H1-conditional stability with explicit Lipshitz constant for a one-dimensional inverse acoustic problem

Abstract - In the present paper we formulate and investigate a one-dimensional inverse acoustic problem in the form of a nonlinear system of Volterra integral equations. We prove the conditional stability of the inverse problem with explicitly given Lipschitz constant depending only on the depth l and on bounds of the norms of the unknown coefficient and inverse problem data. The proposed technique can be applied to prove the local well-posedness of the inverse problem in L2(0, l).

[1]  F. Santosa,et al.  Inverse Problems of Acoustic and Elastic Waves , 1984 .

[2]  K. Habiro,et al.  Algebraic formulae for the q-inverse in a free group , 2001 .

[3]  Xu Bin.  The Degree of Symmetry of Compact Smooth Manifolds , 2001 .

[4]  Sailing He,et al.  An optimization approach to a three‐dimensional acoustic inverse problem in the time domain , 1995 .

[5]  INVERSION OF ACOUSTICAL IMPEDANCE PROFILE BY THE METHODS OF CHARACTERISTICS. , 1982 .

[6]  W. W. Symes.  The Inverse Reflection Problem for a Smoothly Stratified Elastic Medium. , 1979 .

[7]  Guy Chavent,et al.  About the stability of the inverse problem in 1-D wave equations—application to the interpretation of seismic profiles , 1979 .

[8]  V. Romanov,et al.  Multidimensional inverse hyperbolic problem with impulse input and a single boundary measurement , 1999 .

[9]  M. Furuta,et al.  The Seiberg-Witten equations and equivariant e-invariants , 2001 .

[10]  S. Kabanikhin.  Numerical analysis of inverse problems , 1995 .

[11]  Direct and inverse inequalities for the isotropic Lamé system with variable coefficients and applications to an inverse source problem , 2001 .

[12]  S. Morita,et al.  THE PRIMARY APPROXIMATION TO THE COHOMOLOGY OF THE MODULI SPACE OF CURVES AND COCYCLES FOR THE MUMFORD-MORITA-MILLER CLASSES , 1996 .

[13]  W. Symes,et al.  Uniqueness and continuous dependence for a multidimensional hyperbolic inverse problem , 1985 .

[14]  G. Chavent,et al.  Inversion of normal incidence seismograms , 1982 .

[15]  V. Glasko.  Inverse problems in mathematical physics , 1984 .

[16]  Zhang,et al.  Observability inequalities by internal observation and its applications , 2001 .

[17]  Spectral properties of non-symmetric systems of ordinary differential operators , 2001 .

[18]  Masahiro Yamamoto,et al.  Global Uniqueness and Stability for a Class of Multidimensional Inverse Hyperbolic Problems with Two Unknowns , 2003 .

[19]  Kenneth P. Bube,et al.  The One-Dimensional Inverse Problem of Reflection Seismology , 1983 .