Andrew Knyazev
 Phone: 6176217516
 Email:

Position:
Research / Technical Staff
Distinguished Research Scientist, Professor Emeritus University of Colorado Denver, Society for Industrial and Applied Mathematics (SIAM) Fellow 
Education:
Ph.D., Institute of Numerical Mathematics, Russian Academy of Sciences, 1985 
Research Areas:
External Links:
Andrew's Quick Links

Biography
During his 30 years in academia, Andrew contributed to numerical analysis of partial differential equations and computational linear algebra, with emphasis on eigenvalue problems, supported by NSF and DOE awards, and graduated 7 Ph.D. students. Since 2012, his focus at MERL is on algorithms for image & video processing, data sciences, optimal control, material sciences, and numerical simulation of complex phenomena.

News & Events

NEWS Andrew Knyazev (MERL) presents at the SchlumbergerTufts U. Computational and Applied Math Seminar Date: April 10, 2018
MERL Contact: Andrew Knyazev
Research Areas: Data Analytics, Algorithms, Machine Learning, Signal ProcessingBrief Andrew Knyazev, Distinguished Research Scientist of MERL, has accepted an invitation to speak about his work on Big Data and spectral graph partitioning at the SchlumbergerTufts U. Computational and Applied Math Seminar. A primary focus of this seminar series is on mathematical and computational aspects of remote sensing. A partial list of the topics of interest includes: numerical solution of large scale PDEs (a.k.a. forward problems); theory and numerical methods of inverse and illposed problems; imaging; related problems in numerical linear algebra, approximation theory, optimization and model reduction. The seminar meets on average once a month, the location alternates between Schlumberger's office in Cambridge, MA and the Tufts Medford Campus.
Abstract: Data clustering via spectral graph partitioning requires constructing the graph Laplacian and solving the corresponding eigenvalue problem. We consider and motivate using negative edge weights in the graph Laplacian. Preconditioned iterative solvers for the Laplacian eigenvalue problem are discussed and preliminary numerical results are presented.
 Andrew Knyazev, Distinguished Research Scientist of MERL, has accepted an invitation to speak about his work on Big Data and spectral graph partitioning at the SchlumbergerTufts U. Computational and Applied Math Seminar. A primary focus of this seminar series is on mathematical and computational aspects of remote sensing. A partial list of the topics of interest includes: numerical solution of large scale PDEs (a.k.a. forward problems); theory and numerical methods of inverse and illposed problems; imaging; related problems in numerical linear algebra, approximation theory, optimization and model reduction. The seminar meets on average once a month, the location alternates between Schlumberger's office in Cambridge, MA and the Tufts Medford Campus.

NEWS Andrew Knyazev (MERL) invited to 2018 MathWorks Research Summit Date: June 2, 2018  June 4, 2018
Where: Newton, Massachusetts (USA)
MERL Contact: Andrew Knyazev
Research Areas: Algorithms, Advanced Control Systems, Computational Photography, Dynamical Systems, Machine Learning, Predictive ModelingBrief Dr. Andrew Knyazev of MERL has accepted an invitation to participate at the 2018 MathWorks Research Summit. The objective of the Research Summit is to provide a forum for leading researchers in academia and industry to explore the latest research and technology results and directions in computation and its use in technology, engineering, and science. The event aims to foster discussion among scientists, engineers, and research faculty about challenges and research opportunities for the respective communities with a particular interest in exploring crossdisciplinary research avenues.
See All News & Events for Andrew 

Awards

AWARD 2017 Graph Challenge Student Innovation Award Date: August 4, 2017
Awarded to: David Zhuzhunashvili and Andrew Knyazev
MERL Contact: Andrew Knyazev
Research Areas: Data Analytics, Algorithms, Machine LearningBrief David Zhuzhunashvili, an undergraduate student at UC Boulder, Colorado, and Andrew Knyazev, Distinguished Research Scientist at MERL, received the 2017 Graph Challenge Student Innovation Award. Their poster "Preconditioned Spectral Clustering for Stochastic Block Partition Streaming Graph Challenge" was accepted to the 2017 IEEE High Performance Extreme Computing Conference (HPEC '17), taking place 1214 September 2017 (http://www.ieeehpec.org/), and the paper was accepted to the IEEE Xplore HPEC proceedings.
HPEC is the premier conference in the world on the convergence of High Performance and Embedded Computing. DARPA/Amazon/IEEE Graph Challenge is a special HPEC event. Graph Challenge encourages community approaches to developing new solutions for analyzing graphs derived from social media, sensor feeds, and scientific data to enable relationships between events to be discovered as they unfold in the field. The 2017 Streaming Graph Challenge is Stochastic Block Partition. This challenge seeks to identify optimal blocks (or clusters) in a larger graph with known groundtruth clusters, while performance is evaluated compared to baseline Python and C codes, provided by the Graph Challenge organizers.
The proposed approach is spectral clustering that performs block partition of graphs using eigenvectors of a matrix representing the graph. Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method iteratively approximates a few leading eigenvectors of the symmetric graph Laplacian for multiway graph partitioning. Preliminary tests for all static cases for the Graph Challenge demonstrate 100% correctness of partition using any of the IEEE HPEC Graph Challenge metrics, while at the same time also being approximately 5001000 times faster compared to the provided baseline code, e.g., 2M static graph is 100% correctly partitioned in ~2,100 sec. Warmstarts of LOBPCG further cut the execution time 23x for the streaming graphs.
 David Zhuzhunashvili, an undergraduate student at UC Boulder, Colorado, and Andrew Knyazev, Distinguished Research Scientist at MERL, received the 2017 Graph Challenge Student Innovation Award. Their poster "Preconditioned Spectral Clustering for Stochastic Block Partition Streaming Graph Challenge" was accepted to the 2017 IEEE High Performance Extreme Computing Conference (HPEC '17), taking place 1214 September 2017 (http://www.ieeehpec.org/), and the paper was accepted to the IEEE Xplore HPEC proceedings.

AWARD Fellow of the Society for Industrial and Applied Mathematics (SIAM) Date: March 31, 2016
Awarded to: Andrew Knyazev
MERL Contact: Andrew Knyazev
Research Areas: Algorithms, Advanced Control Systems, Decision Optimization, Dynamical Systems, Machine Learning, Predictive Modeling, Wireless Communications & Signal ProcessingBrief Andrew Knyazev selected as a Fellow of the Society for Industrial and Applied Mathematics (SIAM) for contributions to computational mathematics and development of numerical methods for eigenvalue problems.
Fellowship honors SIAM members who have made outstanding contributions to the fields served by the SIAM. Andrew Knyazev was among a distinguished group of members nominated by peers and selected for the 2016 Class of Fellows.
 Andrew Knyazev selected as a Fellow of the Society for Industrial and Applied Mathematics (SIAM) for contributions to computational mathematics and development of numerical methods for eigenvalue problems.

