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A four-dimensional product exactly integrated on a sparse node union

Example

Use the Clenshaw–Curtis levels with polynomial degrees $1$ and $2$ on $[0, 1]$ in four dimensions, and let $Λ_{1} = {ℓ \in N_{0}^{4} : ∣ ℓ ∣_{1} \leq 1}$ . For $f (x) = x_{1}^{2} x_{2} x_{3} x_{4}$ , the Smolyak rule is exact with $48$ distinct nodes; the full tensor product of level-one nodes has $81$ nodes.

Facts & Assumptions

Given: $f (x) = x_{1}^{2} x_{2} x_{3} x_{4}$ on $[0, 1]^{4}$ , with $Q_{0} : = Q_{1, [0, 1]}$ and $Q_{1} : = Q_{2, [0, 1]}$ , the Clenshaw–Curtis rules of polynomial degrees one and two.

[F1]

Smolyak cubature on a finite downward-closed index set: $Q_{Λ_{1}}$ is the sum of the tensor details at $0, e_{1}, e_{2}, e_{3}, e_{4}$ , and its node union is the union of the corresponding level-grid products.

[F2]

Deterministic node merging for Smolyak cubature: each distinct point in that union is counted once, including a point whose combined weight is zero.

[F3]

Clenshaw–Curtis quadrature with the midpoint zero-degree case: on $[0, 1]$ , the $n$ th rule samples at $x_{j} = (1 + \cos (j π / n)) / 2$ for $j = 0, \dots, n$ ; thus $Q_{0}$ is level $n = 1$ and $Q_{1}$ is level $n = 2$ .

[F4]

Polynomial exactness and best-approximation error for Clenshaw–Curtis: the $n$ th Clenshaw–Curtis rule integrates every polynomial of degree at most $n$ exactly.

[F5]

Quarter-turn values and shifts by pi/2 and pi: $\cos (π) = - 1$ and $\cos (π / 2) = 0$ .

[F6]

For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term and The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ : $\int_{0}^{1} x d x = 1 / 2$ and $\int_{0}^{1} x^{2} d x = 1 / 3$ .

[F7]

The integral of a product function on a product rectangle is the product of the two integrals: the integral of a product over a product rectangle is the product of its coordinate integrals.

[F8]

The derivatives of sine and cosine are cosine and minus sine: $\cos (0) = 1$ .

Verification

1.1givenF3F4F5F6F8algebra

The Given rules are the $n = 1$ and $n = 2$ rules. By [F3], their normalized nodes are $(1 + \cos (j π / n)) / 2$ ; [F5] and [F8] give $X_{0} = {0, 1}$ and $X_{1} = {0, 1 / 2, 1}$ . By [F4], these rules are exact through degrees $1$ and $2$ , respectively. Writing the level-zero weights as $w_{0}, w_{1}$ , exactness on $1$ and $x$ gives $w_{0} + w_{1} = 1$ and $w_{1} = 1 / 2$ , hence $Q_{0} (x) = Q_{0} (x^{2}) = 1 / 2$ . Also $Q_{1} (x) = 1 / 2$ and $Q_{1} (x^{2}) = 1 / 3$ by exactness and [F6]. Thus $Δ_{1} (x) = 0$ and $Δ_{1} (x^{2}) = - 1 / 6$ .

2.1givenF1F6F7step 1.1algebra

Since $f$ is a product, its tensor details factor into one-dimensional values. The zero-index term is $Q_{0} (x^{2}) Q_{0} (x)^{3} = 1 / 16$ . The $e_{1}$ term is $Δ_{1} (x^{2}) Q_{0} (x)^{3} = - 1 / 48$ , and each $e_{j}$ term for $j = 2, 3, 4$ vanishes because $Δ_{1} (x) = 0$ . Therefore $Q_{Λ_{1}} (f) = 1 / 16 - 1 / 48 = 1 / 24$ . By [F6]–[F7], $I_{4} (f) = (1 / 3) (1 / 2)^{3} = 1 / 24$ , so the cubature is exact.

3.1givenF1F2step 1.1algebra∎

The node union is $X_{0}^{4}$ together with, for each coordinate $j$ , the $8$ points whose $j$ th coordinate is $1 / 2$ and whose other coordinates are endpoints. Those four midpoint sets are disjoint from one another and from the $16$ corners, so $W (Λ_{1}) = 16 + 4 \cdot 8 = 48$ . The full tensor level-one grid has $∣ X_{1} ∣^{4} = 3^{4} = 81$ nodes.

Depends on

Used by

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Sources

Lloyd N. Trefethen, Is Gauss Quadrature Better than Clenshaw–Curtis?, §2 (standard reference, not scraped)
N. Zabaras, Sparse Grid Collocation Methods, lecture notes, slides 17–26 (standard reference, not scraped)
T. Gerstner and M. Griebel, Sparse Grids, Encyclopedia of Quantitative Finance, §Numerical integration (standard reference, not scraped)