Analyticity

THE INTERPOLATED FACTORED GREEN FUNCTION METHOD

3.2 Analyticity

In order to analyze the above introduced analytic factor, certain notations and conventions are introduced. On one hand, for notational simplicity, but without loss of generality, throughout the remainder of this section it is assumed that the factorization is centered at the origin, i.e., 𝑥_𝑆 = 0; the extension to the general 𝑥_𝑆 ≠ 0 case is, of course, straightforward due to the translation invariance of the Green function under consideration. Incorporating the convention𝑥_𝑆 =0, then, for 0 < 𝜂 < 1, the following sets denoting theanalyticity domainof the analytic factor are considered in the analysis of the factorization.

Definition 5(Analyticity domain). Let𝐵 =𝐵(0, 𝐻) denote an origin-centered box of side𝐻 > 0, as per Definition 4. Let𝜂 > 0. Then, theanalyticity domain 𝐴^𝐻

𝜂 of the analytic factor of a field emitted by the box 𝐵is defined as the subset of the set

𝐴_𝜂 B {(𝑥 , 𝑥⁰) ∈R³×R³ : |𝑥⁰| ≤ 𝜂|𝑥|}

given by

𝐴^𝐻

𝜂 B 𝐴_𝜂∩

R³×𝐵

. (3.6)

Clearly, 𝐴^𝐻

𝜂 is the subset of pairs in 𝐴_𝜂 such that𝑥⁰is restricted to a particular box 𝐵(0, 𝐻). Theorem 4 below implies that, on the basis of an appropriate change of variables which adequately accounts for the analyticity of the analytic factor𝑔_𝑆 up to and including infinity, this factor can be accurately evaluated for (𝑥 , 𝑥⁰) ∈ 𝐴^𝐻

𝜂

by means of a straightforward interpolation rule based on an interpolation mesh in spherical coordinates, which is finite and sparse along the radial direction.

As indicated above, the analytic properties of the factor𝑔_𝑆play a pivotal role in the proposed algorithm. Under the𝑥_𝑆 =0 convention established above, the factors in equation (3.4) become

𝐺(𝑥 ,0) = 𝑒^{𝚤 𝜅|𝑥|}

4𝜋|𝑥| and 𝑔_𝑆(𝑥 , 𝑥⁰) = |𝑥|

|𝑥−𝑥⁰|𝑒^{𝚤 𝜅(|𝑥−𝑥}

0|−|𝑥|)

. (3.7)

In order to analyze the properties of the factor 𝑔_𝑆, we introduce the spherical coordinate parametrization

x˜(𝑟 , 𝜃 , 𝜑) B

©

­

­

«

𝑟sin𝜃cos𝜑 𝑟sin𝜃sin𝜑

𝑟cos𝜃 ª

®

®

®

¬

, 0 ≤𝑟 < ∞, 0≤ 𝜃 ≤ 𝜋, 0≤ 𝜑 <2𝜋, (3.8)

and note that (3.7) may be re-expressed in the form 𝑔_𝑆(𝑥 , 𝑥⁰) = 1

4𝜋

𝑥 𝑟 − ^𝑥0

𝑟

exp

𝚤 𝜅𝑟

𝑥 𝑟

− 𝑥⁰ 𝑟

−1 . (3.9)

The effectiveness of the proposed factorization is illustrated in Figures 3.2a, 3.2b, and 3.2c, where the oscillatory character of the analytic factor𝑔_𝑆and the Green function (1.8) without factorization are compared, as a function of𝑟, for several wavenumbers 𝜅. The slowly-oscillatory character of the factor𝑔_𝑆, even for acoustically large source boxes 𝐵(𝑥_𝑆, 𝐻) as large as twenty wavelengths 𝜆 = 2𝜋/𝜅 (𝐻 = 20𝜆) and starting as close as just 3𝐻/2 away from the center of the source box, is clearly visible in Figure 3.2c; much faster oscillations are observed in Figure 3.2b, even for source boxes as small as two wavelengths in size (𝐻 =2𝜆). Only the real part is depicted in Figures 3.2a, 3.2b, and 3.2c, but, clearly, the imaginary part displays the same behavior.

In addition to the factorization (3.5), the proposed strategy relies on the use of the singularity resolving change of variables

𝑠B ℎ 𝑟

, x(𝑠, 𝜃 , 𝜑) Bx˜(𝑟 , 𝜃 , 𝜑), (3.10) where, once again, 𝑟 = |𝑥| denotes the radius in spherical coordinates and where ℎ denotes the radius of the source box, as in Definition 4. Using these notations, equation (3.9) may be re-expressed in the form

𝑔_𝑆(𝑥 , 𝑥⁰) = 1 4𝜋

𝑥 𝑟 − ^𝑥0

ℎ𝑠

exp

𝚤 𝜅𝑟

𝑥 𝑟

− 𝑥⁰ ℎ 𝑠

−1 . (3.11)

(a) Test setup. The Surrogate Source position 𝑥⁰ gives rise to the fastest possible oscillations along the Measurement line, among all possible source positions within the Source Box.

(b) Real part of the Green function𝐺 in equation (1.8) (without factorization), along the Measurement line depicted in Figure 3.2a, for boxes of various acoustic sizes𝐻.

(c) Real part of the analytic factor 𝑔_𝑆 (equation (3.7)) along the Measurement line depicted in Figure 3.2a, for boxes of various acoustic sizes𝐻.

