Outgoing from [1] and [2] we analyze variational regularization with $L^1$ data fidelity. We investigate discrete models with regular data in the sense that the tails decay subexponentiallly. Therefore, error bounds are provided and numerical simulations of convergence rates are presented.

[1] T. Hohage, F. Werner, Convergence rates for inverse problems with impulsive noise, SIAM J. Numer. Anal., 52: 1203-1221, 2014.

[2] C.König, F. Werner, T. Hohage, Convergence rates for exponentially ill-posed inverse problems with impulsive noise, SIAM J. Numer. Anal., 54: 341-360, 2016.

Three Coefficient Inverse Problems (CIP) will be considered. Respectively, three versions of the globally convergent convexification numerical method will be presented. Global convergence theorems will be outlined and numerical results will be presented. Results outlined below are the first ones for each considered CIP. These CIPs are:

1. CIP for the radiative transport equation with euclidian propagation of particles [1].

2. CIP for the Riemannian radiative transport equation. In this case, particles propagate along geodesic lines between their scattering events [2].

3. Travel Time Tomography Problem in the 3-d case [3]. This is a CIP for the eikonal equation in the 3-d case. First numerical results in 3-d for this CIP will be presented,

[1] M.V. Klibanov, J. Li, L.H. Nguyen, Z. Yang, Convexification numerical method for a coefficient inverse problem for the radiative transport equation, SIAM J. Imaging Sciences, 16;35-63, 2023.

[2] M.V. Klibanov, J. Li, L.H. Nguyen, V.G. Romanov, Z. Yang, Convexification numerical method for a coefficient inverse problem for the Riemannian radiative transfer equation, arxiv: 2212.12593, 2022.

[3] M.V. Klibanov, J. Li, W. Zhang, Numerical solution of the 3-D travel time tomography problem, Journal of Computational Physics, 476:111910, 2023. published online https://doi.org/10.1016/j.jcp.2023.111910.

Piezoelectric materials are an essential component for a wide range of electrical devices. Consequently, the range of possible applications for piezoelectric materials is expansive, encompassing, for example, electronic toothbrushes and microphones, as well as ultrasound imaging and sonar devices.

Simplified, the behaviour in the small signal range can be described by a linearly coupled PDE system for mechanical displacement and electrical potential, which can then be extended by a non-linear PDE system to consider piezoelectric material in high signal range. Since many applications require high precision and also due to the transition to lead-free piezoceramics, a consistent and reproducible characterization of the material parameter set is of very high importance to properly determine the material properties, as the material data provided by the manufacturers often deviate significantly from the real data and are difficult and expensive to measure.

Therefore, this talk will focus on the parameter identification problem for the piezoelectric partial differential equation based on a measured and simulated quantity of the sample. Hence, we will derive the forward operator of this inverse problem generally. Then we will consider this inverse problem using regularization techniques based on all-at-once and reduced iterative methods, and further discuss the connection between the adjoint operators of the all-at-once approach and the adjoint differential equations of the reduced approach. Since several applications exhibit nonlinear material behaviour, the all-at-once approach is of particular interest, especially with respect to computational aspects. Thus, modeling, analysis and the solution of these inverse problems in these different settings by fitting simulated data is the main focus. Finally, numerical examples are provided.

Let $\left\{\xi _{k} \right\}_{k=1}^{\infty } ,$ and $\left\{\zeta _{k} \right\}_{k=1}^{\infty } $ be two independent sequences of random variables defined on any probability space $(\Omega ,\, F,P)$, such that the random variables in each sequence are independent, positive and identically distributed. Now we can construct the stochastic process $X_{1} \left(t\right)$ as follows

$X_{1} \left(t\right)=z-t+\sum _{i=0}^{k-1}\zeta _{i} $, if $\sum _{i=0}^{k-1}\xi _{i} \le t<\sum _{i=0}^{k}\xi _{i} $ ,

where $\xi _{0} =\zeta _{0} =0$. The process $X_{1} \left(t\right)$ is called the semi-Markov random walk process with negative drift, positive jumps. Let this process is delayed by a barrier zero: \[X(t)=X_{1} \left(t\right)-\mathop{\inf }\limits_{0\le s\le t} \left\{0,X_{1} (s)\right\}\]

Now, we introduce the random variable $\tau _{0} =\inf \left\{t:\, \, X(t)=0\right\}$. We set $\tau _{0} =\infty $ if $X(t)>0$for all $t$. It is obvious that the random variable $\tau _{0} $ is the time of the first crossing of the process $X(t)$ into the delaying barrier at zero level. $\tau _{0} $ is called the boundary functional of the semi-Markov random walk process with negative drift, positive jumps.

The aim of the present work is to determine the Laplace transform of the conditional distribution of the random variable $\tau _{0} $. Laplace transform of the conditional distribution of the random variable $\tau _{0} $. by \[L(\theta \left|z\right. )=E\left[e^{-\theta \tau _{0} } \left|X(0)=z\right. \right],\, \, \, \, \theta >0,\, \, z\ge 0.\]

Let us denote the conditional distribution of random variable of $\tau _{0} $ and the Laplace transform of the conditional distribution with \[N(t\left|z\right. )=P\left[\tau _{0} >t\left|X(0)=z\right. \right],\] and \[\tilde{N}(\theta \left|z\right. )=\int _{t=0}^{\infty }e^{-\theta t} N(t\left|z\right. )dt ,\] respectively.

Thus, we can easily obtain that \[\tilde{N}(\theta \left|z\right. )=\frac{1-L(\theta \left|z\right. )}{\theta } \] or, equivalently, \[L(\theta \left|z\right. )=1-\theta \tilde{N}(\theta \left|z\right. ).\]

We construct an integral equation for the $\tilde{N}(\theta \left|z\right. )$. In particular, constructed integral equation reduced to the fractional order differential equation in the class of gamma distributions. The fractional derivatives are described in the Riemann-Liouville sense. Finally , Laplace transform of $\tilde{N}(\theta \left|z\right. )$ is obtained in the form of a threefold sum.