A generalization of Tikhonov's regularization of zero and first order

Structural health monitoring and damage detection using a sub-domain inverse method

“…Although in this study only time data are used, from discrete distributed sensors, it is possible to extend the SDIM to handle spatial distributed data from imaging sensors. Preliminary results from a force identification analysis using a wavelet deconvolution scheme with moiré images taken at a discrete number of times [20] demonstrate that the SDIM can be adapted for damage identification using different types of time or spatial sensors. In this particular case, Toeplitz-like matrices are obtained and the inverse problem can be solved by adapting existing fast or super-fast solvers [21][22][23][24].…”

“…Here, the objective is to find the set of forces {g} that minimize the error functional, where [W] could be a general weighting array on the data and the summation is over all the time steps N . The second term in the summation is a Tikhonov [20] regularization term (with 0 < λ < ∞) which is included to get a robust answer. There is a trade-off relation for the regularization term λ in which a very small value leads to a small regularization effect and the solution tends to a noisy behaviour due to the ill conditioning; on the other hand, large values produce a smoothing effect.…”

“…Adequate selection of this parameter, not only gives a 'good' solution, but also has a filtering effect on noisy data. Finding a 'best' value for λ can be done according to some criterion, which can range from fairly objective to entirely subjective [20]. An algorithm adapted from Bellman's ideas in dynamic programming [21] for time invariant systems provides a time recursive solution for the minimization problem [16,22].…”

Reconstructing dynamic events from time‐limited spatially distributed data

“…This is discussed in more detail later. The regularization method we use is generally called Tikhonov [20][21][22] regularization. Typically, the functional B involves some measures of smoothness that derive from ÿrst or higher derivatives.…”