Nonlinear least-squares inversion

Nonlinear least-squares inversion by the conjugate gradient method This method uses the same ideas as the regularized conjugate gradient method ... 

Exact and Approximate Nonlinear Least Squares Inversion of Dielectric Relaxation Spectra, J. Chem. Phys. 102, 6241-6250. 

Since this monograph is devoted only to the conception of mathematical models, the inverse problem of estimation is not fully detailed. Nevertheless, estimating parameters of the models is crucial for verification and applications. Any parameter in a deterministic model can be sensibly estimated from time-series data only by embedding the model in a statistical framework. It is usually performed by assuming that instead of exact measurements on concentration, we have these values blurred by observation errors that are independent and normally distributed. The parameters in the deterministic formulation are estimated by nonlinear least-squares or maximum likelihood methods. 

The steepest descent method for nonlinear regularized least-squares inversion To solve the problem of minimization of the parametric functional using the steepest descent method, let us calculate the first variation of P (m,d), assuming that the operator A(m) is differentiable, so that... 

Thus, the Newton algorithm for the nonlinear regularized least-squares inversion can be expressed by the formula... 

A weighted three parameter nonlinear least squares analysis of the data arising from either inversion recovery or saturation recovery sequences yield (F/E), D, and E. The constant D depends upon the choice of the experiment that has been selected to follow the time evolution of the magnetization. For an inversion recovery experiment, D is given as -I -I +F/E at t = 0. 

Tarantola, A., 1987, Inverse Problem Theory, Elsevier Tarantola, A., Valette, B., 1982, Generalized nonlinear inverse problems solved using the least squares criterion. Reviews of Geophysics and Space Physics 20, 219... 

Several of the procedures for deriving structural parameters from moments of inertia make use of the method of least squares. Since the relation between moments of inertia and Cartesian coordinates or internal coordinates is nonlinear, an iterative least squares procedure must be used.18 In this procedure an initial estimate of the structural parameters is made and derivatives of the n moments of inertia with respect to each of the k coordinates are calculated based on this estimate. These derivatives make up a matrix D with n rows and k columns. We then define a vector X to be the changes in the k coordinates and a vector B to be the differences between the experimental moments and the calculated moments. We also define a weight matrix W to be the inverse of the ma-... 

To take account of interactions between individual components (association, nonlinearities), calibration using multivariate data analysis is often also carried out with mixtures rather than pure substances. Despite this fact, limitations to this method of assessment are encountered quickly. Therefore, the so-called inverse method using the g-matrix is employed, and either principal component regression (PCR) or the partial least squares (PLS) method is used [6, [114], [116]. In both methods, calibration is carried out not with pure substances, but with various mixtures, which must cover the expected concentration range of all components. Within limits, this can allow for non-linearities ... 

To reduce the requirements of long experimental data records and improve the kernel estimation accuracy, least-squares methods can be used to estimate the discrete kernel values of a finite-order Volterra model, especially when kernel expansions are employed to reduce the number of free parameters (Watanabe Stark 1975). Least-squares solution of this inverse problem requires inversion of a square matrix with dimensions [(M -I- Q -I- 1) /((M -I- 1) Q )]. where M is the kernel memory or the number of basis functions used for kernel expansion (typically a much smaller number) and Q is the (maximum) nonlinear order of the Volterra model. The use of QR decomposition for this purpose offers... 

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