# Convergent numerical methods for parabolic equations with reversed time via a new Carleman estimate

## TL;DR

This paper introduces a new Carleman estimate for second-order parabolic equations with reversed time, enabling stable numerical methods over arbitrary time intervals and extending to quasilinear cases with proven convergence.

## Contribution

It develops a novel Carleman estimate applicable to any time interval, facilitating stable quasi-reversibility methods and global convergence results for reversed time parabolic equations.

## Key findings

- Established a stability estimate for reversed time parabolic equations.
- Proposed a quasi-reversibility numerical method with proven convergence.
- Constructed a globally convex Tikhonov-like functional ensuring convergence.

## Abstract

The key tool of this paper is a new Carleman estimate for an arbitrary parabolic operator of the second order for the case of reversed time data. This estimate works on an arbitrary time interval. On the other hand, the previously known Carleman estimate for the reversed time case works only on a sufficiently small time interval. First, a stability estimate is proven. Next, the quasi-reversibility numerical method is proposed for an arbitrary time interval for the linear case. This is unlike a sufficiently small time interval in the previous work. The convergence rate for the quasi-reversibility method is established. Finally, the quasilinear parabolic equation with reversed time is considered. A weighted globally strictly convex Tikhonov-like functional is constructed. The weight is the Carleman Weight Function which is involved in that Carleman estimate. The global convergence of the gradient projection method to the exact solution is proved for this functional.