AWARD Professor Emeritus University of Colorado Denver Date: January 6, 2016
Awarded to: Andrew Knyazev
MERL Contact: Andrew Knyazev
Research Area: AlgorithmsBrief Andrew Knyazev is awarded the title of Professor Emeritus at the University of Colorado Denver effective 1/31/2016. The award letter from the Chancellor of the University of Colorado Denver provides examples of the record of excellence over 20 years of contributions to the university such as 2008 CU Denver Excellence in Research Award, 2000 Teaching Excellence Award for the college, supervision of Ph.D. students, and two decades of uninterrupted external research funding from the US National Science Foundation and Department of Energy.


MERL Publications
 "Preconditioned Spectral Clustering for Stochastic Block Partition Streaming Graph Challenge", IEEE HPEC Graph Challenge, DOI: 10.1109/HPEC.2017.8091045, September 2017, pp. 16. ,
 "Preconditioned warmstarted NewtonKrylov methods for MPC with discontinuous control", SIAM Conference on Control and its Applications, July 2017. ,
 "Signal reconstruction via operator guiding", International Conference on Sampling Theory and Applications (SampTA), July 2017. ,
 "A Brief Theory of Guided Signal Reconstruction", International Conference on Sampling Theory and Applications (SampTA), July 2017. ,
 "Recent implementations, applications, and extensions of the Locally Optimal Block Preconditioned Conjugate Gradient method (LOBPCG)", Householder Symposium on Numerical Linear Algebra, June 2017. ,

Other Publications
 "Absolute Value Preconditioning for Symmetric Indefinite Linear Systems", SIAM Journal on Scientific Computing, Vol. 35, No. 2, pp. A696A718, 2013.BibTeX
 @Article{Vecharynski2013,
 author = {Vecharynski, E. and Knyazev, A.},
 title = {Absolute Value Preconditioning for Symmetric Indefinite Linear Systems},
 journal = {SIAM Journal on Scientific Computing},
 year = 2013,
 volume = 35,
 number = 2,
 pages = {A696A718},
 url = {http://epubs.siam.org/doi/abs/10.1137/120886686}
 }
,  "Bounds for the Rayleigh Quotient and the Spectrum of SelfAdjoint Operators", SIAM Journal on Matrix Analysis and Applications, Vol. 34, No. 1, pp. 244256, 2013.BibTeX
 @Article{Zhu2013,
 author = {Zhu, P. and Argentati, M. and Knyazev, A.},
 title = {Bounds for the Rayleigh Quotient and the Spectrum of SelfAdjoint Operators},
 journal = {SIAM Journal on Matrix Analysis and Applications},
 year = 2013,
 volume = 34,
 number = 1,
 pages = {244256},
 url = {http://epubs.siam.org/doi/abs/10.1137/120884468}
 }
,  "Angles between infinite dimensional subspaces with applications to the RayleighRitz and alternating projectors methods", Journal of Functional Analysis, Vol. 259, No. 6, pp. 13231345, 2010.BibTeX
 @Article{Knyazev2010,
 author = {Knyazev, A. and Jujunashvili, A. and Argentati, M.},
 title = {Angles between infinite dimensional subspaces with applications to the RayleighRitz and alternating projectors methods},
 journal = {Journal of Functional Analysis},
 year = 2010,
 volume = 259,
 number = 6,
 pages = {13231345},
 note = {Times Cited: 1 Knyazev, Andrew Jujunashvili, Abram Argentati, Merico}
 }
,  "Rayleighritz Majorization Error Bounds With Applications To Fem", Siam Journal on Matrix Analysis and Applications, Vol. 31, No. 3, pp. 15211537, 2010.BibTeX
 @Article{Knyazev2010a,
 author = {Knyazev, A. V. and Argentati, M. E.},
 title = {Rayleighritz Majorization Error Bounds With Applications To Fem},
 journal = {Siam Journal on Matrix Analysis and Applications},
 year = 2010,
 volume = 31,
 number = 3,
 pages = {15211537},
 note = {Times Cited: 2 Knyazev, Andrew V. Argentati, Merico E.}
 }
,  "Gradient Flow Approach To Geometric Convergence Analysis Of Preconditioned Eigensolvers", Siam Journal on Matrix Analysis and Applications, Vol. 31, No. 2, pp. 621628, 2009.BibTeX
 @Article{Knyazev2009,
 author = {Knyazev, A. V. and Neymeyr, K.},
 title = {Gradient Flow Approach To Geometric Convergence Analysis Of Preconditioned Eigensolvers},
 journal = {Siam Journal on Matrix Analysis and Applications},
 year = 2009,
 volume = 31,
 number = 2,
 pages = {621628},
 note = {Times Cited: 3 Knyazev, Andrew V. Neymeyr, Klaus}
 }
,  "Bounds On Changes In Ritz Values For A Perturbed Invariant Subspace Of A Hermitian Matrix", Siam Journal on Matrix Analysis and Applications, Vol. 30, No. 2, pp. 548559, 2008.BibTeX
 @Article{Argentati2008,
 author = {Argentati, M. E. and Knyazev, A. V. and Paige, C. C. and Panayotov, I.},
 title = {Bounds On Changes In Ritz Values For A Perturbed Invariant Subspace Of A Hermitian Matrix},
 journal = {Siam Journal on Matrix Analysis and Applications},
 year = 2008,
 volume = 30,
 number = 2,
 pages = {548559}
 }
,  "Largescale ab initio calculations based on three levels of parallelization", Computational Materials Science, Vol. 42, No. 2, pp. 329336, 2008.BibTeX
 @Article{Bottin2008,
 author = {Bottin, F. and Leroux, S. and Knyazev, A. and Zerah, G.},
 title = {Largescale ab initio calculations based on three levels of parallelization},
 journal = {Computational Materials Science},
 year = 2008,
 volume = 42,
 number = 2,
 pages = {329336},
 note = {Times Cited: 14 Bottin, Francois Leroux, Stephane Knyazev, Andrew Zerah, Gilles}
 }
,  "Observations on degenerate saddle point problems", Computer Methods in Applied Mechanics and Engineering, Vol. 196, No. 3740, pp. 37423749, 2007.BibTeX
 @Article{Knyazev2007,
 author = {Knyazev, A. V.},
 title = {Observations on degenerate saddle point problems},
 journal = {Computer Methods in Applied Mechanics and Engineering},
 year = 2007,
 volume = 196,
 number = {3740},
 pages = {37423749},
 note = {Times Cited: 0 Knyazev, Andrew V.}
 }
,  "Block locally optimal preconditioned eigenvalue Xolvers (BLOPEX) in hypre and PETSc", Siam Journal on Scientific Computing, Vol. 29, No. 5, pp. 22242239, 2007.BibTeX
