Fast Eigenvalue Solutions

Transcription

1 Fast Eigenvalue Solutions Techniques! Steepest Descent/Conjugate Gradient! Davidson/Lanczos! Carr-Parrinello PDF Files will be available

2 Where HF/DFT calculations spend time Guess ρ Form H Diagonalize ρ O(N 4 ) Cutoffs O(N 2 ) O(N 3 ) Difficult to reduce: Lanczos or Conjugate Gradient Currently only important if N > 2000 Yes Did ρ Change? No Done

3 Rayleigh-Ritz Variational Principle Can always map a minimization problem into a diagonalization problem, and vice-versa λ 1 = min u, Au u u, u This isn t surprising to QMers, since we normally interchange the two

4 Iterative Eigenvector Optimization Ron already covered this Eigenvalue problem Hx =Ex (H-D)x = (EI-D)x x = (EI-D) -1 (H-D)x Define an iterative approach x (n+1) = (E (n) I-D) -1 (H-D)x (n) E (n) = x (n) Hx (n) /x (n) x (n) Factor out the correction x (n+1) = x (n) + d (n) d (n) = (E (n) I-D) -1 (H-E (n) I)x (n) D is the diagonal of H

5 The Residual Vector d The residual vector d (n) = (E (n) I-D) -1 (H-E (n) I)x (n)! Represents the direction of greatest change for the eigenvector x.! Known as the steepest descent vector (although this term is also used for (H-E (n) I)x (n) We have a direction, but we don t know how far to move.! Use Steepest Descent or Conjugate Gradient optimization

6 Tends to be slow, because you overshoot the minimum in each direction. Steepest descent optimization 1. Find d1 2. Minimize in 1-d along d1 3. Find d2 4. Minimize in 1-d along d2 5. Etc.

7 Conjugate Gradient optimization 1. Find d1 2. Minimize in 1-d along d1 CG is normally must faster than SD 3. Find conjugate direction d2: d2 d1 = 0 4. Minimize along d2 5. Etc.

8 One-dimensional optimization Given a direction d (SD or CG), how do we optimize?! x (n+1) = x (n) cos(θ) + d sin(θ)! Finding optimal theta normally only requires a few points.! Trig functions preserve normality

9 Orbital/band at a time optimization Trial eigenvector for band Calculate steepest descent vector Orthogonalize to all bands Iterate until converged Determine conjugate direction Orthogonalize to present band and normalize Calculate KS Energy for θ Compute optimum θ Compute new eigenvector

10 Davidson Diagonalization Optimize eigenvectors in a guess subspace Also uses residual vector d (n) = (E (n) I-D) -1 (H-E (n) I)x (n)! Rather than CG or SD, use d to augment subspace! Lanczos technique Derived for Configuration Interactions! N ~ 10,000! K = 1 Lanczos methods can be unstable! Davidson normally more stable, because of the subspace projection.

11 Subspace Optimization Rather than solve a problem in a large space (N ~ 10,000) project onto a smaller space (n ~ 50).! Small space has to contain dominant features of large space Example: Water with 1M basis functions! Only have 5 occupied states " 1s orbital on O " 2 O lone pairs " 2 OH bonds! Project onto a space that spans these five orbitals! Add onto this space until converged

12 Davidson #1: Subspace Transform 1. Choose an initial set of guess vectors {b i, i=1,ng} 2. Transform H into subspace spanned by {b i } G = B T HB 3. Compute eigenvectors v and values λ of G Dense technique (ng is small) If B fully spans the desired eigenvector, one of the eigenvalues λ k will be the exact one. Otherwise, we can iteratively optimize B to make it better. How do we do this? Using the idea of the residual vector d that we saw earlier.

13 Davidson #2: Optimizing the subspace 4. Compute the residual vector d k as before: d = (λ k I-D) -1 (G- λ k I)v k 5. Orthogonalize d to all of the {b i }; normalize d. 6. Set b ng+1 = d 7. Form the additional row and column of G G ng+1,i = B T Hb ng+1,i 8. Diagonalize G, as before. 9. If not converged, go to #4.

14 Davidson Timings

15 Timing Caveats These timings represent a best-case scenario!!! 1 eigenvalue, lots of functions! Your mileage may vary!!!

16 Higher roots with Davidson If we want more than one eigenvalue/vector, the guess vectors {b} converged after computing one are normally a good guess for another. Start with the lowest one, compute to highest. Not a good method to compute all eigenvalues/vectors! End up solving the full problem multiple times!

17 Compare/Contrast CG & Davidson Both use a subspace transform! We normally have the previous iteration s eigenvectors, and we may as well use them Both use the residual vector d! CG uses d to compute the conjugate direction! Dav uses d to augment the subspace Conventional wisdom! Davidson is faster! CG is more stable! Many more people use CG than Davidson

18 Caveats 1. Use a dense technique when n < 500 always. dsyevx from LAPACK is very hard to beat! Dense techniques are much more stable than sparse. 2. Keep an eye on nk/n nk is the number of desired roots, N is the size of the matrix. If nk/n > 0.25, probably best to use dense technique. 3. Precondition! Multiplying by the last iteration s eigenvectors is a good form of precondintioning.

19 Pure Lanczos Techniques (Zunger, 94) 1. Start with a random u 1 2. Solve β i+1 u i+1 = Hu i α i u i β i u i-1 α i = <u i H u i > β i determined by normalization condition of u i 3. Diagonalize tridiagonal matrix M M M α β β β β α β β α O

20 Lanczos #2 4. Check to see how many eigenvalues are converged; add vectors u i until the requisite number of states are converged. 5. The k th eigenvector is given by V k = Σ i b ki u k

21 Pure Lanczos notes Lanczos has a reputation of being unstable! Can also produce duplicate eigenvalues/vectors " Must project out! Can become non-orthogonal! Numerical errors often grow during iterations

22 Car-Parrinello Propagator Technique Different approach! Rather than minimizing the problem, propagate solutions Born-Oppenheimer approximation! Nuclei move much faster than electrons! Fix nuclei, minimize electrons Car-Parrinello technique does not solve for the Born- Oppenheimer surface! Propagated electrons can be shown to lie reasonably close to BO surface

23 Scheme for QM-MD 1. Fix the nuclei, solve for the wave function 2. Compute the forces on the nuclei due to the wave function 3. Propagate the nuclei using Verlet-type scheme 4. Go to #1

24 Car-Parrinello Start from observation that diagonalization and energy minimization are interchangeable Rather than minimizing the electron, propagate it in the same way one propagates the nuclear motions! Electrons must have fictitious masses! Treat as dynamic variables! Much faster than diagonalization Can also dynamically damp the electron to obtain Born-Oppenheimer solution

25 References! Iterative minimization techniques for ab initio total energy calculations. Payne, Teter, Allen, Arias, Joannopoulos. RMP 64, 1045 (1992).! Iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. Davidson. J. Comp. Phys. 17, 87 (1975).! Asymptotic convergence for iterative optimization in electronic structure. Lippert, Sears. PRB 61, (2000).! Large scale electronic structure calculations using the Lanczos method. Wang, Zunger. Comp. Mat. Sci 2, 326 (1994).! Unified approach for MD and DFT Car, Parrinello. PRL 55, 2471 (1985).