"exponentially ill-posed"

Kind of a harsh-sounding term, don't you think? Context:

The authors consider an inverse heat conduction problem in a quarter
plane. The goal of the specific inverse problem is to determine the
surface heat flux in a body from a measured temperature history at a
fixed location inside the body. The standard example is the sideways
heat equation. This type of inverse problem is known to be
exponentially ill-posed.

Fortunately,

The authors propose a Fourier regularization method and prove order
optimal logarithmic stability estimates. A simple numerical example is
presented to verify the analytical results.

Phew!

ha ha ha

"exponentially ill-posed"

Kind of a harsh-sounding term, don't you think? Context:

The authors consider an inverse heat conduction problem in a quarter
plane. The goal of the specific inverse problem is to determine the
surface heat flux in a body from a measured temperature history at a
fixed location inside the body. The standard example is the sideways
heat equation. This type of inverse problem is known to be
exponentially ill-posed.

Fortunately,

The authors propose a Fourier regularization method and prove order
optimal logarithmic stability estimates. A simple numerical example is
presented to verify the analytical results.

Phew!

ha ha ha