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Diffstat (limited to 'content/know/concept/pulay-mixing/index.pdc')
| -rw-r--r-- | content/know/concept/pulay-mixing/index.pdc | 21 |
1 files changed, 14 insertions, 7 deletions
diff --git a/content/know/concept/pulay-mixing/index.pdc b/content/know/concept/pulay-mixing/index.pdc index 9102c0e..8daa54f 100644 --- a/content/know/concept/pulay-mixing/index.pdc +++ b/content/know/concept/pulay-mixing/index.pdc @@ -83,13 +83,14 @@ We thus want to minimize the following quantity, where $\lambda$ is a [Lagrange multiplier](/know/concept/lagrange-multiplier/): $$\begin{aligned} - \braket{R_{n+1}}{R_{n+1}} + \lambda \sum_{m = 1}^n \alpha_m - = \sum_{m=1}^n \alpha_m \Big( \sum_{k=1}^n \alpha_k \braket{R_m}{R_k} + \lambda \Big) + \braket{R_{n+1}}{R_{n+1}} + \lambda \sum_{m = 1}^n \alpha_m^* + = \sum_{m=1}^n \alpha_m^* \Big( \sum_{k=1}^n \alpha_k \braket{R_m}{R_k} + \lambda \Big) \end{aligned}$$ -By differentiating the right-hand side with respect to $\alpha_m$, +By differentiating the right-hand side with respect to $\alpha_m^*$ +and demanding that the result is zero, we get a system of equations that we can write in matrix form, -which is relatively cheap to solve numerically: +which is cheap to solve: $$\begin{aligned} \begin{bmatrix} @@ -107,6 +108,11 @@ $$\begin{aligned} \end{bmatrix} \end{aligned}$$ +From this, we can also see that the Lagrange multiplier +$\lambda = - \braket{R_{n+1}}{R_{n+1}}$, +where $R_{n+1}$ is the *predicted* residual of the next iteration, +subject to the two assumptions. + This method is very effective. However, in practice, the earlier inputs $\rho_1$, $\rho_2$, etc. are much further from $\rho_*$ than $\rho_n$, @@ -121,7 +127,7 @@ You might be confused by the absence of all $\rho_m^\mathrm{new}$ in the creation of $\rho_{n+1}$, as if the iteration's outputs are being ignored. This is due to the first assumption, which states that $\rho_n^\mathrm{new}$ are $\rho_n$ are already similar, -such that they are interchangeable. +such that they are basically interchangeable. Speaking of which, about those assumptions: while they will clearly become more accurate as $\rho_n$ approaches $\rho_*$, @@ -147,8 +153,9 @@ $$\begin{aligned} In other words, we end up introducing a small amount of the raw outputs $\rho_m^\mathrm{new}$, while still giving more weight to iterations with smaller residuals. -Pulay mixing is very effective: -it can accelerate convergence by up to one order of magnitude! +Pulay mixing is very effective for certain types of problems, +e.g. density functional theory, +where it can accelerate convergence by up to one order of magnitude! |