Inverse problem for a nonlocal diffuse optical tomography equation
Philipp Zimmermann
Abstract
Abstract In this article a nonlocal analogue of an inverse problem in diffuse optical tomography is considered. We show that whenever one has given two pairs of diffusion and absorption coefficients <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:math> , such that there holds <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> in the measurement set W and they generate the same DN data, then they are necessarily equal in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> and Ω, respectively. Additionally, we show that the condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mi>W</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mi>W</mml:mi> </mml:msub> </mml:math> is optimal in the sense that without this restriction one can construct two distinct pairs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:math> generating the same DN data.