T SCFM AN E -B Aes12.wfu.edu/pdfarchive/poster/ES12_B_Gavin.pdfANONLINEAR EIGENSOLVER-BASED...
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AN ONLINEAR E IGENSOLVER -B ASED A LTERNATIVE TO T RADITIONAL SCF M ETHODS B RENDAN G AVIN AND E RIC P OLIZZI U NIVERSITY OF M ASSACHUSETTS ,A MHERST [email protected], [email protected] P ROBLEM S OLVING THE K OHN -S HAM E QUATIONS H [n]x i = λ i Sx i , n = X i |x i | 2 Where H[n] is the typical Kohn-Sham Hamiltonian with (for our work here) spinless LDA approxima- tion, x i , λ i are eigenvector/eigenvalue pairs, n is the electron density, and S is the mass matrix re- sulting from a finite elements discretization. T RADITIONAL SCF The Kohn-Sham equations are traditionally solved using self-consistent field methods that follow the general procedure: 1. Find good initial guess n 1 (e.g. with Thomas- Fermi model) 2. Get n 0 = f (n i ), where f (n i ) is the procedure consisting of: (a) Forming the Kohn-Sham hamiltonian H[n i ] by solving poisson’s equation, cal- culating the exchange-correlation opera- tor, etc. (b) Solving the eigenvalue problem H [n i ]x i = λ i Sx i (c) Calculating the new density from the oc- cupied states x i 3. Form n i+1 from some function of n 0 and n i . If kn i+1 - f (n i+1 )k < then stop; otherwise GOTO step 2. S IMPLE M IXING : n i+1 = αn i + (1 - α)n 0 , 0 ≤ α ≤ 1 The value of α is determined heuristically. Typical α =0.1. Convergence is usually very slow. DIIS : A list of m densities is first produced using some other procedure (e.g. simple mixing), and then subsequent n i+1 are given by[2]: n i+1 = m X j =1 c j n i-j + c j r i-j , r i = n i - f (n i ) where c j are the coefficients that minimize k ∑ c j r j k such that ∑ c j =1 . Convergence is known to be super linear. A DVANTAGES Computational efficiency: The primary computa- tional costs of our method are mat-vec mul- tiplication and boundary condition assign- ment in solving Poisson’s equation, both of which can easily be parallelized. No Initial Guess: Our method converges on the correct result for arbitrary initial guess of density and subspace High Convergence rate: Our method converges significantly faster than DIIS, and the conver- gence rate remains high regardless of system size. N ONLINEAR E IGENSOLVER Our method is based on the FEAST algorithm[1], a method for solving linear eigenvalue problems. The nonlinear Kohn-Sham equations are solved in an approximate subspace Q, which is updated at each iteration: 1. Get initial guess for X , the matrix whose columns are the eigenvectors of interest x i , and n, either or both of which can be com- pletely random 2. Get density matrix ρ via numerical complex contour integration: ρ = H C (zS - H [n]) -1 dz, z ∈ C where C is a contour around the part of the real axis where the eigenvalues of the occu- pied states are expected to be found. 3. Form the subspace Q = ρSX 4. Solve the projected, reduced nonlinear eigen- vector problem Q T H [n]Qx q = λ q Q T SQx q by iterating over the density n. This pro- cecure is computationally inexpensive com- pared to performing the full SCF iteration. 5. Calculate x i = Qx q , λ i = λ q , and the new density. If ∑ λ i has converged, exit. Otherwise, GOTO step 2. R ESULTS The following is the result of testing done with LDA all-electron DFT using a third order finite elements discretization. The total energy residual, the relative difference between the calculated total energy at subsequent iterations, is used as the means of gauging convergence. 0 1 2 3 4 5 6 7 8 9 10 -14 10 -13 10 -12 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 Total Energy Residual H2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Iteration CO 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Nonlinear Solver DIIS C6H6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Iteration 10 -14 10 -13 10 -12 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Total Energy Residual DIIS (Good guess) Good Initial Guess Initial Guess n=0.0 C6H6 Nonlinear Solver: Good Initial Guess Vs. Poor Initial guess C60 (Buckminsterfullerine): F UTURE W ORK • Parallelize the solution of the subspace prob- lem Q T H [n]Qx q = λ q Q T SQx q . This will be- come the primary bottleneck for much larger systems. • Update the FEAST package to provide a black-box interface for the algorithm de- scribed above. FEAST 2.0 currently provides black-box fuctionality for steps 2 and 3 de- scribed in "Nonlinear Eigensolver. http://www.ecs.umass.edu/∼polizzi/feast/ R EFERENCES [1] E. Polizzi, "Density-matrix-based algorithm for solving eigenvalue problems," Phys. Rev. B., vol 79 115112, Mar. 2009. [2] T. Rohwedder, R. Schneider, "An analysis for the DIIS acceleration method used in quantum chem- istry calculations," J. Math Chem.,Volume 49, Issue 9, pp 1889-1914, Oct2011.
