Linear Least Squares Sample Clauses

Linear Least Squares. Linear systems that arise from large-scale inverse problems are typically writ- ten as b = Ax true + ε , (1.1) where b ∈ Rm is a known (measured data) vector, A ∈ Rm×n is a matrix describing the forward model, and x true ∈ Rn represents the true solution. The vector ε ∈ Rm represents unknown perturbations in the data (such as noise). Given A and b, the aim is to compute an approximation of x true. We assume that the perturbations are independent and identically distributed with zero mean. With a Gaussian noise model, it is appropriate to consider the least squares formulation: min ||Ax − b||2 . (1.2) Inverse problems of this form arise in many important applications, including image reconstruction, image deblurring, geophysics, parameter identification and inverse scattering. Note that we have not included regularization in (1.2); we discuss this in detail in Section 1.2. Furthermore, for any norms not specified in this dissertation, we assume the 2-norm.
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