# Homogenization theory

**"Complexity reduction in many particles systems with random initial data"**(with P.E. Jabin, M. Potomkin),__SIAM/ASA Journal of Uncertainty Quantification__, 4(1), pp. 446-474 (2016), pdf.**"Effective rheological properties in semidilute bacterial suspensions"**(with M. Potomkin and S. D. Ryan),__Bulletin of Mathematical Biology__, 78(3), pp. 580-615 (2016), pdf.**"A two scale Gamma-convergence approach for random non-convex homogenization"**(with E. Sandier and S. Serfaty),*submitted*(2016), pdf.**"Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization"**(with H. Owhadi, L. Zhang),__ESAIM: Mathematical Modelling and Numerical Analysis.__Special issue, 48 (2), pp. 517-552 (2014), pdf.**"Vortex phase separation in mesoscopic superconductors"**(with O. Iaroshenko, V. Rybalko, V. M. Vinokur),__Scientific Reports: Nature Publishing Group__**3**(2013), pdf.**"Effective viscosity of puller-like microswimmers: a renormalization approach"**(with S. Gluzman and D.A. Karpeev),__J. Royal Society Interface__**10**:89 (2013), pdf.**"Homogenized description of multiple Ginzburg-Landau vortices pinned by small holes"**(with V. Rybalko), special issue__NHM__**8**:1 pp.115-130 (2013), pdf.**"Effective viscosity of bacterial suspensions: A three-dimensional PDE model with stochastic torque"**(with B.M. Haines, I.S. Aranson, D.A. Karpeev),__Comm. Pure Appl. Anal.__, v. 11(1), pp. 19-46 (2012), pdf.**"Viscosity of bacterial suspensions: Hydrodynamic interactions and self-induced noise"**(with S.D. Ryan, B.M. Haines, F. Ziebert, I.S. Aranson), Rapid Communication to__Phys. Rev. E__, 83 050904(R) (2011), pdf.**"Effective shear viscosity and dynamics of suspensions of micro-swimmers from small to moderate concentrations"**(with V. Gyrya, K. Lipnikov, I. Aranson),__J. Math. Biology__, v. 62 (5), pp. 707-740 (2011), pdf.**"Global Energy Matching Method for Atomistic to Continuum Modeling of Self-Assembling Biopolymer Aggregates"**(with Lei Zhang, M.V. Fedorov, H. Owhadi),__Multiscale Model. Simul.__v. 8, n. 5, pp. 1958-1980 (2010), pdf.**"Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast"**(with H. Owhadi),__Arch. Rat. Mech. Anal.__, v. 198, n. 2, pp. 677-721 (2010), pdf.**"A three-dimensional model for the effective viscosity of bacterial suspensions"**(with B.M. Haines, A. Sokolov, I.S. Aranson, D.A. Karpeev),__Physical Review E__, v. 80, pp.041922 (2009), pdf.**"Fictitious Fluid Approach and Anomalous Blow-up of the Dissipation Rate in a 2D Model of Concentrated Suspensions"**(with Y. Gorb and A. Novikov),__Arch. Rat. Mech. Anal.__, v. 193, n. 3, pp. 585-622, (2009), DOI:10.1007/s00205-008-0152-2, pdf.**"Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain"**(with P. Mironescu),__Networks and Heterogenous Media__, v.3, n.3, pp. 461-487 (2008), pdf.**"Effective Viscosity of Dilute Bacterial Suspensions: A Two-Dimensional Model"**(with B. Haines, I. Aronson, and D. Karpeev),__Physical Biology__, 5:4, 046003 (9pp) (2008), pdf.**"The Homogenized Model of Small Oscillations of Complex Fluids"**(with M. Berezhnyy and E. Khruslov),__Networks and Heterogenous Media__, v.3, n.4, pp. 831-862 (2008), pdf.**"Strong and weak blow up of the viscous dissipation rates for concentrated suspensions"**(with A. Panchenko),__Journal of Fluid Mechanics__, v. 578, pp. 1-34 (2007), pdf.**"Asymptotic analysis of an array of absolutely conducting holes"**(with G. Cardone, Y. Gorb, and G. Panasenko),__Networks and Heterogenous Media__, 1:3, pp. 353-377 (2006), pdf.**"Continuum limit for three-dimensional mass-spring networks and discrete Korn's inequality"**(with M. Berezhnyy),__Journal of Mechanics and Physics of Solids__, 54:3, pp. 635-669 (2006), pdf.**"Network approximation for effective viscosity of concentrated suspensions with complex geometry"**(with L. Borcea and A. Panchenko),__SIAM Journal on Mathematical Analysis__, 36:5, pp 1580-1628 (2005), pdf.**"Asymptotics of the Effective Conductivity of Composites with Closely Spaced Inclusions of Optimal Shape"**(with Y. Gorb),__Quarterly Journal of Mechanics and Applied Mathematics__, 58 (1), pp. 83-106 (2005), pdf.**"Homogenization of a Ginzburg-Landau functional"**(with D. Cioranescu and D. Golovaty),__C. R. Acad. Sci.__, Paris, 340/1 pp 87-92 (2005), pdf.**"Network approximation for effective viscosity of concentrated suspensions with complex geometry"**(with L. Borcea and A. Panchenko),__SIAM Journal on Mathematical Analysis__, 36:5, pp 1580-1628 (2005), pdf.**"Increase and Decrease of the Effective Conductivity of Two Phase Composites due to Polydispersity"**(with V. Mityushev),__Journal of Statistical Physics__, 118, 3/4 pp 481-509 (2005), pdf.**"Homogenization of a Ginzburg-Landau model for a nematic liquid crystal with inclusions"**(with D. Cioranescu and D. Golovaty),__Journal des Mathematiques Pures et Appliquees__, 84:1, pp. 97-136 (2005), pdf.