 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-326/2/2/c/solution

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

With exact data

f = K u^{†}

, the iteration error obeys

u^{(k)} - u^{†} = - (I + τ K^{*} K)^{- k} u^{†} .

u^{†} = K^{*} v

∥ u^{(k)} - u^{†} ∥^{2} = \sum_{j} \frac{σ _{j}^{2}}{( 1 + τ σ _{j}^{2} ) ^{2 k}} ∣ ⟨ v, w_{j} ⟩ ∣^{2} .

(1 + τ σ^{2})^{k} \geq 1 + k τ σ^{2}

. Also

1 + k τ σ^{2} \geq 2 k τ σ

. Therefore

\frac{σ}{( 1 + τ σ ^{2} ) ^{k}} \leq \frac{1}{2 k τ} .

Summing the squared coefficients and using Bessel inequality yields

∥ u^{(k)} - u^{†} ∥ \leq \frac{∥ v ∥}{2 k τ} = O (k^{- 1/2}) .

This is an a priori error rate from the source condition for quadratic regularization, with constant independent of

k

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