Logarithmic convexity of evolution equations and application to inverse problems

TL;DR

This paper reviews logarithmic convexity properties of evolution equations and their applications to inverse problems, including classical, analytic, and fractional cases, with explicit estimates and practical examples.

Contribution

It provides a unified analysis of logarithmic convexity for various evolution equations and extends results to fractional and non-symmetric cases with explicit estimates.

Findings

Explicit estimate for analytic semigroups used in inverse problems

Application to Ornstein-Uhlenbeck equations demonstrated

Extension of results to time-fractional evolution equations

Abstract

We review some results on the logarithmic convexity for evolution equations, a well-known method in inverse and ill-posed problems. We start with the classical case of self-adjoint operators. Then, we analyze the case of analytic semigroups. In this general case, we give an explicit estimate, which will be used to study inverse problems for initial data recovery. We illustrate our abstract result by an application to the Ornstein-Uhlenbeck equations. We discuss both analytic and non-analytic semigroups. We conclude with recent results for the time-fractional evolution equations with the Caputo derivative of order $0<\alpha <1$. We start with symmetric evolution equations. Then, we show that the results extend to the non-symmetric case for diffusion equations, provided that a gradient vector field generates the drift coefficient. Finally, some open problems will be mentioned.