Methods of Solving Ill-Posed Problems
By Suresh B. Srinivasamurthy. 2012
313 citations recorded by the source corpus.
Many physical problems can be formulated as operator equations of the form Au = f. If these operator equations are ill-posed, we then resort to finding the approximate solutions numerically. Ill-posed problems can be found in the fields of mathematical analysis, mathematical physics, geophysics, medicine, tomography, technology and ecology. The theory of ill-posed problems was developed in the 1960's by several mathematicians, mostly Soviet and American. In this report we review the methods of solving ill-posed problems and recent developments in this field. We review the variational regularization method, the method of quasi-solution, iterative regularization method and the dynamical systems method. We focus mainly on the dynamical systems method as it is found that the dynamical systems method is more efficient than the regularization procedure.
Subfields: Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/3c2b5570e41e5a9c713ac9d193f4dbcb11cff332
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In Mathematical Physics
Citations in the corpus
- A systematic review on overfitting control in shallow and deep neural networks (2021, background): “When these systems are ill-posed (Tikhonov and Arsenin 1977), the solution is unstable, and the corresponding learning model becomes overfitted (Bejani and Ghatee 2020b).”
- Sparse and Stable Portfolio Selection With Parameter Uncertainty (2015, unknown): “If one define X = √ γ ̂1/2, y = 1√ γ ̂−1/2μ̂0, and = √ 2λ̂1/2g , this is equivalent to a ridge regression or Tikhonov regularization problem (Tikhonov and Arsenin 1979): ŵ = argmin w {||y − Xw||2 + || w||2}, where the solution is given by ŵ = [XT X + T ]−1XT y = (γ ̂ + 2λ̂g)−1μ̂0.”