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
