Inverse problems
Objective. Infer hidden parameters from indirect observations.
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Learn: the key idea
Inverse problems infer hidden model parameters from indirect, noisy observations, which is often ill-posed because small data errors can cause large, unstable changes in the inferred parameter. Tikhonov regularisation stabilises this by minimising ‖Ax−b‖² + λ‖x‖² instead of just the data misfit, trading a little bias for much reduced sensitivity to noise, where λ controls the trade-off. For example, estimating a source strength from a noisy measurement by simple division can give wildly different answers for tiny noise changes, whereas adding a regularisation term damps this instability. Choosing λ, often via cross-validation, balances fitting the data against keeping the solution stable.
Worked example
An inverse problem estimates a parameter x from the model 2x = b with regularisation minimising (2x−b)² + λx². For b = 10 and λ = 1, find the regularised estimate of x using the normal equation (4+λ)x = 2b.
- 1Write the regularised normal equation (4+λ)x = 2b for this linear model.
- 2Substitute λ = 1 and b = 10: (4+1)x = 2(10), giving 5x = 20.
- 3Divide both sides by 5: x = 20/5.
- 4Simplify 20/5 = 4, the regularised parameter estimate.
Answer: 4
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Start practisingCommon mistake. A common mistake is solving an ill-posed inverse problem by direct inversion of the forward operator without any regularisation, which amplifies measurement noise into wildly unstable parameter estimates.
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