 @Article{Knyazev2007a,
 author = {Knyazev, A. V. and Argentati, M. E. and Lashuk, I. and Ovtchinnikov, E. E.},
 title = {Block locally optimal preconditioned eigenvalue Xolvers (BLOPEX) in hypre and PETSc},
 journal = {Siam Journal on Scientific Computing},
 year = 2007,
 volume = 29,
 number = 5,
 pages = {22242239}
 }
,  "Preconditioned Eigensolver LOBPCG in hypre and PETSc", 2007.BibTeX
 @Misc{Lashuk2007,
 author = {Lashuk, I. and Argentati, M.E. and E., Ovchinnikov and Knyazev, A.V.},
 title = {Preconditioned Eigensolver LOBPCG in hypre and PETSc},
 year = 2007,
 volume = 55,
 pages = {635642},
 publisher = {Springer}
 }
,  "Large scale linear and nonlinear eigenvalue problems  Preface", Linear Algebra and Its Applications, Vol. 415, No. 1, pp. 12, 2006.BibTeX
 @Article{Bai2006,
 author = {Bai, Z. J. and Knyazev, A. and van der Vorst, H. A.},
 title = {Large scale linear and nonlinear eigenvalue problems  Preface},
 journal = {Linear Algebra and Its Applications},
 year = 2006,
 volume = 415,
 number = 1,
 pages = {12},
 note = {Times Cited: 0 Bai, ZJ Knyazev, A van der Vorst, HA}
 }
,  "Majorization for changes in angles between subspaces, Ritz values, and graph Laplacian spectra", Siam Journal on Matrix Analysis and Applications, Vol. 29, No. 1, pp. 1532, 2006.BibTeX
 @Article{Knyazev2006,
 author = {Knyazev, A. V. and Argentati, M. E.},
 title = {Majorization for changes in angles between subspaces, Ritz values, and graph Laplacian spectra},
 journal = {Siam Journal on Matrix Analysis and Applications},
 year = 2006,
 volume = 29,
 number = 1,
 pages = {1532},
 note = {Times Cited: 6 Knyazev, Andrew V. Argentati, Merico E. 12th ILAS Conference JUN, 2005 CANADA}
 }
,  "On proximity of Rayleigh quotients for different vectors and Ritz values generated by different trial subspaces", Linear Algebra and Its Applications, Vol. 415, No. 1, pp. 8295, 2006.BibTeX
 @Article{Knyazev2006a,
 author = {Knyazev, A. V. and Argentati, M. E.},
 title = {On proximity of Rayleigh quotients for different vectors and Ritz values generated by different trial subspaces},
 journal = {Linear Algebra and Its Applications},
 year = 2006,
 volume = 415,
 number = 1,
 pages = {8295},
 note = {Times Cited: 4 Knyazev, AV Argentati, ME}
 }
,  "Steepest descent and conjugate gradient methods with variable preconditioning", Siam Journal on Matrix Analysis and Applications, Vol. 29, No. 4, pp. 12671280, 2006.BibTeX
 @Article{Knyazev2006b,
 author = {Knyazev, A. V. and Lashuk, I.},
 title = {Steepest descent and conjugate gradient methods with variable preconditioning},
 journal = {Siam Journal on Matrix Analysis and Applications},
 year = 2006,
 volume = 29,
 number = 4,
 pages = {12671280},
 note = {Times Cited: 0 Knyazev, Andrew V. Lashuk, Ilya}
 }
,  "New a priori fem error estimates for eigenvalues", Siam Journal on Numerical Analysis, Vol. 43, No. 6, pp. 26472667, 2006.BibTeX
 @Article{Knyazev2006c,
 author = {Knyazev, A. V. and Osborn, J. E.},
 title = {New a priori fem error estimates for eigenvalues},
 journal = {Siam Journal on Numerical Analysis},
 year = 2006,
 volume = 43,
 number = 6,
 pages = {26472667},
 note = {Times Cited: 16 Knyazev, AV Osborn, JE}
 }
,  "6th IMACS International Symposium on Iterative Methods in Scientific Computing 2730 March 2003, University of Colorado at Denver, Denver, Colorado, USA  Introduction", Applied Numerical Mathematics, Vol. 54, No. 2, pp. 105106, 2005.BibTeX
 @Article{Billups2005,
 author = {Billups, S. and Knyazev, A. and Mandel, J.},
 title = {6th IMACS International Symposium on Iterative Methods in Scientific Computing 2730 March 2003, University of Colorado at Denver, Denver, Colorado, USA  Introduction},
 journal = {Applied Numerical Mathematics},
 year = 2005,
 volume = 54,
 number = 2,
 pages = {105106},
 note = {Times Cited: 0 Billups, S Knyazev, A Mandel, J}
 }
,  "Towards a dynamic data driven application system for wildfire simulation", 2005, Computational Science  Iccs 2005, Pt 2.BibTeX
 @Inbook{Mandel2005,
 author = {Mandel, J. and Bennethum, L. S. and Chen, M. S. and Coen, J. L. and Douglas, C. C. and Franca, L. P. and Johns, C. J. and Kim, M. J. and Knyazev, A. V. and Kremens, R. and Kulkarni, V. and Qin, G. and Vodacek, A. and Wu, J. J. and Zhao, W. and Zornes, A.},
 title = {Towards a dynamic data driven application system for wildfire simulation},
 booktitle = {Computational Science  Iccs 2005, Pt 2},
 year = 2005,
 editor = {Sunderam, V. S. and VanAlbada, G. D. and Sloot, P. M. A. and Dongarra, J. J.},
 volume = 3515,
 series = {Lecture Notes in Computer Science},
 pages = {632639},
 note = {Times Cited: 3 Mandel, J Bennethum, LS Chen, MS Coen, JL Douglas, CC Franca, LP Johns, CJ Kim, MJ Knyazev, AV Kremens, R Kulkarni, V Qin, G Vodacek, A Wu, JJ Zhao, W Zornes, A 5th International Conference on Computational Science (ICCS 2005) MAY 2225, 2005 Atlanta, GA}
 }
,  "A geometric theory for preconditioned inverse iteration III: A short and sharp convergence estimate for generalized eigenvalue problems", Linear Algebra and Its Applications, Vol. 358, pp. 95114, 2003.BibTeX
 @Article{Knyazev2003,
 author = {Knyazev, A. V. and Neymeyr, K.},
 title = {A geometric theory for preconditioned inverse iteration III: A short and sharp convergence estimate for generalized eigenvalue problems},
 journal = {Linear Algebra and Its Applications},
 year = 2003,
 volume = 358,
 pages = {95114},
 note = {Times Cited: 31 Knyazev, AV Neymeyr, K 3rd International Workshop on Accurate Solution of Eigenvalue Problems JUL 0306, 2002 HAGEN, GERMANY}
 }
,  "Lavrentiev regularization plus Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients", Mathematics of Computation, Vol. 72, No. 241, pp. 1740, 2003.BibTeX
 @Article{Knyazev2003a,
 author = {Knyazev, A. and Widlund, O.},
