TY - JOUR

T1 - Recovering a potential in damped wave equation from Neumann-to-Dirichlet operator

AU - Romanov, Vladimir

AU - Hasanov, Alemdar

N1 - Funding Information: The first author was supported by Mathematical Center in Akademgorodok at Novosibirsk State University (the agreement with Ministry of Science and High Education of the Russian Federation number 075-15-2019-1613). The research of the second author has been supported by the Scientific and Technological Research Council of Turkey (TUBITAK)

PY - 2020/11

Y1 - 2020/11

N2 - The inverse coefficient problem of recovering the potential q(x) in the damped wave equation m(x)utt + μ(x)ut = (r(x)ux)x + q(x)u, (x, t) ∈ ΩT := (0, ℓ) × (0, T) subject to the boundary conditions r(0)ux(0, t) = f(t), u(ℓ, t) = 0, from the Dirichlet boundary measured output ν(t) :=u(0, t), t ∈ (0, T] is studied. A detailed microlocal analysis of regularity of the direct problem solution in the subdomains defined by the characteristics as well as along these characteristics is provided. Based on this analysis, necessary regularity results and energy estimates are derived. It is proved that the Dirichlet boundary measured output uniquely determines the potential q(x) in the interval [0, h(T/2)] and this solution belongs to C(0, h(T/2)) with T < T∗, where h(z) is the root of the equation z = ∫h(z)0 √ m(x)/r(x) dx, T∗ = 2 ∫ℓ0 √m(x)/r(x) dx. Moreover, the global uniqueness theorem is proved. Compactness, invertibility and Lipschitz continuity of the Neumann-to-Dirichlet operator Φ [·] :Q ∪ C(0, ℓ) → L2(0, T), Φf [q](t) :=u(0, t; q) is proved. This allows us to prove an existence of a quasi-solution of the inverse problem defined as a minimum of the Tikhonov functional J(q) :=(1/2) ||Φf [·]. ν||2L2(0,T) as well as its Fréchet differentiability. An explicit formula for the Fréchet gradient is derived by making use of the unique solution to corresponding adjoint problem. The proposed approach is leads to very effective gradient based computational identification algorithm.

AB - The inverse coefficient problem of recovering the potential q(x) in the damped wave equation m(x)utt + μ(x)ut = (r(x)ux)x + q(x)u, (x, t) ∈ ΩT := (0, ℓ) × (0, T) subject to the boundary conditions r(0)ux(0, t) = f(t), u(ℓ, t) = 0, from the Dirichlet boundary measured output ν(t) :=u(0, t), t ∈ (0, T] is studied. A detailed microlocal analysis of regularity of the direct problem solution in the subdomains defined by the characteristics as well as along these characteristics is provided. Based on this analysis, necessary regularity results and energy estimates are derived. It is proved that the Dirichlet boundary measured output uniquely determines the potential q(x) in the interval [0, h(T/2)] and this solution belongs to C(0, h(T/2)) with T < T∗, where h(z) is the root of the equation z = ∫h(z)0 √ m(x)/r(x) dx, T∗ = 2 ∫ℓ0 √m(x)/r(x) dx. Moreover, the global uniqueness theorem is proved. Compactness, invertibility and Lipschitz continuity of the Neumann-to-Dirichlet operator Φ [·] :Q ∪ C(0, ℓ) → L2(0, T), Φf [q](t) :=u(0, t; q) is proved. This allows us to prove an existence of a quasi-solution of the inverse problem defined as a minimum of the Tikhonov functional J(q) :=(1/2) ||Φf [·]. ν||2L2(0,T) as well as its Fréchet differentiability. An explicit formula for the Fréchet gradient is derived by making use of the unique solution to corresponding adjoint problem. The proposed approach is leads to very effective gradient based computational identification algorithm.

KW - Damped wave equation

KW - Existence of a quasisolution

KW - Fr\'{e}chet gradient

KW - Neumann-to-Dirichlet operator

KW - Recovering a potential

KW - Uniqueness of the inverse problem solution

KW - STABILITY

KW - uniqueness of the inverse problem solution

KW - existence of a quasi-solution

KW - UNIQUENESS

KW - damped wave equation

KW - COEFFICIENTS

KW - recovering a potential

UR - http://www.scopus.com/inward/record.url?scp=85096757173&partnerID=8YFLogxK

U2 - 10.1088/1361-6420/abb8e8

DO - 10.1088/1361-6420/abb8e8

M3 - Article

AN - SCOPUS:85096757173

VL - 36

JO - Inverse Problems

JF - Inverse Problems

SN - 0266-5611

IS - 11

M1 - 115011

ER -