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Mixed-derivative bounds for dyadic Simpson details

Statement

Use the dyadic composite-Simpson hierarchy of Nested one-dimensional quadrature levels and differences. Let $d \geq 1$ . For $S = {j_{1} < \dots < j_{k}} \subseteq {1, \dots, d}$ , define $D_{S} f : = \partial_{x_{j_{k}}}^{4} \dots \partial_{x_{j_{1}}}^{4} f,$ so these derivatives are applied in increasing coordinate order; set $D_{\emptyset} f = f$ . Assume every $D_{S} f$ is continuous on the closed cube, with the one-sided endpoint convention of Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ , and choose finite bounds $M_{S} \geq 0$ such that $∣ D_{S} f (x) ∣ \leq M_{S}$ on $[0, 1]^{d}$ . For $ℓ \in N_{0}^{d}$ , put $S (ℓ) : = {j : ℓ_{j} \geq 1}$ and $δ_{ℓ} (f) : = (Δ_{ℓ_{1}}^{(1)} \otimes \dots \otimes Δ_{ℓ_{d}}^{(d)}) (f) .$ Then $∣ δ_{ℓ} (f) ∣ \leq M_{S (ℓ)} {(\frac{17}{2880})}^{∣ S (ℓ) ∣} 16^{- \sum_{j \in S (ℓ)} ℓ_{j}} .$ Moreover, for every finite $F \subseteq N_{0}^{d}$ , $\sum_{ℓ \in F} ∣ δ_{ℓ} (f) ∣ \leq \sum_{S \subseteq {1, \dots, d}} M_{S} {(\frac{17}{43200})}^{∣ S ∣} < \infty .$

Facts & Assumptions

Given: $d \geq 1$ and a function with the continuous mixed fourth partials and finite bounds stated above.

[F1]

Nested one-dimensional quadrature levels and differences: $Q_{ℓ} = Q_{S} (\cdot; 0, 1, 2^{ℓ + 1})$ , $Δ_{0} = Q_{0}$ , and $Δ_{ℓ} = Q_{ℓ} - Q_{ℓ - 1}$ for $ℓ \geq 1$ .

[F2]

Composite Simpson error bound for a four-times continuously differentiable function: on $[0, 1]$ , for an even panel count $N \geq 2$ , $∣ \int_{0}^{1} g - Q_{S} (g; 0, 1, N) ∣ \leq (180 N^{4})^{- 1} ∥ g^{(4)} ∥_{\infty}$ for $g \in C^{4} [0, 1]$ .

[F3]

Composite Simpson rule on an even uniform grid: the two-panel level-zero rule has nonnegative weights $1 / 6, 4 / 6, 1 / 6$ , which sum to one.

[F4]

Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ : the stated continuous mixed derivatives exist through the required coordinatewise orders, with one-sided endpoint derivatives.

[F5]

Triangle inequality for finite sums: the absolute value of a finite sum is at most the sum of its absolute values.

[F6]

Laws of finite sums and finite products: finite sums may be regrouped and distribute over products, and are monotone in their terms.

[F7]

For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges states the series starting at $k = 0$ : for ratio $1 / 16$ , $\sum_{k \geq 0} (1 / 16)^{k} = 16 / 15$ . By Laws of integer exponents, $(1 / 16)^{k} = 16^{- k}$ . Removing the $k = 0$ term, which is $1$ , gives $\sum_{r \geq 1} 16^{- r} = 1 / 15$ ; since all terms are nonnegative, every finite partial sum starting at $r = 1$ is at most $1 / 15$ .

Proof

1.1givenF1F2F3F5algebra

For $ℓ \geq 1$ , use [F2] on levels $ℓ$ and $ℓ - 1$ , inserting the exact integral between their values. With panel counts $2^{ℓ + 1}$ and $2^{ℓ}$ , [F5] gives $∣ Δ_{ℓ} g ∣ \leq (2^{- 4 (ℓ + 1)} / 180 + 2^{- 4 ℓ} / 180) ∥ g^{(4)} ∥_{\infty} = (17 / 2880) 16^{- ℓ} ∥ g^{(4)} ∥_{\infty}$ . For $ℓ = 0$ , [F3] gives $∣ Δ_{0} g ∣ = ∣ Q_{0} g ∣ \leq ∥ g ∥_{\infty}$ .

2.1givenF1F4step 1.1F6algebra

Apply the one-dimensional bounds successively in the coordinates. A finite quadrature functional commutes with derivatives in any other coordinate because it is a finite weighted sum. In each active coordinate $j$ , apply step 1.1 to the section after the earlier-coordinate derivatives have been taken; the needed fourth derivative is the corresponding sequential mixed derivative. In each inactive coordinate use the level-zero supremum bound. Multiplying the bounds gives $M_{S (ℓ)} (17 / 2880)^{∣ S (ℓ) ∣} 16^{- \sum_{j \in S (ℓ)} ℓ_{j}}$ . The stated continuous mixed derivatives justify each section estimate, and [F6] justifies the finite tensor expansions.

3.1givenstep 2.1F6F7algebra∎

Fix finite $F$ and group indices by $S = S (ℓ)$ . For a fixed nonempty $S$ , enlarge its levels to the finite box $1 \leq ℓ_{j} \leq R_{j}$ where $R_{j}$ is the largest such coordinate level in that group. By step 2.1 and [F6], the sum over that box is at most $M_{S} \prod_{j \in S} ((17 / 2880) \sum_{r = 1}^{R_{j}} 16^{- r}) \leq M_{S} (17 / 43200)^{∣ S ∣}$ , using [F7]. For $S = \emptyset$ , the only possible index is $0$ and its term is at most $M_{\emptyset}$ . Summing over all subsets proves the finite-subsum claim; the right side is finite.

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Sources

T. Gerstner and M. Griebel, Sparse Grids, Encyclopedia of Quantitative Finance, §§Classical Sparse Grids and Numerical Integration (standard reference, not scraped)
T. Gerstner and M. Griebel, Dimension-Adaptive Tensor-Product Quadrature, §2 (standard reference, not scraped)
Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, §5.6 (standard reference, not scraped)