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Full-Text Articles in Computer Engineering
Why Curvature In L-Curve: Combining Soft Constraints, Uram Anibal Sosa Aguirre, Martine Ceberio, Vladik Kreinovich
Why Curvature In L-Curve: Combining Soft Constraints, Uram Anibal Sosa Aguirre, Martine Ceberio, Vladik Kreinovich
Departmental Technical Reports (CS)
In solving inverse problems, one of the successful methods of determining the appropriate value of the regularization parameter is the L-curve method of combining the corresponding soft constraints, when we plot the curve describing the dependence of the logarithm $x$ of the mean square difference on the logarithm $y$ of the mean square non-smoothness, and select a point on this curve at which the curvature is the largest. This method is empirically successful, but from the theoretical viewpoint, it is not clear why we should use curvature and not some other criterion. In this paper, we show that reasonable scale-invariance …
Constraint-Related Reinterpretation Of Fundamental Physical Equations Can Serve As A Built-In Regularization, Vladik Kreinovich, Juan Ferret, Martine Ceberio
Constraint-Related Reinterpretation Of Fundamental Physical Equations Can Serve As A Built-In Regularization, Vladik Kreinovich, Juan Ferret, Martine Ceberio
Departmental Technical Reports (CS)
Many traditional physical problems are known to be ill-defined: a tiny change in the initial condition can lead to drastic changes in the resulting solutions. To solve this problem, practitioners regularize these problem, i.e., impose explicit constraints on possible solutions (e.g., constraints on the squares of gradients). Applying the Lagrange multiplier techniques to the corresponding constrained optimization problems is equivalent to adding terms proportional to squares of gradients to the corresponding optimized functionals. It turns out that many optimized functionals of fundamental physics already have such squares-of-gradients terms. We therefore propose to re-interpret these equations -- by claiming that they …