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First, by using re-sampling algorithm encoded derivative regularization, the prior model is translated into a pseudo training dataset, and then the dataset is trained by the Gaussian processes.
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Based on Kuwatani et al. (2014a), we used first-derivative regularization for the smoothness constraint.
Alternative Investment and Derivatives Regulation.
Hence on every iteration, the contrastive divergence algorithm is applied followed by one step of gradient descent using the derivative of the two regularization terms.
The Caputo fractional derivative is a sort of regularization in the time origin for the Riemann-Liouville fractional derivative.
Because the regularization penalizes the derivative and not the actual value of the displacement fields, it does not penalize an arbitrary 3D displacement vector that is added to all displacements in each displacement field.
We used Tikhonov regularization of the derivative (for smoothness) using the added penalty term: γ ∑ i (Δ ψ i ) 2, where i indexes the value of L for which ψ was estimated and γ = 1/20 was determined through ad hoc methods.
Derivatives obtained using Total-Variation Regularization method suppressed the noise and provide the better estimates in Fig. 2c.
The second baseline is a modification of the ITK-demon algorithm using regularization of the second derivative of the flow instead of the first order, which has been shown to provide better convergence properties for certain types of volumes (Fischer and Modersitzki, 2004).
This finds regularized solutions (Tikhonov Regularization) to the two-dimensional inverse problem for MT data using the method of nonlinear conjugate gradients (NLCG) to minimize an objective function that penalizes data residuals and second spatial derivatives of resistivity.
Regularization of the third time derivative alone leads to highly oscillating field behavior.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com