From 1993–1994, Knyazev held a visiting position at the Courant Institute of Mathematical Sciences of New York University, collaborating with Olof B.

[1] From 1994 until retirement in 2014, he was a Professor of Mathematics at the University of Colorado Denver, supported by the National Science Foundation[2] and United States Department of Energy grants.

Knyazev is mostly known for his work in numerical solution of large sparse eigenvalue problems, particularly preconditioning[10] and the iterative method LOBPCG.

[12] Knyazev collaborated with John Osborn [13] on the theory of the Ritz method in the finite element method context and with Nikolai Sergeevich Bakhvalov (Russian: Николай Серге́евич Бахвалов) (Erdős number 3 via Leonid Kantorovich) on numerical solution of elliptic partial differential equations with large jumps in the main coefficients.

[14] Jointly with his Ph.D. students, Knyazev pioneered using majorization for bounds in the Rayleigh–Ritz method (see[15] and references there) and contributed to the theory of angles between flats.