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Second-order bias for the kernel particle gradient
Statement
Let f∈C³(R^d) have bounded third derivative and define D_εf(x)=∫f(y)∇W_ε(x−y)dy. Then for every ε>0 and x∈R^d,
where ||D³f||∞ is the supremum of the operator norm of the trilinear derivative.
Facts & Assumptions
Given: The kernel W, ε>0, and f∈C³ with bounded third derivative.
Scaling identities for kernel moments and derivatives: W_ε(z)=ε^(−d)W(z/ε), so ∇W_ε(z)=ε^(−d−1)∇W(z/ε).
Derivative moments of a compact radial kernel: ∫∇W=0, ∫z_j∂_kW=−δ_jk, ∫z_jz_l∂_kW=0, and ∫|z|³|∇W|=m₃.
Multivariable Taylor formula with a Lagrange remainder along a line segment: a C³ function has a degree-two Taylor expansion on each line segment with the third derivative remainder at an interior point.
A compactly supported Riemann integrand admits the global change-of-variables formula from a diffeomorphism near the relevant compact preimage: let U⊂R^d be open and T:U→R^d be injective C¹ with invertible derivative. If g:R^d→R is compactly supported Riemann integrable with supp(g)⊂T(U), then ∫{R^d}g(y)dy=∫{R^d}h(u)du, where h(u)=g(T(u))|det DT(u)| on U and h=0 outside U. Apply this scalar identity componentwise to the compactly supported vector integrand, with U=R^d and the onto map T(u)=x−εu.
A continuous real function on a compact Jordan measurable set is Riemann integrable over that set: a continuous real integrand on a compact Jordan box is Riemann integrable.
Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in : on a bounding rectangle, integration is linear and monotone.
Cauchy-Schwarz with its equality case, the triangle inequality for , the parallelogram law and polarisation: the Euclidean inner product satisfies Cauchy-Schwarz and the Euclidean norm satisfies its norm axioms.
Proof
The scalar components of f(y)∇W_ε(x−y) are continuous and compactly supported, so [F5] makes them Riemann integrable on a bounding box. Apply [F4] with y=x−εu, whose absolute Jacobian determinant is ε^d, and use [F1] to obtain D_εf(x)=ε^(−1)∫f(x−εu)∇W(u)du.
For each x,u, [F3] gives f(x−εu)=f(x)−ε∇f(x)·u+(ε²/2)D²f(x)[u,u]+R(x,u), with |R(x,u)|≤(ε³/6)||D³f||∞|u|³.
Integrate the expansion of step 1.2 against ε^(−1)∇W(u). Linearity [F6] and [F2] make the constant and quadratic terms vanish; the linear term in coordinate k equals −Σ_j∂_jf(x)∫u_j∂_kW(u)du=∂_kf(x). Thus the integrated polynomial part is ∇f(x).
Put G(u)=R(x,u)∇W(u), a continuous compactly supported vector function, and v=∫G componentwise. If v≠0 set a=v/||v||₂; then ||a||₂=1 and ||v||₂=∫⟨a,G(u)⟩≤∫|⟨a,G(u)⟩|≤∫||G(u)||₂ by linearity, monotonicity, and [F7]. If v=0 the same upper bound is immediate. The pointwise remainder estimate in step 1.2 and [F2] therefore give ||v||₂≤(ε³/6)||D³f||∞∫|u|³|∇W(u)|du=(ε³m₃/6)||D³f||∞. Multiplication by ε^(−1) from step 1.1 proves the stated bound.
Depends on
- Scaled symmetric compactly supported kernel family
- Scaling identities for kernel moments and derivatives
- Derivative moments of a compact radial kernel
- Multivariable Taylor formula with a Lagrange remainder along a line segment
- A compactly supported Riemann integrand admits the global change-of-variables formula from a diffeomorphism near the relevant compact preimage
- A continuous real function on a compact Jordan measurable set is Riemann integrable over that set
- Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $\mathbb{R}^m$
- Cauchy-Schwarz $\lvert\langle x,y\rangle\rvert \le \lVert x\rVert_2\lVert y\rVert_2$ with its equality case, the triangle inequality for $\lVert\cdot\rVert_2$, the parallelogram law and polarisation
Used by
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Sources
- Volker Springel, Smoothed Particle Hydrodynamics in Astrophysics, Annual Review of Astronomy and Astrophysics 48 (2010), author full text (standard reference, not scraped)
- Benjamin Seibold, MIT OpenCourseWare 18.336, Numerical Methods for Partial Differential Equations, Lecture 25: Particle Methods (Spring 2009) (standard reference, not scraped)