DIIS (direct inversion in the iterative subspace or direct inversion of the iterative subspace), also known as Pulay mixing, is an extrapolation technique. DIIS was developed by Peter Pulay in the field of computational quantum chemistry with the intent to accelerate and stabilize the convergence of the Hartree–Fock self consistent field method.^[1]

At a given iteration, the approach constructs a linear combination of approximate error vectors from previous iterations. The coefficients of the linear combination are determined so to best approximate, in a least squares sense, the null vector. The newly determined coefficients are then used to extrapolate the function variable for the next iteration.

[edit] Details

At each iteration, an approximate error vector, e_i, corresponding to the variable value, p_i is determined. After sufficient iterations, a linear combination of m previous error vectors is constructed

The DIIS method seeks to minimize the norm of e_m+1 under the constraint that the coefficients sum to one. This is done by a Lagrange multiplier technique. Introducing an undetermined multiplier λ, a Lagrangian is constructed as

Equating the derivatives of L, with respect to the coefficients and the multiplier, equal to zero, leads to m + 1 linear equations to be solved for the m coefficients. The coefficients are then used to update the function variable as

[edit] References

[edit] External links

• The Mathematics of DIIS

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