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For each example, we provide a model formulation that can be solved with standard software within reasonable time.
Usually, this is too high-dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic Monte Carlo method.
Here, it is shown how a few nonlinear ordinary first order differential equations, which may be solved with standard integration methods, can be used as a close approximation for both dynamic responses and steady state solutions.
The above optimization problem can be solved with standard optimization software, i.e., second-order cone programming (SOCP).
The variably saturated flow equation can be solved with standard retention functions or multimodal relative permeability curves for unsaturated flow in porous and fractured systems.
When the vector x is real, the LASSO problem (2), or its equivalent formulation (3), can be solved with standard quadratic programming techniques [15].
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A mixed integer linear programming formulation is developed which is solved with standard Branch and Bound techniques.
Each of the two integral equations associated with the double inversion scheme, Eq. (2) for finding \ \Delta \epsilon \prime\prime\), and Eq. (4) for finding Δf(E), were solved with standard discretization and regularization techniques.
To gain insight into the complexity of the problem at hand, an extensive computational study is performed with randomly generated instances that are solved with standard mathematical optimization software.
This system can then easily be solved with a standard numerical scheme.
The resulting dynamic nonlinear programs include equality constraints that constitute a low-order system of coupled ordinary differential equations and algebraic equations, which can then be solved with combination of standard temporal discretization and nonlinear programming techniques.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com