 title = {Lavrentiev regularization plus Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients},
 journal = {Mathematics of Computation},
 year = 2003,
 volume = 72,
 number = 241,
 pages = {1740},
 note = {Times Cited: 9 Knyazev, A Widlund, O}
 }
,  "Analysis of transmission problems on Lipschitz boundaries in stronger norms", Journal of Numerical Mathematics jnma, Vol. 11, No. 3, pp. 225234, 2003.BibTeX
 @Article{Knyazev2003b,
 author = {Knyazev, Andrew V},
 title = {Analysis of transmission problems on Lipschitz boundaries in stronger norms},
 journal = {Journal of Numerical Mathematics jnma},
 year = 2003,
 volume = 11,
 number = 3,
 pages = {225234}
 }
,  "Efficient solution of symmetric eigenvalue problems using multigrid preconditioners in the locally optimal block conjugate gradient method.", Electronic Transactions on Numerical Analysis, Vol. 15, pp. 3855, 2003.BibTeX
 @Article{Knyazev2003c,
 author = {Knyazev, Andrew V. and Neymeyr, Klaus},
 title = {Efficient solution of symmetric eigenvalue problems using multigrid preconditioners in the locally optimal block conjugate gradient method.},
 journal = {Electronic Transactions on Numerical Analysis},
 year = 2003,
 volume = 15,
 pages = {3855}
 }
,  "Extension theorems for Stokes and Lame equations for nearly incompressible media and their applications to numerical solution of problems with highly discontinuous coefficients", Numerical Linear Algebra with Applications, Vol. 9, No. 2, pp. 115139, 2002.BibTeX
 @Article{Bakhvalov2002,
 author = {Bakhvalov, N. S. and Knyazev, A. V. and Parashkevov, R. R.},
 title = {Extension theorems for Stokes and Lame equations for nearly incompressible media and their applications to numerical solution of problems with highly discontinuous coefficients},
 journal = {Numerical Linear Algebra with Applications},
 year = 2002,
 volume = 9,
 number = 2,
 pages = {115139},
 note = {Times Cited: 1 Bakhvalov, NS Knyazev, AV Parashkevov, RR}
 }
,  "Principal angles between subspaces in an Abased scalar product: Algorithms and perturbation estimates", Siam Journal on Scientific Computing, Vol. 23, No. 6, pp. 20082040, 2002.BibTeX
 @Article{Knyazev2002,
 author = {Knyazev, A. V. and Argentati, M. E.},
 title = {Principal angles between subspaces in an Abased scalar product: Algorithms and perturbation estimates},
 journal = {Siam Journal on Scientific Computing},
 year = 2002,
 volume = 23,
 number = 6,
 pages = {20082040},
 note = {Times Cited: 26 Knyazev, AV Argentati, ME}
 }
,  "Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method", Siam Journal on Scientific Computing, Vol. 23, No. 2, pp. 517541, 2001.BibTeX
 @Article{Knyazev2001,
 author = {Knyazev, A. V.},
 title = {Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method},
 journal = {Siam Journal on Scientific Computing},
 year = 2001,
 volume = 23,
 number = 2,
 pages = {517541},
 note = {Times Cited: 95 Knyazev, AV 6th Copper Mountain Conference APR 0307, 2000 BOULDER, CO}
 }
,  "Preconditioned eigensolversan oxymoron? %Z Large scale eigenvalue problems (Argonne, IL, 1997)", Electron. Trans. Numer. Anal., Vol. 7, pp. 104123 (electronic), 1998.BibTeX
 @Article{Knyazev1998,
 author = {Knyazev, Andrew V.},
 title = {Preconditioned eigensolversan oxymoron? %Z Large scale eigenvalue problems (Argonne, IL, 1997)},
 journal = {Electron. Trans. Numer. Anal.},
 year = 1998,
 volume = 7,
 pages = {104123 (electronic)}
 }
,  "New estimates for Ritz vectors", Mathematics of Computation, Vol. 66, No. 219, pp. 985995, 1997.BibTeX
 @Article{Knyazev1997,
 author = {Knyazev, A. V.},
 title = {New estimates for Ritz vectors},
 journal = {Mathematics of Computation},
 year = 1997,
 volume = 66,
 number = 219,
 pages = {985995},
 note = {Times Cited: 16 Knyazev, AV}
 }
,  "A subspace preconditioning algorithm for eigenvector/eigenvalue computation", Advances in Computational Mathematics, Vol. 6, No. 2, pp. 159189, 1996.BibTeX
 @Article{Bramble1996,
 author = {Bramble, J. H. and Pasciak, J. E. and Knyazev, A. V.},
 title = {A subspace preconditioning algorithm for eigenvector/eigenvalue computation},
 journal = {Advances in Computational Mathematics},
 year = 1996,
 volume = 6,
 number = 2,
 pages = {159189},
 note = {Times Cited: 33 Bramble, JH Pasciak, JE Knyazev, AV}
 }
,  "Fictitious domain methods and computation of homogenized properties of composites with a periodic structure of essentially different components", 1994, Numerical methods and applications.BibTeX
 @Inbook{Bakhvalov1994,
 author = {Bakhvalov, N. S. and Knyazev, A. V.},
 title = {Fictitious domain methods and computation of homogenized properties of composites with a periodic structure of essentially different components},
 booktitle = {Numerical methods and applications},
 year = 1994,
 pages = {221266},
 address = {Boca Raton, FL},
 publisher = {CRC}
 }
,  "A Korntype inequality and orthogonal decompositions in matrix spaces", Russian Acad. Sci. Math. Doklady, Vol. 48, No. 1, pp. 127129, 1994.BibTeX
 @Article{Bakhvalov1994a,
 author = {Bakhvalov, N. S. and Knyazev, A. V. and Eglit, M.},
 title = {A Korntype inequality and orthogonal decompositions in matrix spaces},
 journal = {Russian Acad. Sci. Math. Doklady},
 year = 1994,
 volume = 48,
 number = 1,
 pages = {127129}
 }
,  "Preconditioned iterative methods in a subspace for linear algebraic equations with large jumps in the coefficients", 1994, Domain decomposition methods in scientific and engineering computing (University Park, PA, 1993).BibTeX
 @Inbook{Bakhvalov1994b,
 author = {Bakhvalov, Nikolai S. and Knyazev, Andrew V.},
 title = {Preconditioned iterative methods in a subspace for linear algebraic equations with large jumps in the coefficients},
 booktitle = {Domain decomposition methods in scientific and engineering computing (University Park, PA, 1993)},
 year = 1994,
 volume = 180,
