Publications
Research monographs
1. Milstein,
G.N. and Tretyakov, M.V. (2004) Stochastic Numerics
for Mathematical Physics. Series: Scientific Computation, Springer, XX+ 596 p., 48 illus. ISBN: 3-540-21110-1.
2. Milstein,
G.N. and Tretyakov, M.V. (2021) Stochastic Numerics
for Mathematical Physics. Series: Scientific Computation, Springer, XXV+ 736 p Revised and expanded Second Edition.
Textbook
Tretyakov,
M.V. (2013) Introductory Course on Financial Mathematics, Imperial
College Press, X+ 266 p. ISBN: 978-1-908977-38-0.
Journal Articles - published
1. Fedotov, S.P. and Tret'yakov,
M.V. (1988): Stationary regimes of heterogeneous chemical reaction at existence
of external noises. Khimicheskay Fizika,
V. 7, No. 11, pp. 1533-1537 (in Russian).
2. Tret'yakov, M.V. and Fedotov, S.P.
(1990): Stationary regimes of heterogeneous chemical reaction at existence of
white Poissonian noise. Khimicheskay
Fizika, V. 9, No. 2, pp. 252-257 (in Russian).
3. Fedotov, S.P. and Tret'yakov, M.V.
(1991): About stochastic ignition of a particle. Khimicheskay
Fizika, V. 10, No. 2, pp. 238-241 (in Russian).
4. Fedotov, S.P.
and Tret'yakov, M.V. (1991): Stochastic criteria for
ignition of single particles. Combust. Sci. and Tech., V. 78,
No. 1-3, pp. 1-6.
5. Buyevich, Yu.A., Tret'yakov,
M.V., and Fedotov, S.P. (1991): Model of stochastic growth of new phase islands
on solid surface. Dokl. Akad.
Nauk SSSR, V. 321, No. 5, pp. 1005-1009 (in Russian);
translation in Dokl. Phys. Chem., V. 6, pp. 852-854
(1991).
6. Buyevich, Yu.A.
and Tret'yakov, M.V. (1992):
Covering of solid surface by a new phase under conditions of metastability reduction by growing nuclei. Dokl. Akad. Nauk,
V. 323, No. 2, pp. 306-310 (in Russian).
7. Tret'yakov, M.V. (1992): Influence of colored noise on combustion of a particle. Ingenerno-Fizicheskii Journal, V. 63, No. 2, pp. 156-160
(in Russian). Translated
in English: Journal of Engineering Physics and Thermophysics,
pp 771-774.
8. Buyevich, Yu.A., Fedotov, S.P., and Tret'yakov,
M.V. (1993): Heterogeneous reaction affected by external noise. Physica A, V. 198, Nos. 3/4,
pp. 354-367.
9. Buyevich,
Yu.A. and Tret'yakov, M.V. (1994): The initiating rate of a
heterogeneous reaction. Journal de Physique II, V. 4, pp.
1605-1616.
10. Milshtein,
G.N. and Tret'yakov,
M.V. (1994): Numerical solution of differential equations with colored noise. J. Stat. Phys., V. 77, Nos. 3/4, pp.
691-715.
11. Tret'yakov, M.V. and Tret'jakov,
S.V. (1994): Numerical integration of Hamiltonian systems with external noise.
Phys. Lett. A, V. 194, pp. 371-374.
12. Milstein, G.N.
and Tret'yakov, M.V. (1997): Mean-square numerical
methods for stochastic differential equations with small noises. SIAM J. Sci. Comput., V. 18, No. 4, pp. 1067-1087.
13. Milstein,
G.N. and Tret'yakov, M.V. (1997): Numerical methods
in the weak sense for stochastic differential equations with small noise. SIAM
J. Numer. Anal., V. 34, No. 6, pp. 2142-2167.
14. Tretyakov, M.V.
(1998): Numerical technique for studying stochastic resonance. Physical Review
E, V. 57, No. 4, pp. 4789-4794.
