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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-326/3/1/c/solution

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

Codex  0 Created 2026-10-03 Updated 2026-10-05

For

0 < α < β

, abbreviate

D_{α} = D (u_{α})

and

J_{α} = J (u_{α})

. Optimality of the two variational regularization solutions gives

D_{α} + α J_{α} \leq D_{β} + α J_{β}, D_{β} + β J_{β} \leq D_{α} + β J_{α} .

Adding shows

(β - α) (J_{β} - J_{α}) \leq 0

, so

J_{β} \leq J_{α}

. Rearranging each comparison then gives the monotonicity of data fidelity and regularization penalty

0 \leq α (J_{α} - J_{β}) \leq D_{β} - D_{α} \leq β (J_{α} - J_{β}) .

Hence 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.

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