 series = {Contemp. Math.},
 pages = {157162},
 address = {Providence, RI},
 publisher = {Amer. Math. Soc.}
 }
,  "Preconditioned Gradienttype Iterative Methods In A Subspace For Partial Generalized Symmetrical Eigenvalue Problems", SIAM Journal on Numerical Analysis, Vol. 31, No. 4, pp. 12261239, 1994.BibTeX
 @Article{Knyazev1994,
 author = {Knyazev, A. V. and Skorokhodov, A. L.},
 title = {Preconditioned Gradienttype Iterative Methods In A Subspace For Partial Generalized Symmetrical Eigenvalue Problems},
 journal = {SIAM Journal on Numerical Analysis},
 year = 1994,
 volume = 31,
 number = 4,
 pages = {12261239},
 note = {Times Cited: 15 KNYAZEV, AV SKOROKHODOV, AL}
 }
,  "A Korntype inequality and orthogonal decompositions in matrix spaces", Dokl. Akad. Nauk, Vol. 331, No. 5, pp. 536537, 1993.BibTeX
 @Article{Bakhvalov1993,
 author = {Bakhvalov, N. S. and Knyazev, A. V. and Eglit, M.},
 title = {A Korntype inequality and orthogonal decompositions in matrix spaces},
 journal = {Dokl. Akad. Nauk},
 year = 1993,
 volume = 331,
 number = 5,
 pages = {536537},
 note = {E}
 }
,  "Variational Rayleigh quotient iteration methods for symmetric eigenvalue problem", (EastWest) Journal of Numerical Mathematics, Vol. 1, No. 2, pp. 121129, 1993.BibTeX
 @Article{Knyazev1993,
 author = {Knyazev, A.V. and Sharapov, I.S.},
 title = {Variational Rayleigh quotient iteration methods for symmetric eigenvalue problem},
 journal = {(EastWest) Journal of Numerical Mathematics},
 year = 1993,
 volume = 1,
 number = 2,
 pages = {121129}
 }
,  "Iterative methods of gradient type in a subspace for a partial symmetric generalized eigenvalue problem", Russian Acad. Sci. Dokl. Mat., Vol. 45, pp. 474478, 1993.BibTeX
 @Article{Knyazev1993a,
 author = {Knyazev, A.V. and Skorokhodov, A.L.},
 title = {Iterative methods of gradient type in a subspace for a partial symmetric generalized eigenvalue problem},
 journal = {Russian Acad. Sci. Dokl. Mat.},
 year = 1993,
 volume = 45,
 pages = {474478}
 }
,  "An efficient iterative method for solving the Lame equations for almost incompressible media and Stokes' equations", Soviet Math. Doklady, Vol. 44, No. 1, pp. 49, 1992.BibTeX
 @Article{Bakhvalov1992,
 author = {Bakhvalov, N.S. and Knyazev, A.V.},
 title = {An efficient iterative method for solving the Lame equations for almost incompressible media and Stokes' equations},
 journal = {Soviet Math. Doklady},
 year = 1992,
 volume = 44,
 number = 1,
 pages = {49}
 }
,  "On an iterative method for finding lower eigenvalues", Russian Journal of Numerical Analysis and Mathematical Modelling, Vol. 7, No. 6, pp. 473486, 1992.BibTeX
 @Article{Dyakonov1992,
 author = {D'yakonov, E.G. and Knyazev, A.V.},
 title = {On an iterative method for finding lower eigenvalues},
 journal = {Russian Journal of Numerical Analysis and Mathematical Modelling},
 year = 1992,
 volume = 7,
 number = 6,
 pages = {473486}
 }
,  "Iterative methods of gradient type in a subspace for a partial symmetric generalized eigenvalue problem", Dokl. Akad. Nauk, Vol. 323, No. 6, pp. 10201023, 1992.BibTeX
 @Article{Knyazev1992,
 author = {Knyazev, A. V. and Skorokhodov, A. L.},
 title = {Iterative methods of gradient type in a subspace for a partial symmetric generalized eigenvalue problem},
 journal = {Dokl. Akad. Nauk},
 year = 1992,
 volume = 323,
 number = 6,
 pages = {10201023}
 }
,  "The Preconditioned Gradienttype Iterative Methods In A Subspace For Partial Generalized Symmetrical Eigenvalue Problem", Doklady Akademii Nauk, Vol. 323, No. 6, pp. 10201023, 1992.BibTeX
 @Article{Knyazev1992a,
 author = {Knyazev, A. V. and Skorokhodov, A. L.},
 title = {The Preconditioned Gradienttype Iterative Methods In A Subspace For Partial Generalized Symmetrical Eigenvalue Problem},
 journal = {Doklady Akademii Nauk},
 year = 1992,
 volume = 323,
 number = 6,
 pages = {10201023},
 note = {Times Cited: 0 KNYAZEV, AV SKOROKHODOV, AL}
 }
,  "A parallel algorithm of subspace iterations and its implementation on a multiprocessor with ring architecture", Russian Journal of Numerical Analysis and Mathematical Modelling, Vol. 7, No. 1, pp. 5561, 1992.BibTeX
 @Article{Knyazev1992b,
 author = {Knyazev, A.V.},
 title = {A parallel algorithm of subspace iterations and its implementation on a multiprocessor with ring architecture},
 journal = {Russian Journal of Numerical Analysis and Mathematical Modelling},
 year = 1992,
 volume = 7,
 number = 1,
 pages = {5561}
 }
,  "Iterative solution of PDE with strongly varying coefficients: algebraic version", 1992.BibTeX
 @Misc{Knyazev1992c,
 author = {Knyazev, Andrey V.},
 title = {Iterative solution of PDE with strongly varying coefficients: algebraic version},
 year = 1992,
 pages = {8589},
 publisher = {Elsevier Science Publishers,Amsterdam, The Netherlands}
 }
,  "An Effective Iterative Method For Solution Of Lame Equations For Almostincompressible Fluids And Stokes Equations", Doklady Akademii Nauk Sssr, Vol. 319, No. 1, pp. 1317, 1991.BibTeX
 @Article{Bakhvalov1991,
 author = {Bakhvalov, N. S. and Knyazev, A. V.},
 title = {An Effective Iterative Method For Solution Of Lame Equations For Almostincompressible Fluids And Stokes Equations},
 journal = {Doklady Akademii Nauk Sssr},
 year = 1991,
 volume = 319,
 number = 1,
 pages = {1317},
 note = {Times Cited: 1 BAKHVALOV, NS KNYAZEV, AV}
 }
,  "Methods for the effective computation of averaged characteristics of composites of periodic structure consisting of essentially different materials", 1991, Computational processes and systems, No. 8 (Russian).BibTeX
 @Inbook{Bakhvalov1991a,
 author = {Bakhvalov, N. S. and Knyazev, A. V.},
 title = {Methods for the effective computation of averaged characteristics of composites of periodic structure consisting of essentially different materials},
 booktitle = {Computational processes and systems, No. 8 (Russian)},