15. Milstein, G.N. and
Tretyakov, M.V. (1999): Simulation of a space-time diffusion. The Annals of
Applied Probability, V. 9, No. 3, pp. 732-779.
16. Milstein,
G.N. and Tretyakov, M.V. (1999): Mean velocity
of noise-induced transport in the limit of weak periodic forcing. Journal of
Physics A: Mathematical and General, V. 32, No. 31, pp. 5795-5805.
17. Milstein,
G.N. and Tretyakov, M.V. (2000): Numerical algorithms for semilinear
parabolic equations with small parameter based on approximation of stochastic
equations. Mathematics of Computation, V. 69,
No. 229, pp. 237-267.
18. Milstein, G.N.
and Tretyakov, M.V. (2000): Numerical analysis of noise-induced regular
oscillations. Physica D: Nonlinear Phenomena, V.140,
Nos. 3-4, pp. 244-256.
19.
Milstein, G.N. and Tretyakov, M.V. (2001): Noise-induced unidirectional transport.
Stochastics and Dynamics, V. 1, No 1, pp. 361-375.
20. Milstein, G.N.
and Tretyakov, M.V. (2001): Numerical solution of Dirichlet
problems for nonlinear parabolic equations by a probabilistic approach. IMA J.
Num. Anal., V. 21, No 4, pp. 887-917.
21. Tretyakov,
M.V. and Fedotov S. (2001): On FKPP equation with Gaussian shear advection. Physica D: Nonlinear Phenomena, V. 159, Nos. 3-4, pp.
191-202.
22. Milstein,
G.N. and Tretyakov, M.V. (2002): The simplest random walks for the Dirichlet problem. Theory of Probability and its Applications V. 47, No 1, pp. 39-58 (Russian); pp. 53-68
(English).
23. Milstein,
G.N., Repin, Yu.M., and Tretyakov, M.V. (2002): Symplectic
integration of Hamiltonian systems with additive noise. SIAM J. Numer. Anal. V. 39, No 6, pp. 2066-2088.
24.
Milstein, G.N. and Tretyakov, M.V. (2002): A probabilistic approach to solution
of the Neumann problem for nonlinear parabolic equations. IMA J. Numer. Anal., V. 22, No 4, pp. 599-622.
25. Milstein,
G.N., Repin, Yu.M., and Tretyakov, M.V. (2002): Numerical methods for
stochastic systems preserving symplectic structure.
SIAM J. Numer. Anal., V. 40, No 4, pp. 1583-1604.
26.
Milstein, G.N. and Tretyakov, M.V. (2003): Quasi-symplectic
methods for Langevin-type equations. IMA J. Numer. Anal., V. 23, No 4, pp. 593-626.
27. Milstein, G.N. and
Tretyakov, M.V. (2004): Evaluation of conditional Wiener integrals by
numerical integration of stochastic differential equations. Journal of
Computational Physics, V. 197, No 1, pp. 275-298.
28.
Milstein, G.N. and Tretyakov, M.V. (2005): Numerical integration of stochastic
differential equations with nonglobally Lipschitz
coefficients. SIAM J. Numer. Anal., V. 43, No3,
1139-1154.
29. Milstein,
G.N. and Tretyakov, M.V. (2005): Numerical analysis of Monte Carlo evaluation
of Greeks by finite differences. Journal of
Computational Finance, V. 8, No 3, pp. 1-33.
30. Milstein,
G.N. and Tretyakov, M.V. (2006): Numerical algorithms for forward-backward
stochastic differential equations. SIAM J. Sci. Comp., V. 28, No 2, pp.
561-582.
31. Milstein, G.N. and Tretyakov,
M.V. (2007): Discretization of forward-backward stochastic differential
equations and related quasilinear parabolic equations. IMA J Numer. Anal. V. 27, No 1, pp. 24-44.
32. Milstein, G.N.
and Tretyakov, M.V. (2007): Computing ergodic limits for Langevin
equations. Physica D: Nonlinear Phenomena, V. 129, No
1, pp. 81-95.
33.
