Optimal Control of the Principal Coefficient in a Scalar Wave Equation
- PDF / 1,453,518 Bytes
- 33 Pages / 439.37 x 666.142 pts Page_size
- 1 Downloads / 184 Views
Optimal Control of the Principal Coefficient in a Scalar Wave Equation Christian Clason1
· Karl Kunisch2,3 · Philip Trautmann2
Accepted: 12 November 2020 © The Author(s) 2020
Abstract We consider optimal control of the scalar wave equation where the control enters as a coefficient in the principal part. Adding a total variation penalty allows showing existence of optimal controls, which requires continuity results for the coefficientto-solution mapping for discontinuous coefficients. We additionally consider a socalled multi-bang penalty that promotes controls taking on values pointwise almost everywhere from a specified discrete set. Under additional assumptions on the data, we derive an improved regularity result for the state, leading to optimality conditions that can be interpreted in an appropriate pointwise fashion. The numerical solution makes use of a stabilized finite element method and a nonlinear primal–dual proximal splitting algorithm. Keywords Optimal control · Wave equation · Total variation regularization · Primal-dual splitting Mathematics Subject Classification 49A22 · 49K20 · 49M29
1 Introduction This work is concerned with an optimal control problem for the scalar wave equation where the control enters as the spatially varying coefficient in the principal part.
B
Christian Clason [email protected] Karl Kunisch [email protected] Philip Trautmann [email protected]
1
Faculty of Mathematics, University Duisburg-Essen, 45117 Essen, Germany
2
Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria
3
Radon Institute, Austrian Academy of Sciences, Linz, Austria
123
Applied Mathematics & Optimization
Informally, we consider the problem ⎧ 1 ⎪ ⎪ By − yd 2O + R(u) ⎪ ⎨ min u,y 2 ⎪ s.t. ytt − div(u∇ y) = f , y(0) = y0 , ∂t y(0) = y1 , ⎪ ⎪ ⎩ u ≤ u ≤ u almost everywhere (a.e.),
(1)
where yd is a given (desired or observed) state, B is a bounded linear observation operator mapping to the observation space O, R is a regularization term, 0 < u < u are constants, and f , y0 , and y1 (as well as boundary conditions) are given suitably. A precise statement is deferred to Sect. 2. Such problems occur, e.g., in acoustic tomography for medical imaging [1] and non-destructive testing [2] as well as in seismic inversion [3]. In the latter, the goal is the determination of a “velocity model” (as described by the coefficient u) of the underground in a region of interest from recordings (“seismograms”, modeled by yd ) of reflected pressure waves generated by sources on or near the surface (entering the equation via f , y0 , y1 , or inhomogeneous boundary conditions). If the region contains multiple different materials like rock, oil, and gas, the velocity model changes rapidly or may even have jumps between material interfaces. In the stationary case, the question of existence of solutions to problem (1) under only pointwise constraints and regularization has received a tremendous amount of attention. However, it was answere
Data Loading...