 year = 1991,
 pages = {5294},
 address = {Moscow},
 publisher = {``Nauka''}
 }
,  "Iterative methods for solving equations with highly varying coefficients", 1991, Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Moscow, 1990).BibTeX
 @Inbook{Bakhvalov1991b,
 author = {Bakhvalov, N. S. and Knyazev, A. V. and Kobelkov, G. M.},
 title = {Iterative methods for solving equations with highly varying coefficients},
 booktitle = {Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Moscow, 1990)},
 year = 1991,
 pages = {197205},
 address = {Philadelphia, PA},
 publisher = {SIAM}
 }
,  "Efficient computation of averaged characteristics of composites of a periodic structure of essentially different materials", Soviet Math. Doklady, Vol. 42, No. 1, pp. 5762, 1991.BibTeX
 @Article{Bakhvalov1991c,
 author = {Bakhvalov, N.S. and Knyazev, A.V.},
 title = {Efficient computation of averaged characteristics of composites of a periodic structure of essentially different materials},
 journal = {Soviet Math. Doklady},
 year = 1991,
 volume = 42,
 number = 1,
 pages = {5762}
 }
,  "Iterative methods for solving equations with highly varying coefficients", Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, 1991, pp. 197205.BibTeX
 @Inproceedings{Bakhvalov1991d,
 author = {Bakhvalov, N.S. and Knyazev, A.V. and Kobel'kov, G.M.},
 title = {Iterative methods for solving equations with highly varying coefficients},
 booktitle = {Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations},
 year = 1991,
 pages = {197205},
 publisher = {SIAM}
 }
,  "A preconditioned conjugate gradient method for eigenvalue problems and its implementation in a subspace", 1991, Numerical treatment of eigenvalue problems, Vol. 5 (Oberwolfach, 1990).BibTeX
 @Inbook{Knyazev1991,
 author = {Knyazev, A. V.},
 title = {A preconditioned conjugate gradient method for eigenvalue problems and its implementation in a subspace},
 booktitle = {Numerical treatment of eigenvalue problems, Vol. 5 (Oberwolfach, 1990)},
 year = 1991,
 volume = 96,
 series = {Internat. Ser. Numer. Math.},
 pages = {143154},
 address = {Basel},
 publisher = {Birkhauser}
 }
,  "On Exact Estimates Of The Convergence Rate Of The Steepest Ascent Method In The Symmetrical Eigenvalue Problem", Linear Algebra and Its Applications, Vol. 154, pp. 245257, 1991.BibTeX
 @Article{Knyazev1991a,
 author = {Knyazev, A. V. and Skorokhodov, A. L.},
 title = {On Exact Estimates Of The Convergence Rate Of The Steepest Ascent Method In The Symmetrical Eigenvalue Problem},
 journal = {Linear Algebra and Its Applications},
 year = 1991,
 volume = 154,
 pages = {245257},
 note = {Times Cited: 3 KNYAZEV, AV SKOROKHODOV, AL}
 }
,  "A New Iterative Algorithm For Solution Of Problems Emerging In The Method Of Fictious Fluxes For Ellipticequations", Doklady Akademii Nauk Sssr, Vol. 312, No. 4, pp. 783786, 1990.BibTeX
 @Article{Bakhvalov1990,
 author = {Bakhvalov, N. S. and Kniazev, A. V.},
 title = {A New Iterative Algorithm For Solution Of Problems Emerging In The Method Of Fictious Fluxes For Ellipticequations},
 journal = {Doklady Akademii Nauk Sssr},
 year = 1990,
 volume = 312,
 number = 4,
 pages = {783786},
 note = {Times Cited: 1 BAKHVALOV, NS KNIAZEV, AV}
 }
,  "Effective Computation Of Averaged Periodic Structure Composites Consisting Of Essentially Different Components", Doklady Akademii Nauk Sssr, Vol. 313, No. 4, pp. 777781, 1990.BibTeX
 @Article{Bakhvalov1990a,
 author = {Bakhvalov, N. S. and Kniazev, A. V.},
 title = {Effective Computation Of Averaged Periodic Structure Composites Consisting Of Essentially Different Components},
 journal = {Doklady Akademii Nauk Sssr},
 year = 1990,
 volume = 313,
 number = 4,
 pages = {777781},
 note = {Times Cited: 3 BAKHVALOV, NS KNIAZEV, AV}
 }
,  "A new iterative algorithm for solving problems of the fictitious flow method for elliptic equations", Soviet Math. Doklady, Vol. 41, No. 3, pp. 481485, 1990.BibTeX
 @Article{Bakhvalov1990b,
 author = {Bakhvalov, N.S. and Knyazev, A.V.},
 title = {A new iterative algorithm for solving problems of the fictitious flow method for elliptic equations},
 journal = {Soviet Math. Doklady},
 year = 1990,
 volume = 41,
 number = 3,
 pages = {481485}
 }
,  "A new iterative algorithm for solving the fictitious fluxes method problems for elliptic equations", Equadiff 7: 7th Czechoslovak Conference on Differential Equations and Their Applications, Kurzweil, Jaroslav, Eds., 1990, vol. 118, pp. 225227.BibTeX
 @Inproceedings{Bakhvalov1990c,
 author = {Bakhvalov, N.S. and Knyazev, A.V.},
 title = {A new iterative algorithm for solving the fictitious fluxes method problems for elliptic equations},
 booktitle = {Equadiff 7: 7th Czechoslovak Conference on Differential Equations and Their Applications},
 year = 1990,
 editor = {Kurzweil, Jaroslav},
 volume = 118,
 pages = {225227},
 publisher = {BSB BG Teubner Verlagsgesellschaft}
 }
,  "The TempleLehmanntype methods in iterative algorithms", 1990.BibTeX
 @Misc{Knyazev1990,
 author = {Knyazev, A.V. and Lebedev, V.I. and Skorokhodov, A.L.},
 title = {The TempleLehmanntype methods in iterative algorithms},
 year = 1990,
 volume = 5,
 number = {4/5},
 pages = {265273}
 }
,  "Preconditioned Iterative Methods In Subspace For Solving Linearsystems With Indefinite Coefficient Matrices And Eigenvalue Problems", Soviet Journal of Numerical Analysis and Mathematical Modelling, Vol. 4, No. 4, pp. 283310, 1989.BibTeX
 @Article{KNYAZEV1989,
 author = {KNYAZEV, AV and SKOROKHODOV, AL},
 title = {Preconditioned Iterative Methods In Subspace For Solving Linearsystems With Indefinite Coefficient Matrices And Eigenvalue Problems},
 journal = {Soviet Journal of Numerical Analysis and Mathematical Modelling},
 year = 1989,
 volume = 4,
 number = 4,
 pages = {283310}
 }
,  Matrichnyi analiz, ``Mir'', 1989.BibTeX
 @Book{Khorn1989,
