Computational linear algebra – the Lanczos and Arnoldi methods, the extended Krylov subspace method, the rational Krylov subspace method; numerical solution of PDEs by means of the Spectral Lanczos/Arnoldi Decomposition Methods and other linear algebra methods;
the theory of rational approximation and its applications to constructing optimal finite difference grids for solution of PDEs;
inverse spectral problems;
numerical solution of ill-posed geophysical problems with the use of variational regularization.
Main publications:
V. L. Druskin, L. A. Knizhnerman, “Error estimates for the simple Lanczos process when computing functions of symmetric matrices and eigenvalues”, J. Comput. Math. and Mathem. Phys., 31:7 (1991), 970–983
L. A. Knizhnerman, “Computation of functions of unsymmetric matrices by means of the Arnoldi method”, J. Comput. Math. and Mathem. Phys., 31:1 (1991), 5–16
V. Druskin and L. Knizhnerman, “Extended Krylov subspaces: approximation of the matrix square root and related functions”, SIAM J. Matrix Anal. Appl., 19:3 (1998), 755–771
V. Druskin and L. Knizhnerman, “Gaussian spectral rules for second order finite-difference schemes”, Numer. Algorithms, 25:1-4 (2000), 139–159
V. Druskin, L. Knizhnerman and M. Zaslavsky, “Solution of large scale evolutionary problems using rational Krylov subspaces with optimized shifts”, SIAM J. Sci. Comp., 31:5 (2009), 3760–3780
M. A. Botchev, L. A. Knizhnerman, “Adaptive residual-time restarting for Krylov subspace matrix exponential evaluations”, Keldysh Institute preprints, 2019, 127, 28 pp.
2009
2.
L. A. Knizhnerman, “Padé–Faber Approximation of Markov Functions on Real-Symmetric Compact Sets”, Mat. Zametki, 86:1 (2009), 81–94; Math. Notes, 86:1 (2009), 81–92
L. A. Knizhnerman, “Gauss–Arnoldi quadrature for $\bigl\langle(zI-A)^{-1}\varphi,\varphi\bigr\rangle$ and rational Padé-type approximation for Markov-type functions”, Mat. Sb., 199:2 (2008), 27–48; Sb. Math., 199:2 (2008), 185–206
A. Greenbaum, V. L. Druskin, L. A. Knizhnerman, “On solving indefinite symmetric linear systems by means of the Lanczos method”, Zh. Vychisl. Mat. Mat. Fiz., 39:3 (1999), 371–377; Comput. Math. Math. Phys., 39:3 (1999), 350–356
L. A. Knizhnerman, “The simple Lanczos procedure: Estimates of the error of the Gauss quadrature formula and their applications”, Zh. Vychisl. Mat. Mat. Fiz., 36:11 (1996), 5–19; Comput. Math. Math. Phys., 36:11 (1996), 1481–1492
L. A. Knizhnerman, “The quality of approximations to a well-isolated eigenvalue, and the arrangement of “Ritz numbers” in a simple Lanczos process”, Zh. Vychisl. Mat. Mat. Fiz., 35:10 (1995), 1459–1475; Comput. Math. Math. Phys., 35:10 (1995), 1175–1187
L. A. Knizhnerman, “Error bounds in Arnoldi's method: The case of a normal matrix”, Zh. Vychisl. Mat. Mat. Fiz., 32:9 (1992), 1347–1360; Comput. Math. Math. Phys., 32:9 (1992), 1199–1211
V. L. Druskin, L. A. Knizhnerman, “Error bounds in the simple Lanczos procedure for computing functions of
symmetric matrices and eigenvalues”, Zh. Vychisl. Mat. Mat. Fiz., 31:7 (1991), 970–983; U.S.S.R. Comput. Math. Math. Phys., 31:7 (1991), 20–30
L. A. Knizhnerman, V. Z. Sokolinskii, “Some estimates of rational trigonometric sums and sums of Legendre symbols”, Uspekhi Mat. Nauk, 34:3(207) (1979), 199–200; Russian Math. Surveys, 34:3 (1979), 203–204
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