Convergence rates for Tikhonov regularization of a coefficient identification problem

Abstract This paper investigates the convergence rates for Tikhonov regularization of the problem for identifying the coefficient <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>a</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> a(x)\in L^{\infty}(\Omega) in the frequency-domain acoustic wave equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mrow> <m:mi>div</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo lspace="0.167em">⁢</m:mo> <m:mrow> <m:mo rspace="0.167em">∇</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>−</m:mo> <m:mrow> <m:mi>b</m:mi> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mi>f</m:mi> </m:mrow> </m:math> -\mathrm{div}(a\nabla u)-bu=f in general dimensional spaces; here <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mi>d</m:mi> </m:msup> </m:mrow> </m:math> x\in\Omega\subset\mathbb{R}^{d} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> d\geq 1 , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>b</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> b(x)&gt;0 . We assume that we know the imprecise measurement data of 𝑢 in the subdomain <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="normal">Ω</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>⊂</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:math> \Omega_{1}\subset\Omega with a measurement error of level <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>δ</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> \delta&gt;0 , while 𝑢 satisfies the general Robin boundary condition on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo rspace="0em">∂</m:mo> <m:msub> <m:mi mathvariant="normal">Ω</m:mi> <m:mn>1</m:mn> </m:msub> </m:mrow> </m:math> \partial\Omega_{1} . We propose to regularize this problem by minimizing a new functional and prove that the functional attains a unique global minimum on the admissible set of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>a</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math