 375-393

Savostyanov, DV. and Tyrtyshnikov, EE., (2009). Approximate multiplication of tensor matrices based on the individual filtering of factors. Computational Mathematics and Mathematical Physics. 49 (10), 1662-1677

Oseledets, IV., Savostyanov, DV. and Tyrtyshnikov, EE., (2009). Fast Simultaneous Orthogonal Reduction to Triangular Matrices. SIAM Journal on Matrix Analysis and Applications. 31 (2), 316-330

Savostyanov, DV., (2009). Fast Revealing of Mode Ranks of Tensor in Canonical Form. Numerical Mathematics: Theory, Methods and Applications. 2 (4), 439-444

Oseledets, IV., Savostyanov, DV. and Tyrtyshnikov, EE., (2009). Linear algebra for tensor problems. Computing. 85 (3), 169-188

Oseledets, IV., Savostianov, DV. and Tyrtyshnikov, EE., (2008). Tucker Dimensionality Reduction of Three-Dimensional Arrays in Linear Time. SIAM Journal on Matrix Analysis and Applications. 30 (3), 939-956

Flad, H-J., Khoromskij, BN., Savostyanov, DV. and Tyrtyshnikov, EE., (2008). Verification of the cross 3D algorithm on quantum chemistry data. Russian Journal of Numerical Analysis and Mathematical Modelling. 23 (4), 329-344

Book chapters (2)

Quiñones-Valles, D., Dolgov, S. and Savostyanov, D., (2019). Tensor Product Approach to Quantum Control. In: Integral Methods in Science and Engineering. Springer International Publishing. 367- 379. 9783030160760

Dolgov, SV. and Savostyanov, DV., (2015). Corrected One-Site Density Matrix Renormalization Group and Alternating Minimal Energy Algorithm. In: Lecture Notes in Computational Science and Engineering. Springer International Publishing. 335- 343. 9783319107042

Conferences (2)

Savostyanov, D. and Oseledets, I., (2011). Fast adaptive interpolation of multi-dimensional arrays in tensor train format

Goreinov, SA., Savostyanov, DV., Tyrtyshnikov, EE. and ACAD, E., (2009). Tensor and Toeplitz Structures Applied to Direct and Inverse 3D Electromagnetic Problems

Contact

d.savostyanov@essex.ac.uk
+44 (0) 1206 876065

Location:

2.411, Colchester Campus

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