His major contributions include preconditioned iterative methods, in particular the ICCG (incomplete
Cholesky conjugate gradient) method (developed together with Koos Meijerink), a version of preconditioned conjugate gradient method,[8][9]
the BiCGSTAB[2] and (together with Kees Vuik) GMRESR[10]Krylov subspace methods and (together with Gerard Sleijpen) the Jacobi-Davidson method[11]
for solving ordinary, generalized, and nonlineareigenproblems.
He has analyzed convergence behavior of the conjugate gradient[12] and Lanczos methods. He has also developed a number of preconditioners for parallel computers,[13] including truncated Neumann series preconditioner, incomplete twisted factorizations, and the incomplete factorization based on the so-called "vdv" ordering.
He is the author of the book Iterative Krylov Methods for Large Linear systems[14]
and one of the authors of the Templates projects for linear problems[15]
and eigenproblems.[16]
^ ab
H.A. van der Vorst (1992), "Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems", SIAM J. Sci. Stat. Comput., 13 (2): 631–644, doi:10.1137/0913035, hdl:10338.dmlcz/104566
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J.A. Meijerink; H.A.van der Vorst (1977), "An Iterative Solution Method for Linear Systems of Which the Coefficient Matrix is a Symmetric M-Matrix", Math. Comp., 31 (137): 148–162, doi:10.2307/2005786, JSTOR2005786
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H.A. van der Vorst (1981), "Iterative solution methods for certain sparse linear systems with a non-symmetric matrix arising from PDE-problems", J. Comput. Phys., 44 (1): 1–19, Bibcode:1981JCoPh..44....1V, doi:10.1016/0021-9991(81)90034-6
^
G.L.G. Sleijpen; H.A. van der Vorst (1996), "A Jacobi-Davidson iteration method for linear eigenvalue problems", SIAM J. Matrix Anal. Appl., 17 (2): 401–425, CiteSeerX10.1.1.50.2569, doi:10.1137/S0895479894270427
