Stability estimates for the solutions to inverse extremal problems for the Helmholtz equation
Tóm tắt
The inverse problems are under study for the Helmholtz equation describing acoustic scattering at a three-dimensional inclusion. Some optimization method reduces these problems to the inverse extremum problems with variable refraction index and boundary source density as controls. We prove that these problems are solvable and derive the optimality systems that describe necessary optimality conditions. Analysis of the optimality systems leads us to some sufficient conditions on the input data ensuring the uniqueness and stability of optimal solutions.
Tài liệu tham khảo
S. A. Cummer and D. Schurig, “One Path to Acoustic Cloaking,” New J. Phys. 9, 45 (2007).
S. A. Cummer, B. I. Popa, D. Schurig, et al., “Scattering Theory Derivation of a 3D Acoustic Cloaking Shell,” Phys. Rev. Letters 100.024301 (2008).
V. G. Romanov, “The Inverse Diffraction Problem for Acoustic Equations,” Dokl. Akad. Nauk, Ross. Akad. Nauk 431(3), 319–322 (2010) [Dokl. Math. 81 (2), 238–240 (2010)].
G. V. Alekseev and V. G. Romanov, “One Class of Nonscattering Acoustic Shells for a Model of Anisotropic Acoustics,” Sibirsk. Zh. Industr. Mat. 14(2), 15–20 (2011) [J. Appl. Indust. Math. 6 (1), 1–5 (2012)].
A. E. Dubinov and L. A. Mytareva, “Masking the Material Bodies by the Wave Flow Method,” Uspekhi Fiz. Nauk 180(5), 475–501 (2010) [Phys.-Uspekhi 53 (5), 455–479 (2010)].
G. V. Alekseev and R. V. Brizitskii, “Theoretical Analysis of Boundary Control Extremal Problems for Maxwell’s Equations,” Sibirsk. Zh. Industr. Mat. 14(1), 3–16 (2011) [J. Appl. Indust. Math. 5 (4), 478–490 (2011)].
G. V. Alekseev, R. V. Brizitskii, and V. G. Romanov, “Stability Estimates for Solutions of Boundary Control Problems for Maxwell’s Equations With Mixed Boundary Conditions,” Dokl. Akad. Nauk, Ross. Akad. Nauk 447(1), 7–12 (2012) [Dokl.Math. 86 (3), 733–737 (2012)].
G. V. Alekseev, “Optimization in Problems of Masking Material Bodies by the Wave Flow Method,” Dokl. Akad. Nauk, Ross. Akad. Nauk 449(6), 652–656 (2013).
G. V. Alekseev, “Cloaking via Impedance Boundary Condition for the 2-D Helmholtz Equation,” Appl. Analysis (2013), DOI: 10.1080/00036811.2013.768340.
L. Beilina and M. V. Klibanov, “A Globally Convergent Numerical Method for a Coefficient Inverse Problem,” SIAM J. Sci. Comput. 31, 478–509 (2008).
L. Beilina and M. V. Klibanov, “A New Approximate Mathematical Model for Global Convergence for a Coefficient Inverse Problem With Backscattering Data,” J. Inverse Ill-Posed Problems 20, 513–565 (2012).
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer, Berlin, 2013).
G. V. Alekseev, “Control Problems for Stationary Equations of Magnetohydrodynamics,” Dokl. Akad. Nauk, Ross. Akad. Nauk 395(3), 322–325 (2004) [Dokl.Math. 69 (2), 310–313 (2004)].
A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems (Nauka, Moscow, 1974; North-Holland, Amsterdam, 1979).