The small worlds of exact solutions
We want to solve models exactly - find a closed form solution. But the world of closed form solutions is usually small.
Continuum models must often be discretized in order to obtain a numerical solution. You need them to solve models of material flows, chemical reactors, conservation of energy, macro economics, quant finance…The selection of the numerical scheme is critical for the numerical stability.
You may think of decomposing the problem into subproblems where exact solutions are possible and recompose the results…but this is very ambitious in high-dimensianal (multi-factor) problems
But even proven approaches can create problems in extreme regimes.
Take a Partial Differential Equation (PDE) of the reaction-convection-diffusion type. You'll find Finite Element solvers preferable. But if the PDE is convection dominated, you need some special stabilization techniques to obtain acceptable results from the FE solver.
These decision points are critical and problem dependent.
A problem is well-posed in the senesce of Jaques Hadamard, if mathematical models of physical phenomena have the following properties
- A solution exists
- The solution is unique
- The solution's behavior changes continuously with the initial conditions
Problems that are not well-posed in the sense of Hadamard are ill-posed.
Inverse problems are often ill-posesd. They ask for the conversation of final data into information about the system that has produced them. Noises in data can lead to results that are pure nonsense. (About 1500 people a week are confused with criminals at the world's airports face recognizers)
Calibration of models - identifying its parameters from final data - is an inverse problem.
But there is hope. There are stabilization techniques helping to obtain stable parameters. To apply the correct one is really critical.
When Monte Carlo is the only choice
Solving high dimensional (stochastic) models MC methods may be the only reasonable choice? But (time) constraints may force you to decide using the much better performing, but less random, Quasi Monte Carlo technique or apply a variance reduction techniques.
How to compare this approach with the possibility to transform the stochastic model into a PDE…?
These decisions may influence you model evaluation and simulation regime significantly.
It's hard working in the data salt mines
Its impossible to create a model? Not even a stochastic? A data driven method is the only choice? It's critical to check
- What truth does your data set represent?
- What are raw data?
- What machine learning method fits best for the purpose?
- Do you have enough data samples?
- Does the selected methods generalize enough?
To make your decisions, you need to know, how to perform cross-model validation…and other tests requiring deep knowledge in machine learning…avoiding critical moments.
System or user action?
There are much more critical points related to the problem, the models and their solvers…and there are additional ones arising from the implementation.
In critical moments a system or user action is required.
The two extremes:
A) The system may select all models and methods automatically, correct results... - it may work perfect in a special, predefined world, but produce pure nonsense outside.
B) The system asks users to select and correct them all - it may work well in many cases but they may not be qualified in others and make decisions that accumulate into risky horror.
The innovation will not work, if a reasonable balance between the two is not found. Scenarios across models-methods..."What if" treatment...(subject of the Constructor phase) will help.
Next up…about Solutions