I'd like to konw the difference of the SVD method and the Newton method in solving geophysical problems. Whether they are both a tool for optimizing the overall objective function, but it seems that they are two different classes of methods. I am confused about their definition and how to use them
The SVD method and the Newton method are both numerical methods for solving optimization problems, but they have different approaches and applications. The SVD method stands for singular value decomposition, which is a technique for decomposing a matrix into three smaller matrices that reveal its properties. The SVD method can be used for least squares problems, which aim to find the best fit for a set of data points by minimizing the sum of squared errors. The SVD method can also be used for other purposes, such as dimensionality reduction, data compression, noise reduction, and image processing.
The Newton method is an iterative method for finding the roots of a nonlinear function, that is, the values of the variable that make the function zero. The Newton method uses the first and second derivatives of the function to approximate its behavior near a root and update the estimate in each iteration. The Newton method can be extended to multivariate functions and optimization problems by using the gradient and the Hessian matrix, which are generalizations of the first and second derivatives. The Newton method can be used for solving nonlinear systems of equations, finding local minima or maxima of a function, and performing nonlinear regression.
In geophysics, both methods can be used for modeling and inversion problems that involve complex and nonlinear phenomena. For example, the SVD method can be used for separating seismic diffractions from reflections by measuring the contribution of every singular value. The Newton method can be used for simulating geodynamical models of Stokes flow with viscoplastic rheologies by linearizing the stress-velocity equations. Both methods have advantages and disadvantages depending on the problem characteristics, such as the size, sparsity, smoothness, and convexity of the matrices or functions involved.
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