Fedotov S., Milstein, G.N. and Tretyakov, M.V. (2007): Superdiffusion
of a random walk driven by ergodic Markov process with switching. Journal of
Physics A: Mathematical and Theoretical, V. 40, pp. 5769-5782.
34. Milstein,
G.N. and Tretyakov, M.V. (2009): Practical variance reduction via regression
for simulating diffusions. SIAM J. Numer. Anal. V.
47, No. 2, pp. 887-910.
35.
Milstein, G.N. and Tretyakov, M.V. (2009): Monte Carlo algorithms for backward
equations in nonlinear filtering. Advances in Applied Probability, V. 41, No.
1, pp. 63-100.
36. Milstein,
G.N. and Tretyakov, M.V. (2009): Solving parabolic stochastic partial
differential equations via averaging over characteristics. Math. Comp. V. 78,
No 268, pp. 2075-2106.
37. Ruslan L. Davidchack,
Richard Handel, and M.V. Tretyakov (2009): Langevin
thermostat for rigid body dynamics. J. Chem. Phys. V. 130, Issue 23, pp.
234101-14.
38. Robert J. Hardwick,
Michael V. Tretyakov, and Yuri E. Dubrova (2009): Age-related accumulation of
mutations supports a replication-dependent mechanism of spontaneous mutation at
tandem repeat DNA loci in mice. Molecular Biology and Evolution. V. 26, No 11,
pp. 2647-2654.
39.
Jonathan C. Mattingly, Andrew M. Stuart, and M.V. Tretyakov (2010): Convergence
of numerical time-averaging and stationary measures via Poisson equations. SIAM
J. Numer. Anal. V. 48, No 2, pp. 552-577.
40. J.H.
Mentink, M.V. Tretyakov, A. Fasolino, M.I. Katsnelson, Th. Rasing (2010):
Stable and fast semi-implicit integration of the stochastic Landau-Lifshitz equation. J. Phys.: Condens.
Matter V. 22, Number 17, 176001 (12pp). [included in Highlights of 2010 of
the journal] [SIA and SIB integrators are included in UppASD library]
41.
Dumas, W.M. and Tretyakov, M.V. (2011): Computing conditional Wiener integrals
of functionals of a general form. IMA J. Numer. Anal. V. 31, No 3, pp. 1217-1251.
42. Milstein G.N. and
Tretyakov M.V. (2012): Solving the Dirichlet problem
for Navier-Stokes equations by probabilistic
approach. BIT Numer. Math. V. 52 ,
No 1, pp. 141-153.
43. Z. Zhang, B. Rozovskii, M.V. Tretyakov and G. Karniadakis (2012): A
multi-stage Wiener chaos expansion method for stochastic
advection-diffusion-reaction equations. SIAM J. Sci. Comp. V. 34, No 2, pp.
A914-A936.
44. Bayliss C., Bidmos F., Anjum A., Manchev V., Richards R., Grossier
J-P.; Wooldridge K., Ketley J., Barrow P., Jones M.,
Tretyakov M.V. (2012): Phase variable genes of Campylobacter jejuni exhibit high
mutation rates and specific mutational patterns but mutability is not the major
determinant of population structure during host colonisation. Nucleic Acids
Research V 40, No. 13, pp. 5876-5889.
45. Milstein
G.N. and Tretyakov M.V. (2013): Probabilistic methods for the incompressible Navier‒Stokes equations with space periodic
conditions. Advances in Applied Probability, V. 45, No. 3, pp. 742-772.
46. Tretyakov
M.V. and Zhang Z. (2013): A fundamental mean-square convergence theorem for
SDEs with locally Lipschitz coefficients and its applications. SIAM J. Numer. Anal. V. 51, No 6, pp. 3135-3162.
47. Krivko
M. and Tretyakov M.V. (2014): Numerical integration of Heath-Jarrow-Morton
model of interest rates. IMA J. Numer. Anal. V. 34,
pp. 147-196.
48. Zhang Z., Tretyakov M.V., Rozovskii
B.L. and Karniadakis G. (2014): A recursive sparse
grid collocation method for differential equations with white noise. SIAM J.
