On Ill‐Conditioning of Descent Methods to Recover Initial Data for the Burgers Equation With Small Viscosity

ABSTRACT The identification of the initial condition of a parabolic equation given terminal data is a problem arising in several applications. In this work, we consider the problem for the semilinear parabolic viscous Burgers equation in the vanishing viscosity case. The limiting equation is the nonlinear hyperbolic Burgers equation, for which there is non‐uniqueness backward in time. In both cases, solutions are sought by formulating the identification problem as the minimization of a least squares functional, and gradient descent methods are used for computation. Following this approach, we construct the first and second variations of the functional by the adjoint state approach. We prove that the second variation is continuous with respect to viscosity. Consequently, gradient descent methods of approximation inherit the ill‐conditioning of the nonlinear hyperbolic limit.