A preconditioner (matrix) is a matrix which multiplies (from left or from right, thus left and right preconditioners) the matrix of coefficients of a system which discretizes a differential equation in order to get faster convergence (=smaller condition number) of iterative numerical solution techniques. As a technique, preconditioning is used also for optimization algorithms.

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John W. Pearson, Martin Stoll, Andy Wathen, Regularization-robust preconditioners for time-dependent PDE constrained optimization problems, SIAM J. Matrix Anal. and Appl., 33(4), 1126–1152, 2012 doi

Michele Benzi, Preconditioning techniques for large linear systems: a survey, J. of Computational Physics 182, 418–477 (2002) doipdf