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• W4387161888 abstract "In this chapter, we are concerned with the stability analysis for inverse problems associated with time-harmonic acoustic scattering, where we follow the treatment in [3]. Let $$kin mathbb {R}_+$$ be a wavenumber of the acoustic wave, signifying the frequency of the wave propagation. Let $$Vin L^infty (mathbb {R}^n)$$ , n = 2, 3, be a potential function, which signifies the material parameter of the medium at the point x and is related to the refractive index in our setting. We assume that supp(V ) ⊂ BR, where BR is a central ball of radius $$Rin mathbb {R}_+$$ in $$mathbb {R}^n$$ . That is, the inhomogeneity of the medium is supported inside a given bounded domain of interest. The inhomogeneous medium is often referred to as a scatterer. An incident wave ui is sent to interrogate the medium V , where the scattered wave us is generated. We let u denote the total wave field. The former is an entire solution to the Helmholtz equation (Δ + k2)ui = 0. The aforementioned acoustic scattering problem can be formulate by $$displaystyle {} big (varDelta + k^2(1+V)big ) u = 0quad mbox{in}quad {mathbb R}^n. $$ Moreover, the scattered wave us = u − ui satisfies the Sommerfeld radiation condition $$displaystyle {} vert mathbf xvert ^{frac {n-1}{2}} big (partial _r - mathrm ikbig ) u^s rightarrow 0 $$ uniformly with respect to the angular variable θ := x∕|x| as $$r:= vert mathbf xvert rightarrow infty $$ . Here, ∂r is the derivative along the radial direction from the origin. The radiation condition implies the existence of a far-field pattern. More precisely, there is a real analytic function on the unit sphere at infinity, namely, $$A_{u^i}: mathbb {S}^{n-1}rightarrow {mathbb C}$$ satisfies $$displaystyle {} u(rtheta ) = u^i(rtheta ) + frac {e^{ikr}}{r^{(n-1)/2}} A_{u^i}(theta ) + mathcal {O} Big ( frac {1}{r^{n/2}} Big ) $$ uniformly along the angular variable θ. This function is called the far-field pattern or scattering amplitude of u." @default.
• W4387161888 created "2023-09-30" @default.
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• W4387161888 date "2023-01-01" @default.
• W4387161888 modified "2023-09-30" @default.
• W4387161888 title "Stable Determination of an Acoustic Medium Scatterer by a Single Far-Field Pattern" @default.
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• W4387161888 doi "https://doi.org/10.1007/978-3-031-34615-6_12" @default.
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