Figure 3.2: Surrogate Source factorization test, set up as illustrated in Figure 3.2a.

Figure 3.2c shows that the analytic factor𝑔_𝑆 oscillates much more slowly, even for 𝐻 =20𝜆, than the unfactored Green function does for the much smaller values of 𝐻 considered in Figure 3.2b.

Remark 4. While the source point 𝑥and its norm𝑟 depend on𝑠, the quantity𝑥/𝑟 is independent of𝑟 and therefore also of𝑠.

The introduction of the variable𝑠 gives rise to several algorithmic advantages, all of which stem from the analyticity properties of the function 𝑔_𝑆—as presented in Lemma 1 below and Theorem 4 in Section 3.3. Briefly, these results establish that, for any fixed values𝐻 >0 and𝜂satisfying 0< 𝜂 <1, the function𝑔_𝑆is analytic for (𝑥 , 𝑥⁰) ∈ 𝐴^𝐻

𝜂, with 𝑥-derivatives that are bounded up to and including |𝑥| =∞. As a result (as shown in Section 3.3) the 𝑠change of variables translates the problem of interpolation of𝑔_𝑆 over an infinite𝑟 interval into a problem of interpolation of an analytic function of the variable𝑠over a compact interval in the𝑠variable. The relevant 𝐻-dependent analyticity domains for the function 𝑔_𝑆 for each fixed value of𝐻are described in the following lemma.

Lemma 1. Let𝑥⁰ ∈ 𝐵(𝑥_𝑆, 𝐻) and let 𝑥

0 =x(˜ 𝑟

0, 𝜃

0, 𝜑

0) =x(𝑠

0, 𝜃

0, 𝜑

0)(𝑠

0 = ℎ/𝑟

0) be such that (𝑥

0, 𝑥⁰) ∈ 𝐴^𝐻

𝜂. Then𝑔_𝑆is an analytic function of𝑥around𝑥

0and also an analytic function of (𝑠, 𝜃 , 𝜑) around (𝑠

0, 𝜃

0, 𝜑

0). Further, the function𝑔_𝑆 is an analytic function of (𝑠, 𝜃 , 𝜑) (resp. (𝑟 , 𝜃 , 𝜑)) for 0 ≤ 𝜃 ≤ 𝜋, 0 ≤ 𝜑 < 2𝜋, and for 𝑠 in a neighborhood of𝑠

0 =0(resp. for𝑟 in a neighborhood of𝑟

0 =∞, including 𝑟 =𝑟

0=∞).

Proof. The claimed analyticity of the function 𝑔_𝑆 around𝑥

0 = x(𝑠

0, 𝜃

0, 𝜑

0) (and, thus, the analyticity of𝑔_𝑆around(𝑠

0, 𝜃

0, 𝜑

0)) is immediate since, under the assumed hypothesis, the quantity

𝑥 𝑟

− 𝑥⁰ ℎ 𝑠

, (3.12)

does not vanish in a neighborhood of 𝑥 =𝑥

0. Analyticity around 𝑠

0 = 0 (𝑟

0 = ∞) follows similarly since the quantity (3.12) does not vanish around𝑠=𝑠

0=0.

Corollary 1. Let 𝐻 > 0 be given. Then for all 𝑥⁰ ∈ 𝐵(𝑥_𝑆, 𝐻), the function 𝑔_𝑆(x(𝑠, 𝜃 , 𝜑), 𝑥⁰) is an analytic function of (𝑠, 𝜃 , 𝜑) for 0 ≤ 𝑠 < 1,0 ≤ 𝜃 ≤ 𝜋 and 0≤ 𝜑 < 2𝜋.

Proof. Take𝜂 ∈ (0,1). Then, for 0 ≤ 𝑠 ≤ 𝜂, we have (x(𝑠, 𝜃 , 𝜑), 𝑥⁰) ∈ 𝐴^𝐻

𝜂. The analyticity for 0 ≤ 𝑠 ≤ 𝜂 follows from Lemma 1, and since𝜂 ∈ (0,1) is arbitrary,

the lemma follows.

For a given𝑥⁰∈R³, Corollary 1 reduces the problem of interpolation of the function 𝑔_𝑆(𝑥 , 𝑥⁰) in the 𝑥 variable to a problem of interpolation of a re-parametrized form of the function𝑔_𝑆over a bounded domain—provided that (𝑥 , 𝑥⁰) ∈ 𝐴^𝐻

𝜂, or, in other words, provided that 𝑥 is separated from 𝑥⁰ by a factor of at least 𝜂, for some 𝜂 < 1. In the IFGF algorithm presented in Section 3.6, side-𝐻 boxes 𝐵(𝑥_𝑆, 𝐻) containing sources𝑥⁰are considered, with target points𝑥 at a distance no less than 𝐻 away from 𝐵(𝑥_𝑆, 𝐻). Clearly, a point (𝑥 , 𝑥⁰) in such a configuration necessarily belongs to 𝐴^𝐻

𝜂 with 𝜂 =

√

3/3. Importantly, as demonstrated in the following section, the interpolation quality of the algorithm does not degrade as source boxes of increasingly large side 𝐻 are used, as is done in the proposed multi-level IFGF algorithm (with a single box size at each level), leading to a computing cost per level which is independent of the level box size𝐻.