Sci. Comp. V. 36, No 4, pp. A1652-A1677.
49.
Leimkuhler B., Matthews C., and Tretyakov M.V. (2014): On the
long-time integration of stochastic gradient systems. Proc. R. Soc. A V. 470, No 2170, 20140120, 16 pp.
50. Zhang Z., Tretyakov M.V., Rozovskii B.L. and Karniadakis G.
(2015): Wiener chaos vs stochastic collocation methods for linear
advection-diffusion-reaction equations with multiplicative white noise. SIAM J.
Numer. Anal. V. 53, No 1, pp. 153-183.
51. Davidchack
R.L., Ouldridge T.E., and Tretyakov M.V. (2015): New Langevin
and gradient thermostats for rigid body dynamics. J. Chem. Phys. V. 142, No 14,
144114, 13 pp.
[Langevin
C integrator is implemented in LAMMPS]
52.
Park M. and Tretyakov M.V. (2015): A block circulant
embedding method for simulation of stationary Gaussian random fields on
block-regular grids. Int. J. Uncertainty Quantification, V. 5, No. 6, pp.
527-544.
53. Milstein
G.N. and Tretyakov M.V. (2016): Layer methods for Navier-Stokes
equations with additive noise using simplest characteristics. J. Comp Appl.
Maths. V. 302, pp. 1-23.
54. Park M. and Tretyakov M.V.
(2017): Stochastic resin transfer molding process.
SIAM/ASA Journal on Uncertainty Quantification. Vol. 5, No. 1, pp. 1110-1135.
[also downloadable
from arxiv]
55. Davidchack
R.L., Ouldridge T.E., and Tretyakov M.V. (2017): Geometric integrator for Langevin systems with quaternion-based rotational degrees of
freedom and hydrodynamic interactions. J. Chem. Phys. Vol.147, Issue 22,
224103, 12pp+5pp of Suppl Mat. +videos [also downloadable from arxiv]
56. Matveev M.Y., Ball
F., Jones I.A., Long A.C., Schubel P.J., Tretyakov
M.V. (2018): Uncertainty in geometry of fibre preforms manufactured with
Automated Dry Fibre Placement (ADFP) and its effects on permeability. Journal
of Composite Materials V.52, Issue 16, pp. 2255-2269.
57. Bailey N.Y, Hibberd
S., Power H., Tretyakov M.V. (2018): Evaluation of the
minimum face clearance of a high speed gas lubricated bearing with Navier slip boundary conditions under random excitations.
Journal of Engineering Mathematics Vol. 112, Issue 1, pp. 17-35.
58. Iglesias M., Park M.,
Tretyakov M.V. (2018): Bayesian inversion in resin transfer molding.
Inverse Problems V. 34, No 10, 1005002.
59. Bayliss C.D.,
Fallaize C., Howitt R., Tretyakov M.V. (2019): Mutation and selection in
bacteria: modelling and calibration. Bulletin of Mathematical Biology V. 81, No
3, pp. 639-675.
60. Bailey N.Y, Hibberd S.,
Power H., Tretyakov M.V. (2019): Effect of random forcing on fluid lubricated
bearing. IMA J. Appl. Math. V. 84, Issue 3, pp. 632-649.
61. Milstein G.N. and
Tretyakov M.V. (2020): Mean-square approximation of Navier-Stokes
equations with additive noise in vorticity-velocity formulation. Numerical
Mathematics: Theory, Methods and Applications V. 14, Issue 1, pp. 1-30
62. Matveev,
M.Y., Endruweit, A., Long, A.C., Iglesias, M.A, Tretyakov, M.V. (2021):
Bayesian inversion algorithm for estimating local variations in permeability
and porosity of reinforcements using experimental data. Composites Part A V.
143, 106323.
63. Deligiannidis, D.,
Maurer, S., Tretyakov, M.V. (2021): Random walk algorithm for the Dirichlet problem for parabolic integro-differential
equation. BIT Numerical Mathematics V. 61, pp. 1223-1269.
