Well-posedness conditions in stochastic inversion problems
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Stochastic inversion problems arise when it is wanted to estimate the probability distribution of a stochastic input from indirect observable and noisy information and the limited knowledge of an operator that connects the inputs to the observable output. While such problems are characterized by strong identifiability conditions, well-posedness conditions of "signal over noise" nature should be respected priori to collect observations. In addition to well-known Hadamard' well-posedness condition, a new one is established based on the predictive transmission of input uncertainty to output, which can be interpreted as the result provided by a sensitivity analysis if the problem were solved. This new condition should take part within the input model itself, which adds a constraint in established frequentist or Bayesian methodologies of stochastic inversion. While this article mainly deals with linearizable operators, the lack of constrast typical of linear problems suggest that the proposed condition should be used in more general settings, provided the operator owns diffferentiability properties.
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