Transcript of T SCFM AN E -B Aes12.wfu.edu/pdfarchive/poster/ES12_B_Gavin.pdfANONLINEAR EIGENSOLVER-BASED...
A NONLINEAR EIGENSOLVER-BASED ALTERNATIVETO TRADITIONAL SCF METHODS
BRENDAN GAVIN AND ERIC POLIZZI UNIVERSITY OF MASSACHUSETTS, [email protected], [email protected]
PROBLEMSOLVING THE KOHN-SHAM EQUATIONS
H[n]xi = λiSxi , n =∑i
|xi|2
Where H[n] is the typical Kohn-Sham Hamiltonianwith (for our work here) spinless LDA approxima-tion, xi, λi are eigenvector/eigenvalue pairs, n isthe electron density, and S is the mass matrix re-sulting from a finite elements discretization.
TRADITIONAL SCFThe Kohn-Sham equations are traditionally solvedusing self-consistent field methods that follow thegeneral procedure:
1. Find good initial guess n1 (e.g. with Thomas-Fermi model)
2. Get n′ = f(ni), where f(ni) is the procedureconsisting of:
(a) Forming the Kohn-Sham hamiltonianH[ni] by solving poisson’s equation, cal-culating the exchange-correlation opera-tor, etc.
(b) Solving the eigenvalue problemH[ni]xi = λiSxi
(c) Calculating the new density from the oc-cupied states xi
3. Form ni+1 from some function of n′ and ni. If‖ni+1 − f(ni+1)‖ < ε then stop; otherwiseGOTO step 2.
SIMPLE MIXING:
ni+1 = αni + (1− α)n′ , 0 ≤ α ≤ 1
The value of α is determined heuristically. Typicalα = 0.1. Convergence is usually very slow.
DIIS:
A list of m densities is first produced using someother procedure (e.g. simple mixing), and thensubsequent ni+1 are given by[2]:
ni+1 =m∑j=1
cjni−j + cjri−j , ri = ni − f(ni)
where cj are the coefficients that minimize‖∑cjrj‖ such that
∑cj = 1 . Convergence is
known to be super linear.
ADVANTAGESComputational efficiency: The primary computa-
tional costs of our method are mat-vec mul-tiplication and boundary condition assign-ment in solving Poisson’s equation, both ofwhich can easily be parallelized.
No Initial Guess: Our method converges on thecorrect result for arbitrary initial guess ofdensity and subspace
High Convergence rate: Our method convergessignificantly faster than DIIS, and the conver-gence rate remains high regardless of systemsize.
NONLINEAR EIGENSOLVEROur method is based on the FEAST algorithm[1],a method for solving linear eigenvalue problems.The nonlinear Kohn-Sham equations are solved inan approximate subspace Q, which is updated ateach iteration:
1. Get initial guess for X , the matrix whosecolumns are the eigenvectors of interest xi,and n, either or both of which can be com-pletely random
2. Get density matrix ρ via numerical complexcontour integration:
ρ =∮C(zS −H[n])−1dz, z ∈ C
where C is a contour around the part of thereal axis where the eigenvalues of the occu-pied states are expected to be found.
3. Form the subspace Q = ρSX
4. Solve the projected, reduced nonlinear eigen-vector problem
QTH[n]Qxq = λqQTSQxq
by iterating over the density n. This pro-cecure is computationally inexpensive com-pared to performing the full SCF iteration.
5. Calculate xi = Qxq, λi = λq , and thenew density. If
∑λi has converged, exit.
Otherwise, GOTO step 2.
RESULTSThe following is the result of testing done with LDA all-electron DFT using a third order finite elementsdiscretization. The total energy residual, the relative difference between the calculated total energy atsubsequent iterations, is used as the means of gauging convergence.
0 1 2 3 4 5 6 7 8 910
-14
10-13
10-12
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
Tota
l E
ner
gy R
esid
ual
H2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Iteration
CO
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Nonlinear SolverDIIS
C6H6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Iteration
10-14
10-13
10-12
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Tota
l E
ner
gy R
esid
ual
DIIS (Good guess)
Good Initial GuessInitial Guess n=0.0
C6H6Nonlinear Solver: Good Initial Guess Vs. Poor Initial guess
C60 (Buckminsterfullerine):
FUTURE WORK• Parallelize the solution of the subspace prob-
lem QTH[n]Qxq = λqQTSQxq . This will be-
come the primary bottleneck for much largersystems.
• Update the FEAST package to provide ablack-box interface for the algorithm de-scribed above. FEAST 2.0 currently providesblack-box fuctionality for steps 2 and 3 de-scribed in "Nonlinear Eigensolver.
http://www.ecs.umass.edu/∼polizzi/feast/
REFERENCES[1] E. Polizzi, "Density-matrix-based algorithm forsolving eigenvalue problems," Phys. Rev. B., vol 79115112, Mar. 2009.
[2] T. Rohwedder, R. Schneider, "An analysis for theDIIS acceleration method used in quantum chem-istry calculations," J. Math Chem.,Volume 49, Issue9, pp 1889-1914, Oct2011.
