# Space-Time Methods for Acoustic Waves with Applications to Full Waveform Inversion

Ernesti, Johannes

##### Abstract (englisch):
Classically, wave equations are considered as evolution equations where the derivative with respect to time is treated in a stronger way than the spatial differential operators. This results in an ordinary differential equation (ODE) with values in a function space, e.g. in a Hilbert space, with respect to the spatial variable.
For instance, acoustic waves in a spatial domain $\Omega \subset \mathbb{R}^d$ for a given right-hand side $\mathbf b$ can be considered in terms of the following ODE
\begin{equation*}
\partial_t \mathbf y = A\mathbf y + \mathbf b\quad \text{ in }[0,T]\,,\quad
\mathbf y(0) = \mathbf 0\,,
where the solution $\mathbf y = (p, \mathbf v)$ is an element of the space $\mathrm C^0\big(0,T; \mathcal D(A)\big) \cap \mathrm C^1\big(0,T; \mathrm L_2(\Omega)\big)$ with $\mathcal D(A) \subset \mathrm H^1(\Omega) \times H(\operatorname{div}, \Omega)$. In order to analyze this ODE, space and time are treated separately and hence tools for partial differential equations are used in space and tools for ODEs are used in time. ... mehr