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With respect to actually fitting the models, convergence can readily be achieved either by choosing to include a set of parameters that is known to be identifiable or by using penalization techniques as proposed by Li et al. [2009].
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29 As an example, Hirata et al used penalization to define a disease activity score using serum biomarkers.
With this connection to least squares regression it is straightforward to use penalization terms to impose sparsity on v ˜. (4) (u, v ˜ ^ ) = arg min u, v ˜ ‖ X − u v ˜ ^ T ‖ F 2 + λ P (v ˜ ), where P (v ˜ ) is a penalization term that induces sparsity on v ˜ and λ is a tuning parameter that determines the strength of the penalization.
The straightforward extension of penalizing momentum and energy equations using Brinkman penalization with respective normalized viscous, η, and thermal, ηT, permeabilities produces unsatisfactory results, mostly due to nonphysical wave transmissions into obstacles, resulting in considerable energy and mass losses in reflected waves.
A new multi-objective function is formulated to reduce grayscales since using a penalization parameter for physical property interpolation is ineffective in electromagnetic problems.
We also present an analytical solution of Stokes flow calculated using the penalization which illustrates the error and continuity properties of the approach.
The incompressible Navier Stokes equations are discretized in space onto a fixed cartesian mesh and the displacement of deformable objects through the fluid is taken into account using a penalization method.
Membrane incompressibility is handled by using two penalization parameters per suspended entity: one for deviation of the global area from prescribed value and another for the sum of squares of local strains defined on each vertex.
In [16] the obstacle problem is considered for a linear elliptic operator perturbed by a nonlinearity having at most a quadratic growth on the gradient, and the existence of weak solutions of the variational inequality is obtained by using a penalization method, Schauder's fixed point theorem and a priori estimates.
This method is an extension of PLS regression using a penalization technique.
One such algorithm uses the maximum likelihood criterion adjusted using a penalization term for taking into account model complexity to define breakpoints [ 28].
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