 author = {Khorn, R. and Dzhonson, Ch.},
 title = {Matrichnyi analiz},
 year = 1989,
 pages = 656,
 address = {Moscow},
 publisher = {``Mir''},
 note = {Translated from the English by Kh. D. Ikramov, A. V. Knyazev and E. E. Tyrtyshnikov; Translation edited and with a foreword by Ikramov}
 }
,  Bezoshibochnye vychisleniya, ``Mir'', 1988.BibTeX
 @Book{Gregori1988,
 author = {Gregori, R. and Krishnamurti, E.},
 title = {Bezoshibochnye vychisleniya},
 year = 1988,
 pages = 208,
 address = {Moscow},
 publisher = {``Mir''},
 note = {Metody i prilozheniya. [Methods and applications]; Translated from the English by Kh. D. Ikramov and A. V. Knyazev; Translation edited and with a foreword by Ikramov}
 }
,  Metody TemplyaLimanna v iteratsionnykh algoritmakh, vol. 181, Akad. Nauk SSSR Otdel Vychisl. Mat., 1988.BibTeX
 @Book{Knyazev1988,
 author = {Knyazev, A. V. and Lebedev, V. I. and Skorokhodov, A. L.},
 title = {Metody TemplyaLimanna v iteratsionnykh algoritmakh},
 year = 1988,
 volume = 181,
 pages = 16,
 address = {Moscow},
 publisher = {Akad. Nauk SSSR Otdel Vychisl. Mat.}
 }
,  Modifitsirovannye iteratsionnye metody v podprostranstve dlya resheniya sistem uravnenii so znakoneopredelennymi matritsami i spektralnykh zadach, vol. 205, Akad. Nauk SSSR Otdel Vychisl. Mat., 1988.BibTeX
 @Book{Knyazev1988a,
 author = {Knyazev, A. V. and Skorokhodov, A. L.},
 title = {Modifitsirovannye iteratsionnye metody v podprostranstve dlya resheniya sistem uravnenii so znakoneopredelennymi matritsami i spektralnykh zadach},
 year = 1988,
 volume = 205,
 pages = 48,
 address = {Moscow},
 publisher = {Akad. Nauk SSSR Otdel Vychisl. Mat.}
 }
,  "The Rate Of Convergence Of The Method Of Steepest Descent In A Euclidean Norm", Ussr Computational Mathematics and Mathematical Physics, Vol. 28, No. 5, pp. 195196, 1988.BibTeX
 @Article{Knyazev1988b,
 author = {Knyazev, A. V. and Skorokhodov, A. L.},
 title = {The Rate Of Convergence Of The Method Of Steepest Descent In A Euclidean Norm},
 journal = {Ussr Computational Mathematics and Mathematical Physics},
 year = 1988,
 volume = 28,
 number = 5,
 pages = {195196},
 note = {Times Cited: 0 KNYAZEV, AV SKOROKHODOV, AL}
 }
,  "The rate of convergence of the method of steepest descent in the Euclidean norm", Zh. Vychisl. Mat. i Mat. Fiz., Vol. 28, No. 10, pp. 15771578, 1599, 1988.BibTeX
 @Article{Knyazev1988c,
 author = {Knyazev, A. V. and Skorokhodov, A. L.},
 title = {The rate of convergence of the method of steepest descent in the Euclidean norm},
 journal = {Zh. Vychisl. Mat. i Mat. Fiz.},
 year = 1988,
 volume = 28,
 number = 10,
 pages = {15771578, 1599}
 }
,  "A parallel algorithm for iterations of a subspace and its implementation on a computer with ring architecture", 1987, Numerical methods and mathematical modeling (Russian).BibTeX
 @Inbook{Knyazev1987,
 author = {Knyazev, A. V.},
 title = {A parallel algorithm for iterations of a subspace and its implementation on a computer with ring architecture},
 booktitle = {Numerical methods and mathematical modeling (Russian)},
 year = 1987,
 pages = {115123},
 address = {Moscow},
 publisher = {Akad. Nauk SSSR Otdel Vychisl. Mat.}
 }
,  "Methods for the derivation of some estimates in a symmetric eigenvalue problem", 1987, Computational processes and systems, No. 5 (Russian).BibTeX
 @Inbook{Knyazev1987a,
 author = {Knyazev, A. V.},
 title = {Methods for the derivation of some estimates in a symmetric eigenvalue problem},
 booktitle = {Computational processes and systems, No. 5 (Russian)},
 year = 1987,
 pages = {164173},
 address = {Moscow},
 publisher = {``Nauka''}
 }
,  "Convergence rate estimates for iterative methods for a mesh symmetrie eigenvalue problem", Russian Journal of Numerical Analysis and Mathematical Modelling, Vol. 2, No. 5, pp. 371396, 1987.BibTeX
 @Article{Knyazev1987b,
 author = {Knyazev, A.V.},
 title = {Convergence rate estimates for iterative methods for a mesh symmetrie eigenvalue problem},
 journal = {Russian Journal of Numerical Analysis and Mathematical Modelling},
 year = 1987,
 volume = 2,
 number = 5,
 pages = {371396}
 }
,  "On modified gradient methods for eigenvalue problems", Differentsial'nye Uravneniya, No. 4, pp. 715717, 1987.BibTeX
 @Article{Knyazev1987c,
 author = {Knyazev, A.V.},
 title = {On modified gradient methods for eigenvalue problems},
 journal = {Differentsial'nye Uravneniya},
 year = 1987,
 number = 4,
 pages = {715717}
 }
,  "BIPR5M software code", VANT, ser. Physics and Technology of Nuclear Reactors, No. 5, pp. 3435, 1986.BibTeX
 @Article{Apokorina1986,
 author = {Apokorina, V.S. and Knyazev, A.V.},
 title = {BIPR5M software code},
 journal = {VANT, ser. Physics and Technology of Nuclear Reactors},
 year = 1986,
 number = 5,
 pages = {3435}
 }
,  Computation of eigenvalues and eigenvectors for mesh problems: algorithms and error estimates (In Russian) Vychislenie sobstvennykh znachenii i vektorov v setochnykh zadachakh: algoritmy i otsenki pogreshnosti, Akad. Nauk SSSR Otdel Vychisl. Mat., 1986.BibTeX
 @Book{Knyazev1986,
 author = {Knyazev, A. V.},
 title = {Computation of eigenvalues and eigenvectors for mesh problems: algorithms and error estimates (In Russian) Vychislenie sobstvennykh znachenii i vektorov v setochnykh zadachakh: algoritmy i otsenki pogreshnosti},
 year = 1986,
 pages = 188,
 address = {Moscow},
 publisher = {Akad. Nauk SSSR Otdel Vychisl. Mat.}
 }
,  "An error estimate for the approximation by the RayleighRitz method to the invariant subspace that corresponds to nearby eigenvalues", 1985, Computer architecture and numerical methods (Russian).BibTeX
 @Inbook{Knyazev1985,
 author = {Knyazev, A. V.},
 title = {An error estimate for the approximation by the RayleighRitz method to the invariant subspace that corresponds to nearby eigenvalues},
 booktitle = {Computer architecture and numerical methods (Russian)},
