Cell-Based Empirical Cubature

• Cell-based empirical cubature is a method for constructing high-order, stable numerical integration formulas on finite-element subdomains using scattered sampling points.
• The approach leverages an underdetermined solution space to compute nonnegative weights, ensuring robustness against measurement noise and error amplification.
• Implementation via linear or quadratic programming in finite element codes provides computational efficiency and superior accuracy compared to classical methods.

Cell-based empirical cubature refers to a methodology for constructing stable, high-order cubature formulas on arbitrary cells—typically finite-element subdomains—where data are available only at scattered or non-Gaussian quadrature points. This approach fundamentally enables integration over domains subdivided into cells using experimental, equidistant, or random sampling points, rather than relying on classical cubature theories based on fixed node locations. Its distinguishing feature is the generation of nonnegative weights, providing robustness to measurement error and ensuring stability in numerical integration, especially in large-scale finite-element settings where classical Gaussian nodes are inaccessible (Glaubitz, 2020).

1. Formulation on Single Cells

Given a cell $\Omega_e \subset \mathbb{R}^t$, let $\{x_i\}_{i=1}^N \subset \Omega_e$ denote $N$ sampling sites, which may correspond to experiment-derived or computational candidate points. For a fixed dimension $K$ and basis $\{\phi_j\}_{j=1}^K$ of the function space $V_K$ (commonly polynomials up to degree $d$), the objective is to determine a weight vector $w \in \mathbb{R}^N$ such that the cubature formula

$C[f] := \sum_{i=1}^N w_i f(x_i)$

is exact for all $f \in V_K$. Exactness requires

$$\sum_{i=1}^N w_i \phi_j(x_i) = \int_{\Omega_e} \phi_j(x)\,dx, \quad j = 1,\ldots,K,$$

which can be compactly encoded in matrix form as $A w = b$, with $A_{j,i} = \phi_j(x_i)$ and $b_j = \int_{\Omega_e} \phi_j(x)\,dx$.

2. Underdetermined Solution Space and V_K–Unisolvency

Assuming $K < N$ and that the point set $\{x_i\}$ is $V_K$–unisolvent (no nonzero $p \in V_K$ vanishes on all $x_i$), the interpolation matrix $A$ has rank $K$. Thus, the constraint $A w = b$ admits an affine family of solutions:

$$\mathcal{W} := \{w \in \mathbb{R}^N \mid A w = b\} = w_p + \text{Null}(A),$$

with $\dim \text{Null}(A) = N - K$. Any member $w\in\mathcal{W}$ yields cubature exactness of degree $d$. The degrees of freedom make possible the selection of $w$ with additional desirable properties, such as stability or nonnegativity.

3. Stability and Nonnegativity Criteria

Given noisy observations $f^\epsilon(x_i)$ with uniform error $\epsilon$, the deviation in the cell-based cubature formula obeys the bound

$$|C[f] - C[f^\epsilon]| \leq \|w\|_1 \cdot \|f - f^\epsilon\|_\infty \leq \epsilon\,\|w\|_1,$$

where $\|w\|_1 = \sum_{i=1}^N |w_i|$. Notably, strict nonnegativity implies $\|w\|_1 = \sum w_i = \int 1$, minimizing error amplification. Two stability norms are central:

• $\ell^1$-stability: $\kappa_1(w) = \|w\|_1 = \sum_{i=1}^N |w_i|$
• Weighted $\ell^2$-stability: $\kappa_2(w) = \|R^{-1/2}w\|_2$, with $R = \operatorname{diag}(r_i)$ and $r_i > 0$ chosen proportionally to cell-volume fractions.

4. Classes of Stable High-Order Cubature Formulas

Two principal optimization formulations for weight selection in $\mathcal{W}$ are as follows:

CF Type	Objective	Constraints	Algorithmic Method
$\ell^1$-CF	$\min_{w \in \mathbb{R}^N} \|w\|_1$	$A w = b$, $w \geq 0$	Linear program (LP), e.g., MATLAB linprog
LS-CF	$\min_{w \in \mathbb{R}^N} \|R^{-1/2}w\|_2$	$A w = b$, $w \geq 0$	Quadratic program or minimal-norm least-squares (QR)

The $\ell^1$-CF ("basis-pursuit cubature," Editor's term) minimizes the total variation of the weights under exactness and nonnegativity constraints, focusing on stability relative to data noise. Conversely, LS-CF achieves a unique closed-form solution for weights by minimizing the weighted $\ell^2$ norm, subject to the same constraints; nonnegativity may necessitate a reduction in polynomial degree.

5. Implementation in Finite Element Codes

For a domain partitioned into $M$ cells $\{\Omega_e\}$:

• Precomputation: Generate $N$ candidate sample sites in each cell.
• Build $A^{(e)}_{j,i} = \phi_j(x_i^{(e)})$ and $b_j^{(e)} = \int_{\Omega_e} \phi_j dx$.
• Solve LP or QP for nonnegative weights $w^{(e)}$.
• Run-time: For each cell, approximate integrals using $\sum_i w_i^{(e)} f(x_i^{(e)})$, then aggregate over all cells.

Computational complexity per cell:

• $\ell^1$-CF: LP size $N$, $K$ equality constraints; worst-case $O(N^3)$, practical improvements via sparsity.
• LS-CF: QR-based weighted least-squares, $O(N K^2)$.

The empirical relation $N \approx C K^s$ governs required sample counts for nonnegative cubature, with $s \approx 1.5$–$2$ in common cases and $K = \dim P_d \sim O(d^q)$, maintaining feasibility for moderate-$d$ 2D or 3D applications. All cells are independent, enabling trivial data parallelism (Glaubitz, 2020).

6. Numerical Performance, Sample Complexity, and Accuracy

Empirical investigations on canonical domains (square $\Omega = [-1,1]^2$ and ball $B_2$) reveal the dependence of minimal sample requirements $N = C K^s$ on point distribution and cubature type:

Domain	CF Type	Sample Type	$s$	$C$
Square	LS-CF	Equidist	1.9	0.30
Square	LS-CF	Random	0.90	23
Square	LS-CF	Halton	1.3	1.4
Square	$\ell^1$-CF	Equidist	1.5	0.36
Square	$\ell^1$-CF	Random	1.4	0.74
Square	$\ell^1$-CF	Halton	1.6	0.27

Testing against Monte Carlo, quasi-Monte Carlo, and tensor-Legendre cubature for smooth functions on cubes and balls demonstrates that both LS and $\ell^1$ cubatures systematically outperform MC at equivalent sampling sites, with accuracy approaching that of optimized Gaussian-product rules for large $N$. This suggests robust performance for cell-based empirical cubature in practical settings.

7. Stepwise Implementation Recipe and Contextual Significance

• Step 1: Select desired exactness degree $d$, set $K = \dim P_d$, and choose $N > K$ sample locations per cell.
• Step 2: Construct $A$ and $b$ for each cell.
• Step 3: Solve for nonnegative weights using LS or $\ell^1$ programs; if infeasible, decrease $d$.
• Step 4: Record cell weights and site locations.
• Step 5 (run-time): For each cell, compute local sum $\sum_i w_i^{(e)} f(x_i^{(e)})$ and aggregate.

Cell-based empirical cubature offers a rigorous framework for integrating experimental or computational data on arbitrary meshes without requiring classical Gaussian quadrature nodes. The resulting formulas are stable, high-order, and employ nonnegative weights, making them ideal for large-scale finite-element or experimental-data integration contexts (Glaubitz, 2020).