**"The Effective Conductivity of Densely Packed High Contrast Composites with Inclusions of Optimal Shape"**(with Y. Gorb)__Proceedings of Continuum Models and Discrete Systems__(CMDS10) Conference, Tel-Aviv, Israel (2004), pdf.**"Homogenized Non-Newtonian Viscoelastic Rheology of a Suspension of Interacting Particles in a Viscous Newtonian Fluid"**(with E. Ya. Khruslov)__SIAM Journal of Applied Mathematics__, 64:3, pp. 1002-1034 (2004), pdf.**"Discrete Network Approximation for Highly-Packed Composites with Irregular Geometry in Three Dimensions"**(with Y. Gorb and A. Novikov), in__Lecture Notes in Computational Science and Engineering__, editors B. Engquist, P. Lotstedt and O. Runborg, Springer Verlag (2004), pdf.**"Homogenized Non-Newtonian Viscoelastic Rheology of a Suspension of Interacting Particles in a Viscous Newtonian Fluid"**(with E. Ya. Khruslov)__SIAM Journal of Applied Mathematics__, 64:3, pp. 1002-1034 (2004), pdf.-
**"Geometric patterns and effective conductivity of highly packed two-phase composites"**(with A. Novikov),__Homogenization 2001__, (Naples), 113-129 GAKUTO Internat. Ser. Math. Sci. Appl, 18, Tokyo (2003). **"Error of the network approximation for densely packed composites with irregular geometry"**(with A. Novikov),__SIAM Journal on Mathematical Analysis__, 34(2), pp. 385-408 (2002), pdf.**"Network Approximation in the Limit of Small Interparticle Distance of the Effective Properties of a High-Contrast Random Dispersed Com- posite"**(with A. Kolpakov),__Archive for Rational Mechanics and Analysis__, 159, pp. 179-227 (2001), pdf.**"Homogenization of harmonic maps with large number of vortices and applications in superconductivity and superfluidity"**(with E. Ya. Khruslov),__Advances in Differential Equations__, v. 6/2, pp. 229-256 (2001), pdf.**"Homogenization. In Memory of Sergei Kozlov."**Edited by V. Berdichevsky, V. Jikov and G. Papanicolaou",__Series on Advances in Mathematics for Applied Sciences__, 50, pp. xiv+418 (1999), pdf.**"Homogenization of the Ginzburg-Landau functional with a surface energy term"**,__Asymptotic Analysis__, 37-59, 21 (1999), pdf.**"Homogenization of harmonic maps and superconducting composites"**(with E. Ya. Khruslov),__SIAM Journal of Applied Mathematics__, 59, 5 pp. 1892-1916 (1999), pdf.**"Effective Properties of Superconducting and Superfluid Composites"**,__International Journal of Modern Physics B__, 12, 29, 3063-3073 (1999), pdf.**"Renormalization Group Technique for Asymptotic Behavior of a Thermal Diffusive Model with Critical Nonlinearity"**(with J. Xin), in__Pitman Research Notes__, No. 324 "Recent Development in Evolution Equations", 76-85 (1995), pdf.**"Effective Elastic Moduli of a Soft Medium with Hard Polygonal Inclusions and Extremal Behavior of Effective Poisson's Ratio"**(with K.S. Promislow),__Journal of Elasticity__, 40:1, 45-73 (1995), pdf.**"Exact Result for the Effective Conductivity of a Continuum Percolation Model"**(with K. Golden),__Physical Review B__, 50 (4):2114-2117 (1994), pdf.**"Asymptotics of the Homogenized Moduli for the Elastic Chess-Board Composite"**(with S. Kozlov),__Archive for Rational Mechanics and Analysis__, 118, 95-112 (1992), pdf.**"The averaging of the diffusion equation in porous medium with weak absorption"**(with M. Goncharenko),__Journal of Soviet Mathematics__, 53, No. 5:3428-3435 (1990), pdf.-
**"Operational separation of variables in problems of short wave asymptotic behavior for differential equations with fast oscillating coefficients"**(with S. Yu. Dobrokhotov),__Sov. Phys. Dokl.__, 32 (9):714-716 (1987). -
**"Homogenization and short wave asymptotics for solutions of an initial value problem for the Schroedinger equation with a rapidly oscillating potential"**(with S. Yu. Dobrokhotov),__Uspekhy Mat. Nauk.__, 41 (4):195-196 (in Russian, 1986). -
**"Asymptotic behavior of solutions of the Dirichlet boundary value problem in elasticity for domains with fine-grained boundaries"**__Uspekhy Mat. Nauk.__, 38, No. 6 (234), 107-108 (in Russian, 1983). -
**"Averaging of elasticity equations in domains with fine-grained boundaries, Part 1-2"**,__Functional theory, functional analysis and applications__, Kharkov, 39:16-25 - Part 1; 40:16-23 - Part 2 (in Russian, 1983). -
**"An asymptotic description for a thin plate with a large number of small holes"**,__Ukrainian Academy of Science Reports, Ser. A.__, No. 10:5-8 (in Russian, 1983). -
**"Averaging of boundary value problems for higher order differential operators in domains with holes"**(with I. Yu. Chudinovich),__Soviet Math. Dokl.__, 28, No. 2:427-430 (1983). -
**"An averaged description of an elastic medium with a large number of small absolutely solid inclusions"**(with A. Okhotsimskii),__Soviet Phys. Dokl.__, 28 (1):81-84 (1983). -
**"On the vibration of an elastic body with a large number of small holes"**__Ukrainian Academy of Science reports, Ser. A__, 2:3-5 (in Russian, 1983).