2015/05/11 by Nicolae Cîndea, Cindea, Nicolae, Arnaud Münch +1
Computer Science · Earth and Planetary Sciences · Engineering · Mathematics · Medicine · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Medical Imaging Techniques and Applications #Numerical methods in inverse problems #Optimization and Control (math.OC) #Seismic Imaging and Inversion Techniques #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1505.02566
openalex publication_date 2015/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a direct method allowing to solve numerically inverse type\nproblems for linear hyperbolic equations. We first consider the reconstruction\nof the full solution of the wave equation posed in \Ω\× (0,T) -\n\Ω a bounded subset of \ℝN - from a partial boundary\nobservation. We employ a least-squares technique and minimize the L2-norm of\nthe distance from the observation to any solution. Taking the hyperbolic\nequation as the main constraint of the problem, the optimality conditions are\nreduced to a mixed formulation involving both the state to reconstruct and a\nLagrange multiplier. Under usual geometric optic conditions, we show the\nwell-posedness of this mixed formulation (in particular the inf-sup condition)\nand then introduce a numerical approximation based on space-time finite\nelements discretization. We prove the strong convergence of the approximation\nand then discuss several examples for N=1 and N=2. The problem of the\nreconstruction of both the state and the source term is also addressed.\n