DIIS (direct inversion in the iterative subspace or direct inversion of the iterative subspace), also known as Pulay mixing, is a technique for
extrapolating
In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between know ...
the solution to a set of linear equations by directly minimizing an error residual (e.g. a
Newton–Raphson
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-v ...
step size) with respect to a linear combination of known sample vectors. DIIS was developed by
Peter Pulay in the field of computational
quantum chemistry
Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contribution ...
with the intent to accelerate and stabilize the
convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
* "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that united the four Weir ...
of the
Hartree–Fock self-consistent field method.
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
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the r ...
sense, the
null vector
In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which .
In the theory of real bilinear forms, definite quadratic forms a ...
. The newly determined coefficients are then used to extrapolate the function variable for the next iteration.
Details
At each iteration, an approximate error vector, , corresponding to the variable value, is determined. After sufficient iterations, a linear combination of previous error vectors is constructed
:
The DIIS method seeks to minimize the norm of under the constraint that the coefficients sum to one. The reason why the coefficients must sum to one can be seen if we write the trial vector as the sum of the exact solution () and an error vector. In the DIIS approximation, we get:
:
We minimize the second term while it is clear that the sum coefficients must be equal to one if we want to find the exact solution.
The minimization is done by a
Lagrange multiplier
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied ...
technique. Introducing an undetermined multiplier , a Lagrangian is constructed as
:
Equating zero to the derivatives of with respect to the coefficients and the multiplier leads to a system of
linear equation
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...
s to be solved for the coefficients (and the Lagrange multiplier).
:
Moving the minus sign to , results in an equivalent symmetric problem.
:
The coefficients are then used to update the variable as
:
Citations
References
*
* {{Cite journal, doi=10.1007/s10910-011-9863-y, title=An analysis for the DIIS acceleration method used in quantum chemistry calculations, year=2011, last1=Rohwedder, first1=Thorsten, last2=Schneider, first2=Reinhold, journal=Journal of Mathematical Chemistry, volume=49, issue=9, pages=1889, citeseerx=10.1.1.461.1285, s2cid=51759476
See also
*
GMRES In mathematics, the generalized minimal residual method (GMRES) is an iterative method for the numerical solution of an indefinite nonsymmetric system of linear equations. The method approximates the solution by the vector in a Krylov subspace wi ...
External links
The Mathematics of DIIS
Quantum chemistry
Computational chemistry
Numerical linear algebra