Fast approximately application of Greens function of Hamiltonian - PowerPoint PPT Presentation

Fast approximately application of Green’s function of Hamiltonian using symbol compression Song Mei ICME, Stanford May 1, 2015 Joint work with Lin Lin and Lexing Ying. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei (ICME, Stanford) Fast approximately application of Green’s function of Hamiltonian using symbol compression May 1, 2015 1 / 25

− − − − Electronic Structure Calculation 10 20 30 Given the location of the atom 40 50 nuclear(Si atoms). 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei (ICME, Stanford) Fast approximately application of Green’s function of Hamiltonian using symbol compression May 1, 2015 2 / 25

Electronic Structure Calculation 10 20 30 Given the location of the atom 40 50 nuclear(Si atoms). 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 A slice of the potential 1.4 10 1.2 20 1 30 0.8 40 0.6 The potential. 50 0.4 60 0.2 0 70 − 0.2 80 − 0.4 90 − 0.6 100 10 20 30 40 50 60 70 80 90 100 A slice of the density − 3 x 10 10 3.5 20 3 30 2.5 40 The density of the electrons. 50 2 60 1.5 70 1 80 90 0.5 100 10 20 30 40 50 60 70 80 90 100 . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei (ICME, Stanford) Fast approximately application of Green’s function of Hamiltonian using symbol compression May 1, 2015 2 / 25

Electronic Structure Calculation Kohn-Sham density functional theory. Given the location of the nuclears, i.e. given effective potential V [ ρ ]( x ) which depends on the local density of electrons. Solve a nonlinear eigenvalue problem: ( ) − ∆ + V [ ρ ]( x ) ψ i ( x ) = ε i ψ i ( x ) , i = 1 , 2 , . . . , N e , ∫ R 3 ψ ∗ i ( x ) ψ j ( x ) dx = δ ij , (1) N e ∑ | ψ i ( x ) | 2 . ρ ( x ) = i =1 ψ i is the electron’s orbit, ε i is the energy level. Use self-consistent iteration to solve this problem. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei (ICME, Stanford) Fast approximately application of Green’s function of Hamiltonian using symbol compression May 1, 2015 3 / 25

Notations ρ is the solution of the DFT. ˆ Hamiltonian H = − ∆ + V [ˆ ρ ]. Its eigenfunctions ψ i , eigenvalues ε i , with ε 1 ≤ ε 2 ≤ ε 3 ≤ . . . . Projection to lower energy orbits P = ∑ N c j =1 ψ j ψ ∗ j . N c is a number we define afterwards. j > N c ψ j ψ ∗ Projection to upper energy orbits Q = I − P = ∑ j . Fourier transform F . G ε usually represents Green’s function of Hamiltonian with a shift G ε = ( H − ε I ) − 1 = ∑ i ψ i ψ ∗ i / ( ε i − ε ). Number of electrons N e . Number of grid points N . Sometimes continuous, sometimes discrete. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei (ICME, Stanford) Fast approximately application of Green’s function of Hamiltonian using symbol compression May 1, 2015 4 / 25

Goal and motivation We would like to approximately fast apply G ε = ( H − ε I ) − 1 = ∑ N i =1 ψ i ψ ∗ i / ( ε i − ε ), i.e. solve ( H − ϵ I ) u = g . Applications: Perturbation theory for excited states, correlation energy calculation, etc. In these applications, we usually need to apply G ε to a lot of O ( NN e ) RHS. May not require the result to be very accurate. Solving O ( NN e ) times of linear equation is not acceptable. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei (ICME, Stanford) Fast approximately application of Green’s function of Hamiltonian using symbol compression May 1, 2015 5 / 25

Intuition Intuition: for high energy electrons, the orbit is like plane wave, i.e., ψ j ( x ) ≈ e 2 π ijx . We write G ε as j ≤ N c ψ j ψ T j > N c ψ j ψ T G ε = G ε 1 + G ε 2 = ∑ j / ( ε j − ε ) + ∑ j / ( ε j − ε ), where N c is a given number to split G ε 1 and G ε 2 . e 2 π ij ( x − x ′ ) Then G ε 2 ( x , x ′ ) ≈ ∑ . The latter expression is rank 1 in j > N c ε j − ε symbol representation. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei (ICME, Stanford) Fast approximately application of Green’s function of Hamiltonian using symbol compression May 1, 2015 6 / 25

Symbol representation For an operator A , its pseudodifferential symbol a ( x , ξ ) is defined as ∫ e 2 π ix · ξ a ( x , ξ )ˆ ( Af )( x ) = f ( ξ ) d ξ. (2) In matrix form, a = ( A · F − 1 ) ./ F − 1 , where F the fourier matrix. ∂ For A = ∂ x , its symbol representation is a ( x , ξ ) = 2 π i ξ . For A = − ∆, its symbol representation is a ( x , ξ ) = 4 π 2 | ξ | 2 . e 2 π ij ( x − x ′ ) The symbol representation of G = ∑ is j > N c ε j − ε 1 symbol ( G ) = ∑ j > N c δ ( ξ − ε j ), which is rank 1. ξ − ε In matlab form, G = ones ( N , 1) · [1 ./ ( ε 1: N − ε ) . ∗ (1 : N > N c )]. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei (ICME, Stanford) Fast approximately application of Green’s function of Hamiltonian using symbol compression May 1, 2015 7 / 25

Solving an optimization problem We write j ≤ N c ψ j ψ T j > N c ψ j ψ T G ε = G ε 1 + G ε 2 = ∑ j / ( ε j − ε ) + ∑ j / ( ε j − ε ). e 2 π ij ( x − x ′ ) Then G ε 2 ( x , x ′ ) ≈ ∑ . The latter expression is rank 1 in j > N c ε j − ε symbol. We hope symbol ( G ε 2 ) is low rank. We cannot perform SVD on symbol ( G ε 2 ) directly, it is expensive to calculate symbol ( G ε 2 ). To get the low rank approximation of the symbol of G ε 2 , we approximately solve the following optimization problem: r ∑ diag ( u k ) F − 1 diag ( v k ) F ) QR − G ε 2 R ∥ 2 { u k , v k } r k =1 = argmin u k , v k ∥ Q ( F . k =1 (3) Here, Q is a projection, F is the Fourier matrix, R = randn ( N , N R ) is a random matrix, and N R = O ( r ) is a small constant. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei (ICME, Stanford) Fast approximately application of Green’s function of Hamiltonian using symbol compression May 1, 2015 8 / 25

The optimization problem The optimization problem is r ∑ { u k , v k } r diag ( u k ) F − 1 diag ( v k ) F ) QR − G ε 2 R ∥ 2 k =1 = argmin u k , v k ∥ Q ( F . k =1 (4) This is a non-convex problem which is hard to solve. We propose an algorithm to approximately solve it. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei (ICME, Stanford) Fast approximately application of Green’s function of Hamiltonian using symbol compression May 1, 2015 9 / 25

The optimization problem Result : Given G ε 2 applying on several random right hand side, calculate the symbol compression of G ε 2 . Set N R = 30. Generate random R = randn ( N , N R ); Apply G ε 2 to R to get GR . This is done by solving GR = CG ( H − ε I , QR ); for k = 1 : r do Approximately solve ∥ Q ( diag ( u k ) F − 1 diag ( v k ) F ) QR − GR ∥ 2 using alternating least square; GR ← GR − Q ( diag ( u k ) F − 1 diag ( v k ) F ) QR . end . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei (ICME, Stanford) Fast approximately application of Green’s function of Hamiltonian using symbol compression May 1, 2015 10 / 25