Geodesic
Behavior of the misfit functional for a~one-dimensional hyperbolic inverse problem
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 2 (1999) no. 2, pp. 137-160

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In this paper we investigate the behavior of the misfit functional for a one-dimensional hyperbolic inverse problem when an unknown coefficient stands by a lowest term of a differential equation. Assuming an existence of an inverse problem solution we prove a uniqueness of a stationary point of the functional. If the minimization sequence belongs to a bounded set, we show that the following estimates of the convergence rate for the suggested method of the descent $$ J[q_k]\le J[q_0]\exp\{-c(k-1)\},\quad\|q_k-q_*\|^2_{L_2[-T,T]}\le CJ[q_0]\exp\{-c(k-1)\} $$ takes place. 
@article{SJVM_1999_2_2_a3,
     author = {A. L. Karchevsky},
     title = {Behavior of the misfit functional for a~one-dimensional hyperbolic inverse problem},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {137--160},
     publisher = {mathdoc},
     volume = {2},
     number = {2},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_1999_2_2_a3/}
}
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A. L. Karchevsky. Behavior of the misfit functional for a~one-dimensional hyperbolic inverse problem. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 2 (1999) no. 2, pp. 137-160. http://geodesic.mathdoc.fr/item/SJVM_1999_2_2_a3/