Forward maps, identifiability, and ill-posedness

Write F:D→Y and y=F(x†). The object x† may be a coefficient field, image, source term, or finite parameter vector. The data space and its norm specify what small observational error means. Choosing a stronger object norm can make inversion unstable even when a weaker norm would permit continuity. Constraints such as positivity or compact support are part of the admissible class, rather than conclusions established by a reconstruction program. They can alter both uniqueness and stability.

Hadamard well-posedness asks for existence, uniqueness, and continuous dependence of the solution on the data in the specified topologies. A noisy datum need not lie in the exact range, so even a uniquely solvable noiseless equation can become inconsistent. Least squares changes the question to approximation of the observed data; it does not automatically make the minimizer unique or stable. A small residual only says that the chosen forward model reproduces the measurements closely, not that the object is close to truth.

For a bounded linear map A, injectivity means its nullspace is zero. If h lies in the nullspace, x and x+h give exactly the same data whenever both are admissible. No reduction of measurement noise can separate that pair using A alone. A penalty can select one representative, but that selection uses an additional convention or prior. Adding an independent measurement may remove the nullspace; merely running the same algorithm longer cannot create the missing observation.

A small but nonzero singular value gives a different situation. The corresponding object direction is theoretically identifiable, but its effect on the data is tiny. Dividing by that singular value amplifies noise. An invertible finite matrix has a continuous inverse, so its problem is formally well posed, yet the stability constant can be impractically large. For compact operators with infinite-dimensional range, singular values may tend to zero and the inverse on the range is unbounded. Finite discretizations then reveal worsening conditioning as the resolution increases.

The noise bound should account for errors in the measurements and, when appropriate, discrepancy between the assumed forward model and the physical process. If the actual data are produced by a different operator, forcing the residual below the measurement noise alone can fit model error. Calibration errors, omitted physics, and discretization error can dominate at precisely the level where an inversion appears most precise. A guarantee conditional on a correct forward map should be reported with that condition attached.

Experimental design changes what information the map carries. A second viewing angle, a different excitation, or an additional sensor can rotate or shrink its nullspace and improve its weakest singular directions. This is a different intervention from regularization, which changes how existing information is interpreted. Constraints can also restore identifiability on a restricted class without making the unrestricted map injective. A sound result therefore states both the observation scheme and the class on which uniqueness or a stability estimate has actually been proved.