 year = 1985,
 pages = {1618},
 address = {Moscow},
 publisher = {Akad. Nauk SSSR Otdel Vychisl. Mat.}
 }
,  "Modified gradient methods and their block analogues for the problems", 1985, Computer architecture and numerical methods (Russian).BibTeX
 @Inbook{Knyazev1985a,
 author = {Knyazev, A. V.},
 title = {Modified gradient methods and their block analogues for the problems},
 booktitle = {Computer architecture and numerical methods (Russian)},
 year = 1985,
 pages = {1932},
 address = {Moscow},
 publisher = {Akad. Nauk SSSR Otdel Vychisl. Mat.}
 }
,  "Sharp a priori error estimates for the RayleighRitz method with no assumptions on fixed sign or compactness", Mat. Zametki, Vol. 38, No. 6, pp. 900907, 958, 1985.BibTeX
 @Article{Knyazev1985b,
 author = {Knyazev, A. V.},
 title = {Sharp a priori error estimates for the RayleighRitz method with no assumptions on fixed sign or compactness},
 journal = {Mat. Zametki},
 year = 1985,
 volume = 38,
 number = 6,
 pages = {900907, 958}
 }
,  "Sharp a priori error estimates of inequality type in the RayleighRitz method for symmetric spectral problems without assumptions on signdefiniteness or compactness", 1985, Methods of numerical and applied mathematics (Russian).BibTeX
 @Inbook{Knyazev1985c,
 author = {Knyazev, A. V.},
 title = {Sharp a priori error estimates of inequality type in the RayleighRitz method for symmetric spectral problems without assumptions on signdefiniteness or compactness},
 booktitle = {Methods of numerical and applied mathematics (Russian)},
 year = 1985,
 pages = {7488},
 address = {Moscow},
 publisher = {Akad. Nauk SSSR Vychisl. Tsentr}
 }
,  "Sharp A Priori Errorestimates Of The Rayleighritz Method Without Assumptions Of Fixed Sign Or Compactness", Mathematical Notes, Vol. 38, No. 56, pp. 9981002, 1985.BibTeX
 @Article{Knyazev1985d,
 author = {Knyazev, A. V.},
 title = {Sharp A Priori Errorestimates Of The Rayleighritz Method Without Assumptions Of Fixed Sign Or Compactness},
 journal = {Mathematical Notes},
 year = 1985,
 volume = 38,
 number = {56},
 pages = {9981002},
 note = {Times Cited: 1 KNYAZEV, AV}
 }
,  "The KatoTemple block algorithm for the estimation of eigenvalues: justification and application", 1985, Conjugate equations and perturbation theory in problems of mathematical physics (Russian).BibTeX
 @Inbook{Knyazev1985e,
 author = {Knyazev, A. V.},
 title = {The KatoTemple block algorithm for the estimation of eigenvalues: justification and application},
 booktitle = {Conjugate equations and perturbation theory in problems of mathematical physics (Russian)},
 year = 1985,
 pages = {127135},
 address = {Moscow},
 publisher = {Akad. Nauk SSSR Otdel Vychisl. Mat.}
 }
,  Parallel Algorithms for Solving Some Stationary Problems of Mathematical Physics, Parallelnye algoritmy resheniya nekotorykh statsionarnykh zadach matematicheskoi fiziki (In Russian) With a preface by G. I. Marchuk, Akad. Nauk SSSR Otdel Vychisl. Mat., 1984.BibTeX
 @Book{Lebedev1984,
 author = {Lebedev, V. I. and Bakhvalov, N. S. and Agoshkov, V. I. and Baburin, O. V. and Knyazev, A. V. and Shutyaev, V. P.},
 title = {Parallel Algorithms for Solving Some Stationary Problems of Mathematical Physics, Parallelnye algoritmy resheniya nekotorykh statsionarnykh zadach matematicheskoi fiziki (In Russian) With a preface by G. I. Marchuk},
 year = 1984,
 pages = 143,
 address = {Moscow},
 publisher = {Akad. Nauk SSSR Otdel Vychisl. Mat.}
 }
,  O nekotorykh dvukhstupenchatykh metodakh nakhozhdeniya granits spektra lineinogo puchka matrits, vol. 3749/16, Inst. Atomn. Energii, 1983.BibTeX
 @Book{Knyazev1983,
 author = {Knyazev, A. V.},
 title = {O nekotorykh dvukhstupenchatykh metodakh nakhozhdeniya granits spektra lineinogo puchka matrits},
 year = 1983,
 volume = {3749/16},
 pages = 29,
 address = {Moscow},
 publisher = {Inst. Atomn. Energii}
 }
,  "Estimates for convergence and analysis of the optimality of iterative methods for the simultaneous computation of several eigenvectors", 1983, Computational methods in linear algebra (Russian) (Moscow, 1982).BibTeX
 @Inbook{Knyazev1983a,
 author = {Knyazev, A. V. and Lebedev, V. I.},
 title = {Estimates for convergence and analysis of the optimality of iterative methods for the simultaneous computation of several eigenvectors},
 booktitle = {Computational methods in linear algebra (Russian) (Moscow, 1982)},
 year = 1983,
 pages = {94114},
 address = {Moscow},
 publisher = {Akad. Nauk SSSR Otdel Vychisl. Mat.}
 }
,  "A group iteration method for finding the lowest eigenvalues", Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet., No. 2, pp. 2934, 81, 1982.BibTeX
 @Article{Dyakonov1982,
 author = {Dyakonov, E. G. and Knyazev, A. V.},
 title = {A group iteration method for finding the lowest eigenvalues},
 journal = {Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet.},
 year = 1982,
 number = 2,
 pages = {2934, 81}
 }
,  "Group iterative method for finding lowerorder eigenvalues", Moscow Univ., Ser. 15, Comput. Math. Cyber., No. 2, pp. 3240, 1982.BibTeX
 @Article{Dfyakonov1982,
 author = {Dyakonov, E.G. and Knyazev, A.V.},
 title = {Group iterative method for finding lowerorder eigenvalues},
 journal = {Moscow Univ., Ser. 15, Comput. Math. Cyber.},
 year = 1982,
 number = 2,
 pages = {3240}
 }
,  Ob odnom iteratsionnom metode nakhozhdeniya mladshikh sobstvennykh chisel, vol. 26, Akad. Nauk SSSR Otdel Vychisl. Mat., 1981.BibTeX
 @Book{Dyakonov1981,
 author = {Dyakonov, E. G. and Knyazev, A. V.},
 title = {Ob odnom iteratsionnom metode nakhozhdeniya mladshikh sobstvennykh chisel},
 year = 1981,
 volume = 26,
 pages = 19,
 address = {Moscow},
 publisher = {Akad. Nauk SSSR Otdel Vychisl. Mat.}
 }
,
 "Absolute Value Preconditioning for Symmetric Indefinite Linear Systems", SIAM Journal on Scientific Computing, Vol. 35, No. 2, pp. A696A718, 2013.