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Given , it has been observed empirically (see Gropp and Keyes ) that there often exists an optimal value of which minimizes the total computational time for solving the problem. A small provides a better, but more expensive, coarse grid approximation, and requires solving more, but smaller, subdomain solves. A large has the opposite effect. For model problems, the optimal can be determined for both sequential and parallel implementations (see Chan and Shao ). In practice, it may pay to determine a near optimal value of empirically if the preconditioner is to be re-used many times. However, there may also be geometric constraints on the range of values that can take.