Stable Inversions

Overview

In DQMC, to obtain the equal-times Green's function we commonly perform the inversion

\[G = \left[1 + B\right]^{-1}.\]

Similarily, we compute

\[G = \left[A + B\right]^{-1}\]

for the time-displaced Green's function. The following methods are exported to facilitate these tasks.

inv_one_plus, inv_one_plus!
inv_sum, inv_sum!

When function names are suffixed with _loh, i.e. inv_one_plus_loh, a more sophisticated method is used for numerical stabilization (see the paper linked above for more details).

Details

`inv_one_plus`

\[\begin{aligned} G &= [\mathbb{1} + UDX]^{-1} \\ &= [U\underbrace{(U^\dagger X^{-1} + D)}_{udx}X]^{-1}\\ &= [(Uu)d(xX)]^{-1}\\ &= U_r D_r X_r, \end{aligned}\]

with $U_r = (xX)^{-1}$, $D_r = d^{-1}$, $X_r = (Uu)^{-1}$.

\[\begin{aligned} G &= \left[\mathbb{1} + U_L D_L T_L \left( U_R D_R T_R \right)^\dagger \right]^{-1} \\ &= \left[\mathbb{1} + U_L \underbrace{\left( D_L \left( T_L T_R^\dagger \right) D_R \right)}_{udt} U_R^\dagger \right]^{-1} \\ &= \left[\mathbb{1} + U D T \right]^{-1}, \end{aligned}\]

with $U=U_Lu$, $D=d$, and $T=tU_R^\dagger$. The remaining computation is then performed as in the single argument version of inv_one_plus above.

`inv_one_plus(::SVD, ::SVD)`

\[\begin{aligned} G &= \left[\mathbb{1} + U_L D_L V_L^\dagger U_R D_R V_R^\dagger \right]^{-1} \\ &= \left[\mathbb{1} + U_L \underbrace{\left( D_L \left( V_L^\dagger U_R \right) D_R \right)}_{udv^\dagger} V_R^\dagger \right]^{-1} \\ &= \left[\mathbb{1} + U D V^\dagger \right]^{-1}, \end{aligned}\]

with $U=U_Lu$, $D=d$, and $V=v V_R$. The remaining computation is then performed as in the single argument version of inv_one_plus above.

`inv_one_plus_loh`

\[\begin{aligned} G &= [\mathbb{1} + UDX]^{-1}\\ &= [\mathbb{1} + UD_mD_pX]^{-1}\\ &= [(X^{-1} D_p^{-1} + U D_m) D_p X]^{-1}\\ &= X^{-1} \underbrace{[D_p^{-1} (\underbrace{X^{-1} D_p^{-1} + UD_m}_{udx})^{-1}]}_{udx} \\ &= U_r D_r X_r, \end{aligned}\]

with $D_m = \min(D, 1)$, $D_p = \max(D, 1)$, $U_r = X^{-1}u$, $D_r = d$, and $X_r = x$. [Loh2005, Loh1989]

`inv_sum`

\[\begin{aligned} G(\tau_1, \tau_2) &= [U_L D_L X_L + U_R D_R X_R]^{-1}\\ &= [U_L \underbrace{(D_L X_L X_R^{-1} + U_L^\dagger U_R D_R)}_{udx} X_R ]^{-1}\\ &= [(U_L u) d^{-1} (x X_R)]^{-1}\\ &= U_r D_r X_r, \end{aligned}\]

where $U_r = (x X_R)^{-1}$, $D_r = d^{-1}$, and $X_r = (U_L u)^{-1}$.

`inv_sum_loh`

\[\begin{aligned} G(\tau_1, \tau_2) &= [U_L D_L X_L + U_R D_R X_R]^{-1} \\ &= [U_L D_{Lm} D_{Lp} X_L + U_R D_{Rm} D_{Rp} X_R]^{-1} \\ &= \left[U_L D_{Lp} \underbrace{\left( \dfrac{D_{Lm}}{D_{Rp}} X_L X_R^{-1} + U_L^\dagger U_R \dfrac{D_{Rm}}{D_{Lp}} \right)}_{udx} X_R D_{Rp} \right]^{-1} \\ &= X_R^{-1} \underbrace{\dfrac{1}{D_{Rp}} [udx]^{-1} \dfrac{1}{D_{Lp}}}_{udx} U_L^\dagger\\ &= U_r D_r X_r, \end{aligned}\]

with $D_{Rm} = \min(D_R, 1)$, $D_{Rp} = \max(D_R, 1)$, $D_{Lm} = \min(D_L, 1)$, $D_{Lp} = \max(D_L, 1)$, $U_r = X_R^{-1} u$, $D_r = d$, and $X_r = x U_L^\dagger$. [Loh2005]

Resources

[Loh2005]

@article{Loh2005,
	author = {Loh, E. Y. and Gubernatis, J. E. and Scalettar, R. T. and White, S. R. and Scalapino, D. J. and Sugar, R. L.},
	title = {{Numerical Stability and the Sign Problem in the Determinant Quantum Monte Carlo Method}},
	journal = {International Journal of Modern Physics C},
	volume = {16},
	number = {08},
	pages = {1319--1327},
	month = {aug},
	year = {2005}
	issn = {0129-1831},
	doi = {10.1142/S0129183105007911},
	url = {http://www.worldscientific.com/doi/abs/10.1142/S0129183105007911},
}

[Loh1989]

@incollection{Loh1989,
	author = {Loh, E. Y. and Gubernatis, J. E. and Scalettar, R. T. and Sugar, R. L. and White, S. R.},
	title = {{Stable Matrix-Multiplication Algorithms for Low-Temperature Numerical Simulations of Fermions}},
	year = {1989}
	pages = {55--60},
	doi = {10.1007/978-1-4613-0565-1_8},
	url = {http://link.springer.com/10.1007/978-1-4613-0565-1{\_}8},
}