Inverse Problems and Parameter Estimation
Infer unknown model inputs from observed outputs, a task that is often ill-posed and sensitive to noise.
Working Backwards
A forward problem computes outputs from known inputs. An inverse problem does the reverse: it infers unknown inputs from observed outputs. Estimating a material property from measured behavior, or a source distribution from measured fields, are inverse problems. They are central to calibration and to interpreting experiments, and they are notoriously harder than the forward problems they invert.
Ill-Posedness
- A solution may not exist, if no inputs reproduce the data exactly.
- A solution may not be unique, if different inputs produce the same output.
- A solution may be unstable, if tiny changes in the data cause large changes in the inferred inputs.
Regularization
The instability of inverse problems is tamed by regularization: adding a preference for solutions that are smooth, small, or otherwise plausible, which suppresses wild input estimates that fit noise. The regularization strength trades off fidelity to the data against plausibility of the solution, and choosing it is a judgment that must be reported, because different choices give different answers.
The Bayesian View
Bayesian inference provides a principled framework for inverse problems: the prior acts as regularization, encoding what plausible inputs look like, and the posterior quantifies how well the data constrains the inputs, including non-uniqueness as a spread or multiple modes in the posterior. This makes explicit what a single regularized point estimate hides, namely how much the answer is driven by data versus by the assumed prior.
In verification and validation, inverse problems appear whenever model parameters are estimated from experiments. The danger is treating a well-fitting but non-unique or noise-sensitive inversion as a firm measurement. Honest practice reports the inferred inputs with their uncertainty and non-uniqueness, and holds out independent data to test whether the inferred parameters predict, rather than merely fit.