6 votes

What are systematic ways of approximating a non-smooth (non-continuously differentiable) system dynamic to be n-smooth?

Two systematic ways of smoothing a function $h$ would be: 1. Join the piecewise smooth parts of your function using Hermite interpolation so that the derivatives are matched to your satisfaction. 2. ...
6 votes
Accepted

zero terminal value of the adjoint based optimal control

You won't like the answer, but it is in fact "This lies in the right functional-analytic framework" — which you have not given! In particular, you did not specify in which function space you are ...
5 votes
Accepted

Algorithms for radiation treatment planning

This problem is actually more of an optimal control problem for a partial differential equation. As a starting point, I would recommend the following books: F. Tröltzsch, Optimal control of partial ...
4 votes

Discrete-time Algebraic Riccati Equation (DARE) solver in C++

If you want a ten-line solution that is decently fast and stable, you can implement yourself the structured doubling algorithm: set up the coupled iteration $A_0 = A, G_0 = G = BR^{-1}B^T, H_0 = Q$ ...
4 votes

Solving numerically an Optimal Control Problem subject to a conservation law (transport equation)

From your comment, it seems like you might want to do some reading about variational calculus and PDE-constrained optimization. Briefly, let's suppose that the solution $\rho$ of the PDE lives in some ...
3 votes
Accepted

Which optimization algorithm to max a single parameter by searching a landscape of five parameters?

I'd do the following: before your tube burns out do an automated parameter sweep of reasonable range for your 5 parameters. That should give you a rough idea of the maxima/minima involved, and which ...
  • 2,151
3 votes
Accepted

What is difference between L2 norm and H2 Norm?

I am not sure about your application -- and we say the $L^2$ norm of a function and not a system. But for simplicity I will explain the concepts for real valued functions. Consider an open domain $\...
  • 254
3 votes

What's the right choice of variable settings for setting up my optimal control problem?

The m.MV() type has additional tuning parameters such as move suppression that is likely contributing to the difference in solution. Also, the ...
3 votes

Position Estimation using 2D multilateration for non-intersecting distances

The problem you describe is an example of a nonlinear least squares problem. You can find documentation for the MATLAB function to solve such a problem along with several references describing the ...
  • 5,784
3 votes

Algorithms for radiation treatment planning

In addition to Brian's reference, there was also a review article in SIAM Review a few years ago that summarized the state of research at the time: http://epubs.siam.org/doi/abs/10.1137/...
2 votes
Accepted

Direct multiple shooting (numerical optimal control)

In the direct multiple shooting method, using the notation from the linked notes by Chachuat, the "extra decision variables" $\mathbf{\xi}_{0}^{k}$ are not present in the objective function, therefore ...
2 votes

Strict Feasibility in Interior Point Methods

Unlike barrier methods, the affine scaling method doesn't use a barrier to push the iterates away from the boundaries of the feasible region. As a result, the iterates can very quickly approach the ...
2 votes
Accepted

Approximation Error in a Finite Difference Approximation of the Square of Derivative

It is true that $C$ depends on $x$ and $h$. This implies that if your function has a very poorly behaved second derivative at $x$, this method will be inaccurate. The dependance on $h$ is also to be ...
  • 1,097
2 votes
Accepted

proper derivation of a functional for a time dependent parameter estimation problem

You want to solve $$ \arg \min_{D(x),k(x)} \mathcal{G}(u;D,k) := \frac{1}{2} \int_{\Omega} \left(u(t=T) - {\bar{u}}(T) \right)^2 d\Omega + \frac{1}{2}\int_{\Omega} D^2 d\Omega + \frac{1}{2}\int_{\...
  • 2,126
2 votes

2-norm and infinty norm of a system in controls

You probably know that matrix norms can be defined by the vector norms in the following way: \begin{equation} ||A||:= \max_{x\neq 0} \frac{||Ax||}{||x||} \end{equation} for a matrix $A$. So you just ...
  • 394
1 vote
Accepted

Objective function for PDE-constrained boundary control problem in cylindrical coordinates

There are a couple of questions here, some of which pertain specifically to the geometry and input data for your problem and some have more to do with PDE-constrained optimization in general. Some of ...
1 vote
Accepted

Numerical solution of non-linear first order partial differential equation (HJB)

I haven't solved the HJB equations myself, but I can think of two things you can try based on my experience with other PDEs. Use an "upwinded" approximation for $J_x$, instead of a central ...
  • 323
1 vote
Accepted

Robust/Tested Solver for incompressible 2D Euler (Fluid dynamics) Equation

Since you are solving inviscid equations, you will need some form of stabilization like SUPG or a DG scheme, to get a robust scheme. You wont need the pressure I presume. I would recommend solving the ...
  • 2,983
1 vote
Accepted

Using two reference values for a scalar variable: What's the name of this type of problem?

Have you thought of barycentric coordinates? There is a unique way to write $x=\alpha A + \beta B$ with $\alpha+\beta=1$. Barycentric coordinates are usually employer in larger dimensions, but seem ...
1 vote

Algorithms for radiation treatment planning

There has been a lot of research in this area over the last 20 years. It's appropriate to start by using search engines such as Google Scholar and Web of Science to look for survey and review ...

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