Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 326 / 3 / 1 / c

For $0<\alpha <\beta$, abbreviate $D_{\alpha }=D(u_{\alpha })$ and $J_{\alpha }=J(u_{\alpha })$. Optimality of the two variational regularization solutions givesAdding shows $(\beta -\alpha )(J_{\beta }-J_{\alpha })\le 0$, so $J_{\beta }\le J_{\alpha }$. Rearranging each comparison then gives the monotonicity of data fidelity and regularization penaltyHence the regulariser value is non-increasing and the data misfit is non-decreasing with the parameter. These inequalities hold for any choices of minimizers at the two distinct parameters, including when uniqueness fails.