A BACKWARD PROBLEM FOR THE HOMOGENEOUS FRACTIONAL HELMHOLTZ EQUATION WITH GAUSSIAN WHITE NOISE MEASUREMENT

Abstract

We investigate a backward final value problem for a homogeneous fractional Helmholtz equation on the strip $\left(0,\pi \right)\times \left[0,1\right]$ The final observations are assumed to be corrupted by Gaussian white noise.

Since the solution operator contains exponentially increasing Fourier multipliers, the problem is severely ill-posed in the sense of Hadamard. A Fourier truncation method is proposed to stabilize the reconstruction. Under Sobolev-type smoothness assumptions on the exact final data, and under two different source conditions imposed on the exact solution, we derive convergence estimates for the expectation of the reconstruction error.

In the  $L^{2}\left(0,\pi \right)$ norm, logarithmic-type convergence is obtained under a weaker exponentially weighted condition, whereas a Hölder-type rate is established under a stronger exponential source condition. An additional estimate in the $L^{2}\left(0,\pi \right)$ norm is also proved. The results show that the truncation method provides a simple and effective regularization strategy for the homogeneous fractional Helmholtz backward problem with Gaussian white noise measurements.

2020 Mathematics Subject Classification. 35J05, 35R11, 47A52, 60G15, 35R30