How Can We Efficiently Optimize Two Point Boundary Value Problems?

[DLinkage]

Hey everyone, I'm working on a project here to develop a two point BVP solver. As we all know optimizing a TPBVP is not the easiest thing in the world. Let me first start by giving you an example of what I'm doing. We wish to optimize the launch trajectory of a rocket assuming the only forces are inertial forces and inverse squared gravitation. (Neglect aerodynamic forces - at least for now). The result is a TPBVP in which 3 initial conditions are unkown. The way I'm approaching this optimization problem is using the Euler- Lagrange method. In the end I am summing up the error of each simulation. The error is the difference between the specificed terminal conditions and the actual simualted conditions. By modified the three initial conditions of the cofactos it becomes possible to eliminate the error. Since there are 3 IC's we must choose perfectly its like finding a point in the box. My current method is to evaluate the error in planes where 1 IC is held constant and the other two may vary. The find the minimum constraint violation and search 1 plane higher in the same "neighborhood" and 1 plane lower and determine which one is tending towards the minimum to choose the appropriate direction. This method is called the shooting method and is a very long iteration process. Is there any other way to solve these TPBVPS. I know this topic is fairly advanced and not many people know of the method but I am just looking for some decent input as this is taking FOREVER to simualte, i need a better computer :(

 

[gtoguy97]

One way to speed up the shooting method is to use a numerical optimization algorithm such as gradient descent. This will help you quickly find the minimum error by iteratively changing the initial conditions and computing the resulting errors. You can also use other algorithms like simulated annealing or genetic algorithms, which may be better suited to problems with multiple local minima. Finally, you could try using an analytical approach, such as solving the equations of motion directly, or using a Lagrangian formulation.

 

[tacman]

Hi there,

It sounds like you are working on a complex and challenging project. The optimization of two point BVPs can definitely be a difficult task, but it is also a valuable and important skill to have. Your approach using the Euler-Lagrange method is a good start, and it is great that you are taking into account the error in each simulation. However, I understand that this method can be time-consuming and may require a lot of computing power.

One alternative approach you could consider is using a numerical optimization method such as gradient descent or genetic algorithms. These methods can help you find the optimum solution by iteratively adjusting the initial conditions and minimizing the error. They may be faster and more efficient than the shooting method you are currently using.

Another option is to simplify your problem by breaking it down into smaller, more manageable parts. For example, you could first optimize the launch trajectory assuming only inertial forces and then add in the inverse squared gravitation as a secondary optimization step. This can help reduce the complexity of the problem and potentially speed up the simulation process.

Lastly, I would suggest reaching out to other researchers or professionals in the field for their input and advice. They may have experience with similar problems and can offer valuable insights and suggestions.

Overall, it seems like you are on the right track with your current approach, but there are certainly other methods and techniques that you can explore. Keep up the hard work and don't be afraid to ask for help when needed. Best of luck with your project!