64. Endruweit, A., Matveev, M.Y., Tretyakov, M.V. (2022): Controlling resin flow in liquid
composite moulding processes through localised irradiation with ultraviolet
light. Polymer Composites V. 43, Issue 11, pp. 8308-8321.
65. Iglesias, M.,
McGrath, D. M., Tretyakov, M. V., Francis, S.T. (2022): Ensemble Kalman inversion for magnetic resonance elastography.
Physics in Medicine & Biology V. 67, Number 23, 235003.
66. Kalise, D., Sharma,
A., Tretyakov, M.V. (2023): Consensus based
optimization via jump-diffusion stochastic differential equations. Mathematical
Models and Methods in Applied Sciences (M3AS) V. 33, No. 2, pp. 289-339 [earlier
version downloadable from arxiv]
67. Leimkuhler, B., Sharma, A., Tretyakov,
M.V. (2023): Simplest random walk for approximating Robin boundary value
problems and ergodic limits of reflected diffusions. The Annals of Applied
Probability V. 33, No. 3, 1904-1960
68. Hinds,
P.D., Tretyakov, M.V. (2023): Neural variance reduction for stochastic
differential equations. J. Comp. Finance V. 27 [a version is downloadable from arxiv]
69. Su, H., Tretyakov, M.V., Newton, D.P. (2024): Deep learning of
transition probability densities for stochastic asset models with applications
in option pricing. Management Science (accepted) (an earlier version is
available at [downloadable from arxiv])
Refereed contributions to books
70. Milstein,
G.N. and Tretyakov, M.V. (2011): Averaging over characteristics with innovation
approach in nonlinear filtering. In Handbook of Nonlinear Filtering (Eds. D.
Crisan and B. Rozovskii) Oxford University Press,
ISBN 978-0-19-953290-2, pp. 892-919.
71.
Stanciulescu,
V.N. and Tretyakov, M.V. (2011): Numerical solution of the Dirichlet
problem for linear parabolic SPDEs based on averaging over characteristics. In
Stochastic Analysis 2010 (Ed. D. Crisan), Springer, ISBN 978-3-642-15357-0, pp.
191-212.
72.
Krivko M. and Tretyakov M.V. (2013): Application of simplest random walk
algorithms for pricing barrier options. In: Recent Developments in
Computational Finance (Eds. T. Gerstner and P.E. Kloeden), World Scientific, ISBN 978-981-4436-42-7, pp. 407-428. [pre-publication
is at arxiv]
73. Bailey N.Y, Hibberd S.,
Power H., Tretyakov M.V. (2019): Effect of uncertainty
in external forcing on a fluid lubricated bearing. In: Cavalca
K., Weber H. (eds)
Proceedings of the 10th International Conference on Rotor Dynamics - IFToMM. IFToMM 2018. Mechanisms
and Machine Science, vol 63. Springer.
Recent preprints
74. Maurer, S., Sharp, T.E., and Tretyakov, M.V. (2021): Pricing FX options
under intermediate currency. [downloadable from arxiv]
Book Editor
Numerical
Mathematics and Advanced Applications. Proceedings of ENUMATH 2011, the 9th
European Conference on Numerical Mathematics and Advanced Applications,
Leicester, September 2011. Cangiani, A.; Davidchack, R.L.;
Georgoulis, E.; Gorban, A.N.; Levesley, J.; Tretyakov, M.V. (Eds.), Springer
2013; VIII, 854 pp. ISBN 978-3-642-33133-6.
Other
Tretyakov
M.V. (2015): Stochastic filtering. In Encyclopedia of
Applied and Computational Mathematics, Editor-in-Chief Bjorn Engquist, Springer, ISBN 978-3-540-70530-7.
Milstein G.N. and Tretyakov M.V.
(2012): Layer methods for Navier-Stokes equations
with additive noise. Research Report No. MA-12-001, Department of Mathematics,
University of Leicester, Leicester (UK), 2012, 35pp. [downloadable from arxiv]