Introduction

Inverse problems are concerned with the determination of causes by knowing the consequences. In modern language, they are concerned with data assimilation that seeks to optimally combine theory with observations. In its abstract formulation, an inverse problem can be described as follows. Let (X, ∥⋅∥X) denote the space of target objects (usually consisting of quantities or geometries) and (Y, ∥⋅∥Y) denote the space of observation data. Let ℱ $$\mathcal {F}$$ denote the model which sends a target object Ω ∈ X to a set of data y ⊂ Y and is determined by the underlying physical process. In its typical form, an inverse problem is given by the following operator equation: ℱ ( Ω ) = y , Ω ∈ X , y ⊂ Y . $$\displaystyle \mathcal {F}(\varOmega )=\boldsymbol y,\quad \varOmega \in X,\ \ \boldsymbol y\subset Y. $$ That is, the inverse problem is to determine Ω by knowledge of ℱ $$\mathcal {F}$$ and y. In (1.1.1), ℱ $$\mathcal {F}$$ is more than often described by a certain PDE (partial differential equation) system and Ω is a certain missing part of ℱ $$\mathcal {F}$$ , say, e.g., the coefficient(s) of the PDE, and hence, the inverse problem is generally nonlinear. Moreover, the inverse problem is usually ill-posed in the sense that a small variation on y may cause a large variation on Ω or even the nonexistence of a true Ω. This shall be more evident in our subsequent discussion. Hence, the mathematical study on inverse problems is intriguing and challenging. We broadly summarize the following six categories.