lambda_method.qmd

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title: $\lambda$ method for discretization effects
author: Simone Romiti
date: last-modified
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        html-math-method: katex
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# Presentation of the method

In some lattice calculations we have 2 (or more) determinations affected by different lattice artifacts,
but that must agree in the continuum limit.
_Example_: in a calculation involving fermions, we may have the result for 2 different regularizations of fermions.

Instead of extrapolating the curves independently,
one can use this physical information and fit them simultaneously with the constraint that they share the same continuum limit.
Even in this case however,
it may be necessary to work out several types of fits,
each including a different combination of discretization effects.
_A posteriori_, one needs to average the models with some information criterion.

The $\lambda$ method presented here is a methodology to bypass the model averaging,
and consider only the constant fit of a combination of the different curves.
Let's say we have a set of $n$ curves $y_i(a)$, where $a$ is the lattice spacing and:
$$\lim_{a \to 0} y_i(a) = y_0 < \infty \, \forall i=1,\ldots,n \, .$$

Then, we can construct a combination such that the discretization effects are minimal,
i.e. such that it is the flattest possible as a function of the lattice spacing:

$$y_\lambda(a) = \sum_i \lambda_i(a) y_i(a) \, , \, \sum_i \lambda_i=1 \, .$$ 

The only constraint is that

$$\lim_{a \to 0} \lambda_i(a) < \infty 
\, , \text{and} \, \lim_{a \to 0} \sum_i \lambda_i(a) = 1 \, . $$

In the following,
we consider $\lambda_i$ of the form:
$$\lambda_i(a) = \lambda_0 + c_i^k f_k(a) \, ,$$
where $f_k(a)$ are a set of functions describing different types of discretization effects.
_Example_: $f_1(a)=a^2$, $f_2(a)=a^2/\log(a)$.

## Algorithm and minimization guesses

Algorithmically, we propose a pairwise reduction approach:
consider sequential pairs of $y_i$ and find the optimal combination each time.
At each step,
given the curves $y_i$, $y_j$,
we find $\lambda_{ij}(a)$ in order to minimimize the average slope squared of the following curve:
$$y_{\lambda_{ij}}(a) = \lambda_{ij}(a) y_i(a) + (1 - \lambda_{ij}(a))y_j(a) \, ,$$
namely by minimizing:
$$R = \sum_{p} [y_{\lambda_{ij}}'(a_p)]^2$$
where $a_p$ are the values of the lattice spacing for which the curves have been evaluated.
It is understood that the derivatives are computed as finite differences.

Once the 1st pair $y_{\lambda_{12}}$ have been found,
we can use this curve to repeat the procedure with $y_3$ and build $y_{\lambda_{123}}$,
and so on.

At each step,
it is relevant to start the minimization from a reasonable guess for the parameters $c^k$.
Here's a procedure that estimates them from the data points.
Without loss of generality, we call the 2 curves $y_1$ and $y_2$,
and use the convention $\lambda(a) = \lambda_0 + \frac{c^k}{2} f_k(a)$.
_Notation_: We use $\braket{\cdot}$ to indicate the average over the data points,
and $\bar{y}=(y_1+y_2)/2$.

**Step 1**: We build $y_{\lambda_0}$:
$$y_{\lambda_0}(a) = \lambda_0 y_1(a) + (1-\lambda_0) y_2(a)$$

and impose $y'_{\lambda_0}(a) = 0$. From this it follows the estimator:
$$\lambda_0 = \braket{-y_2'/(y_1' - y_2')}$$

**Step 2**: We build $y_{\lambda_1}$:
$$y_{\lambda_1}(a) 
= \lambda_1 y_1(a) + (1-\lambda_1) y_2(a) 
= y_{\lambda_0} + c^1 f_1 \bar{y} 
\, .$$

From $y'_{\lambda_1}(a) = 0$ we get:
$$c^1 = \braket{-y_{\lambda_0}'/(f_1 \bar{y})'}$$

**Step k**: 
We proceed until we find all the $c^k$ iteratively. 
Their estimators are:

$$c^k = \braket{- y_{\lambda_{k-1}}'/(f_k \bar{y})'}$$


**Note**: At each step $k$, one can do the minimization using a limited subset of the functions $f_k$ with the guesses found to that point, and then set the $k$-th guess for $c^k$ to the fit value (not just the estimator).
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