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• W2963485162 abstract "We consider the problem to reconstruct a wave speed <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c element-of upper C Superscript normal infinity Baseline left-parenthesis upper M right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>c in C^infty (M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a domain <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M subset-of double-struck upper R Superscript n> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>M subset mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from acoustic boundary measurements modelled by the hyperbolic Dirichlet-to-Neumann map <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Lamda> <mml:semantics> <mml:mi mathvariant=normal>Λ<!-- Λ --></mml:mi> <mml:annotation encoding=application/x-tex>Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We introduce a reconstruction formula for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c> <mml:semantics> <mml:mi>c</mml:mi> <mml:annotation encoding=application/x-tex>c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that is based on the Boundary Control method and incorporates features also from the complex geometric optics solutions approach. Moreover, we show that the reconstruction formula is locally Lipschitz stable for a low frequency component of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c Superscript negative 2> <mml:semantics> <mml:msup> <mml:mi>c</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>c^{-2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> under the assumption that the Riemannian manifold <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper M comma c Superscript negative 2 Baseline d x squared right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>c</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>d</mml:mi> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(M, c^{-2} dx^2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a strictly convex function with no critical points. That is, we show that for all bounded <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C squared> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>C^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> neighborhoods <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper U> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding=application/x-tex>U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c> <mml:semantics> <mml:mi>c</mml:mi> <mml:annotation encoding=application/x-tex>c</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there is a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Superscript 1> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>C^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> neighborhood <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper V> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding=application/x-tex>V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c> <mml:semantics> <mml:mi>c</mml:mi> <mml:annotation encoding=application/x-tex>c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and constants <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C comma upper R greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>C, R > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartLayout 1st Row StartAbsoluteValue script upper F left-parenthesis c overTilde Superscript negative 2 Baseline minus c Superscript negative 2 Baseline right-parenthesis left-parenthesis xi right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C e Superscript 2 upper R StartAbsoluteValue xi EndAbsoluteValue Baseline double-vertical-bar normal upper Lamda overTilde minus normal upper Lamda double-vertical-bar Subscript asterisk Baseline comma xi element-of double-struck upper R Superscript n Baseline comma EndLayout> <mml:semantics> <mml:mtable columnalign=right left right left right left right left right left right left rowspacing=3pt columnspacing=0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em side=left displaystyle=true> <mml:mtr> <mml:mtd> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>F</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>c</mml:mi> <mml:mo>~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>−<!-- − --></mml:mo> <mml:msup> <mml:mi>c</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>C</mml:mi> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> <mml:mi>R</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> </mml:mrow> </mml:msup> <mml:msub> <mml:mrow> <mml:mo symmetric=true>‖</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi mathvariant=normal>Λ<!-- Λ --></mml:mi> <mml:mo>~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant=normal>Λ<!-- Λ --></mml:mi> <mml:mo symmetric=true>‖</mml:mo> </mml:mrow> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msub> <mml:mo>,</mml:mo> <mml:mspace width=1em /> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:annotation encoding=application/x-tex>begin{align*} |mathcal {F}left (widetilde c^{-2} - c^{-2}right )(xi )| le C e^{2R|xi |}left |widetilde Lambda - Lambda right |_*, quad xi in mathbb {R}^n, end{align*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c overTilde element-of upper U intersection upper V> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>c</mml:mi> <mml:mo>~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>U</mml:mi> <mml:mo>∩<!-- ∩ --></mml:mo> <mml:mi>V</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>widetilde c in U cap V</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Lamda overTilde> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi mathvariant=normal>Λ<!-- Λ --></mml:mi> <mml:mo>~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding=application/x-tex>widetilde Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the Dirichlet-to-Neumann map corresponding to the wave speed <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c overTilde> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>c</mml:mi> <mml:mo>~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding=application/x-tex>widetilde c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartAbsoluteValue dot EndAbsoluteValue Subscript asterisk> <mml:semantics> <mml:msub> <mml:mrow> <mml:mo>|</mml:mo> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mo>|</mml:mo> </mml:mrow> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msub> <mml:annotation encoding=application/x-tex>left |cdot right |_*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a norm capturing certain regularity properties of the Dirichlet-to-Neumann maps." @default.
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• W2963485162 title "A Lipschitz stable reconstruction formula for the inverse problem for the wave equation" @default.
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