Optimisation along a curve on a surface

In summary, the conversation discusses an optimization problem on a 2D surface with a curve. The task is to find the minimum of a function along the curve, with the only adjustment being the step length. The person has a closed form analytic expression for the function and its derivatives, as well as knowledge of geodesics, arc elements, and Riemannian metric. They are seeking guidance in the right direction to solve this problem. The surface is also defined on a Riemannian manifold of constant Gaussian curvature. Despite the potential difficulty of only being able to move along a pre-defined curve, the person may try approaching the problem in a simpler Euclidean space before attempting it on the manifold.
  • #1
Vasileios
6
0
Hi everyone,

First of all I am not sure if I have chosen the right category for this posting but this looked the most reasonable out.
I have a problem that I would like to solve but I am not sure where to look for answers. It seems like something other people might have worked with before but I am not sure what these types of problems are called.


(Please do excuse the abuse of any notation since I am only learning now about diff. geometry and such).

Basically it is an optimization problem. I have a 2d surface y= F(σ,κ) in ℝ3 and on that surface I have a curve γ. I do not have a parameterisation for the curve, but it is continuous and I can sample it at any point P (see illustration).
http://img694.imageshack.us/img694/9179/surfacekq.png



My task is:

Given a random initial point P0 on γ, to find the minimum of F along γ.
I can only move along this (blue) curve γ, backwards or forwards. So the only thing I can adjust is my step length λ.

Now of course I can use various heuristics to solve this, but there must be a more elegant way to solve this problem and someone must have seen these class of problems before.

I do have a closed form analytic expression for the whole F(σ,κ) as well as the first and second partial derivatives at each point. Also note that I have expressions for geodesics, arc elements, Riemmanian metric etc.

Can someone point me to the right direction?

Many thanks
V.
 
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  • #2
Vasileios said:
I do have a closed form analytic expression for the whole F(σ,κ) as well as the first and second partial derivatives at each point. Also note that I have expressions for geodesics, arc elements, Riemmanian metric etc.
Hi just a further clarification.
The actual objective surface F is defined on a Riemmanian manifold W of constant Gaussian curvature. So it is more like in the picture
http://img840.imageshack.us/img840/2443/manifoldm.jpg But I can still only move along a line. I cannot chose a direction to get to the global optimum of F, bhttps://www.physicsforums.com/editpost.php?do=editpost&p=3737946ut only the "global optimum" of the curve γ.I guess I could try to look at this problem in the easier case for a Euclidean space and then try the manifold approach. But still, the fact that I can only move on a pre-defined curve gives me a bit of trouble.
 
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1. What is the concept of "optimisation along a curve on a surface"?

The concept of "optimisation along a curve on a surface" refers to finding the maximum or minimum value of a function along a particular curve that lies on a surface. This involves finding the values of the independent variables that result in the highest or lowest value of the function, while also satisfying any constraints imposed by the curve and the surface.

2. Why is optimisation along a curve on a surface important in scientific research?

Optimisation along a curve on a surface is important in scientific research because it allows for the optimization of various processes and systems. This can lead to improvements in efficiency, cost-effectiveness, and overall performance. Additionally, it can help in understanding the behavior of complex systems and in making informed decisions.

3. What are the key steps in solving an optimisation problem along a curve on a surface?

The key steps in solving an optimisation problem along a curve on a surface are: formulating the problem, identifying the objective function and any constraints, finding the critical points of the objective function, determining which critical points lie on the curve, and evaluating the objective function at those points to identify the optimum value.

4. How is optimisation along a curve on a surface different from traditional optimisation?

Optimisation along a curve on a surface is different from traditional optimisation in that it involves finding the maximum or minimum value of a function along a specific curve on a surface, rather than optimizing a function over a set of independent variables. This type of optimisation also takes into account any constraints imposed by the curve and the surface, making it more complex than traditional optimisation problems.

5. What are some applications of optimisation along a curve on a surface in different fields of science?

Optimisation along a curve on a surface has various applications in different fields of science, such as physics, engineering, and economics. For example, it can be used in determining the optimal trajectory for a rocket launch, finding the most efficient design for a bridge, or optimizing production processes in a manufacturing plant. It can also be applied in finance to optimize investment portfolios or in biology to understand the behavior of complex biological systems.

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