Figures of merit for ordinary lattice rules

LatNet Builder supports the following figures of merit for ordinary rank-1 lattices:

The spectral figure of merit [17], [18] using a term-by-term evaluation with an \(\ell_q\) norm:
\(\mathcal D_{\mathfrak u}(P_n)\) is the normalized maximum distance between two successive parallel hyperplanes.
The \(\mathcal P_\alpha\) discrepancy [32], [13], [26] with \(\alpha \in \{2, 4, 6, 8\}\), using a term-by-term evaluation with an \(\ell_q\) norm:
\[ \mathcal{D}^2_{\mathfrak u}(P_n) = \frac{1}{n} \sum_{i = 0}^n \prod_{j \in \mathfrak u} p_{\alpha}(x_{i,j}), \]

where
\[ p_{\alpha}(x) = \frac{- (-4 \pi^2)^{\alpha/2} B_{\alpha}(x)}{\alpha!} \]

with \(B_{\alpha}\) the Bernouilli polynomial of even degree \(\alpha\).

If \( q = 2\), a coordinate-uniform evaluation algorithm can be used. This algorithm drastically reduces the computation time.
The \(\mathcal R_\alpha\) criterion [24], [11], with \(\alpha \in \{1, 2, \dots, 9\}\), using a term-by-term evaluation with an \(\ell_q\) norm:
\[ \mathcal{D}^2_{\mathfrak u}(P_n) = \frac{1}{n} \sum_{i = 0}^n \prod_{j \in \mathfrak u} r_{\alpha, n}(x_{i,j}), \]

where
\[ r_{\alpha, n}(x) = \sum_{h = - \lfloor (n-1)/2 \rfloor }^{\lfloor n/2 \rfloor} \max(1,|h|)^{-\alpha} e^{2 \pi i h x} - 1. \]

If \( q = 2\), a coordinate-uniform evaluation algorithm can be used. This algorithm drastically reduces the computation time.

Figures of merit for polynomial lattice rules

LatNet Builder supports the following figures of merit for polynomial rank-1 lattices:

The \(\mathcal P_{\alpha}\) discrepancy [21] with \(\alpha \in \{2, 4, 6, 8\}\) , using a term-by-term evaluation with an \(\ell_q\) norm:
\[ \mathcal{D}^2_{\mathfrak u}(P_n) = \frac{1}{n} \sum_{i = 0}^n \prod_{j \in \mathfrak u} \omega_{\alpha}(x_{i,j}), \]

where
\[ \omega_{\alpha}(x) = \mu(\alpha) - 2^{(1+\lfloor \log_2(x) \rfloor) (\alpha - 1)}(\mu(\alpha) + 1), \]

where \( \mu(\alpha) = (1 - 2^{1- \alpha})^{-1} \) for even \( \alpha \).

If \( q = 2\), a coordinate-uniform evaluation algorithm can be used. This algorithm drastically reduces the computation time.
The \(\mathcal R\) criterion [25], using term-by-term evaluation with an \(\ell_q\) norm:
\[ \mathcal{D}^2_{\mathfrak u}(P_n) = \frac{1}{n} \sum_{i = 0}^n \prod_{j \in \mathfrak u} \phi'_m(x_{i,j}), \]

where \(m\) is the degree of the modulus and
\[ \phi'_m(x) = \begin{cases} \frac{1}{2}i_0 & \textrm{ if } x = \sum_{i= i_0}^{\infty} \xi_i 2^{-i}, 1 \leq i_0 \leq m, \xi_{i_0} \neq 0 \\ 1 + \frac{1}{2}m & \textrm{otherwise.} \end{cases} \]

If \( q = 2\), a coordinate-uniform evaluation algorithm can be used. This algorithm drastically reduces the computation time.

Figures of merit for interlaced polynomial lattice rules

Figures of merit specific to interlaced polynomial lattice rules [10] are also supported by LatNet Builder.

These figures of merit for interlaced polynomial lattice rules in dimension \(s\) can be seen as figures of merit for the underlying lattice rules in higher dimension \(ds\), where \(d\) is the interlacing factor. If \(P_n\) is the interlaced lattice rule, let \(\bar{P}_n = \left\{(z_{0,1}, \dots, z_{0, ds}), \dots, (z_{n-1,1}, \dots, z_{n-1, ds}) \right\}\) denote the underlying polynomial lattice rule in dimension \(ds\). This point of view also requires to transform (or interlace) weights in dimension \(s\) into weights in dimension \(d s\). Let \( \bar{\gamma} \) denote these interlaced weights.

The figures of merit for interlaced polynomial lattice rules supported by LatNet Builder have the following form:

\[ D(P_n) = \sum_{\emptyset \neq \mathfrak u \subseteq \{1, \dots, d s\}}\left(\bar{\gamma}_{\mathfrak u} D_{\mathfrak u}(\bar{P}_n)\right), \]

The \(B_{\alpha, d, (1)}\) interlaced discrepancy, using an \(\ell_1\) norm with a coordinate-uniform evaluation algorithm:

\[ D_{\mathfrak u}(\bar{P}_n) = \sum_{i = 0}^{n-1} \prod_{j \in \mathfrak u} \phi_{\alpha, d, (1)}(z_{i, j}), \]

where \( \phi_{\alpha, d, (1)}(z) \) is defined by:
\[ \phi_{\alpha, d, (1)}(z) = \frac{1 - 2^{(\min(\alpha, d) -1) \lfloor \log_2(z) \rfloor} (2^{\min(\alpha, d)} -1)}{2^{(\alpha+2)/2} (2^{\min(\alpha, d) - 1} -1) }. \]

For this discrepancy, the interlaced weights are defined as:
\[ \bar{\gamma}_{\mathfrak u} = \gamma_{w(\mathfrak u)} 2^{\alpha (2d-1) |w(\mathfrak u)| / 2}, \]

where \(w\) denotes the weights interlacing operator:
\[ w(\mathfrak u) = \left\{ \left \lceil \frac{j}{d} \right\rceil, j \in \mathfrak u \right\}. \]
The \(B_{d, (2)}\) interlaced discrepancy, using an \(\ell_1\) norm with a coordinate-uniform evaluation algorithm:

\[ D_{\mathfrak u}(\bar{P}_n) = \sum_{i = 0}^{n-1} \prod_{j \in \mathfrak u} \phi_{d, (2)}(z_{i, j}), \]

where \( \phi_{d, (2)}(z) \) is defined by:
\[ \phi_{d, (2)}(z) = \frac{2^{d-1}(1 - 2^{(d -1) \lfloor \log_2(z) \rfloor} (2^{d} -1))}{(2^{d - 1} -1) }. \]

For this discrepancy, the interlaced weights are defined as:
\[ \bar{\gamma}_{\mathfrak u} = \gamma_{w(\mathfrak u)} \prod_{j \in \mathfrak u} 2^{- ( (j-1) \mod d) + 1}, \]

where \(w\) denotes the weights interlacing operator.