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Suppose you are given some functional $S$, (with some 'Lagrangian' $L$) $$S[f]=\int_{t_i}^{t_f}L[f(t),f'(t)]dt$$ and you want to find the function $f$ that minimizes it, subject to some boundary conditions at $t_i$ and $t_f$. To be specific I'll consider the following case $$S[f]=\int_{0}^{t_{max}}\left(f'(t)\right)^2dt,$$ where $f(0)=\pi, f(t_{max})=0$. This is just minimized by a linear $f$, but my ultimate goal is to do this for a different problem which I can't solve analytically.

I wanted to discretize the problem and treat $f$ as a list of large number $n-1$ of variables, and then use FindMinimum in Mathematica for this purpose since I saw it implements the quasi-Newton BFGS approximation. But I am new to Mathematica and can't seem to figure out how to do this in practice.


This is what I've tried:

Setting up an initial trial solution f0 (which is just a shifted exponential in this case),

tMax = 10;
n = 100;
dt = tMax/n;
f0 = Table[\[Pi]/(1 - Exp[-tMax]) (Exp[-(i*dt)] - Exp[-tMax]), {i, 1, 
    n - 1}];

Defining a numerical derivative derList, and the functional action,

derList[f_, delta_] := 
  1/delta Join[{1/2  (f[[2]] - \[Pi])}, 
    ListConvolve[{1/2 , 0, -(1/2 )}, f], {-1/2  f[[Length[f] - 1]]}];
action[f_] := derList[f, 1] . derList[f, 1];

where I don't care so much about overall positive factors multiplying the functional, hence the use of 1 in action rather than something more appropriate like dt.

Now I want to use FindMinimum, but I can't seem to figure out the proper syntax. If I use FindMinimum[action[f], {f, f0}] it does not work. How should I set up this problem?


Just to be clear, in the end you can completely forget the motivation for this problem. action[f] is just some positive real-valued function of n-1 variables given by the components of the list f, and I have initial guesses for each one of those variables which are given by components of f0. But I just do not know how to set up FindMinimum to work with lists in this way. I would rather not copy-paste in an explicit expression for action[f] in terms of components, and write out 99 initial guesses, so I'm just looking for a way to avoid that.

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  • $\begingroup$ What's wrong with finding the Euler-Lagrange equations and using NDSolve? $\endgroup$
    – Carl Woll
    Sep 21, 2021 at 0:02
  • $\begingroup$ @CarlWoll, For the actual problem I want to eventually solve $L$ actually depends on the second derivative of $f$ and the Euler-Lagrange equations leads to a fourth order non-linear ODE. I tried using NDSolve to solve it but got stiffness or singularity errors. I thought an approach using something like gradient flow might be more robust, and was led to the BFGS approximation as something people use in practice for this kind of problem. $\endgroup$
    – octonion
    Sep 21, 2021 at 0:09
  • $\begingroup$ @octonion Do you suppose that f is complex function? $\endgroup$ Sep 21, 2021 at 3:40
  • $\begingroup$ @AlexTrounev, In principle complex saddle points may be interesting too, but the particular types of $S$ that I am considering have a local minimum given by a real $f$. I have a pretty good guess at an approximate $f_0$ that could be used as an initial condition for the problem I am interested in, so I expect (hope) it will converge to this real solution. $\endgroup$
    – octonion
    Sep 21, 2021 at 3:56
  • $\begingroup$ @octonion This is a typical optimization problem, therefore we need to use NMinimize. $\endgroup$ Sep 21, 2021 at 4:19

1 Answer 1

1
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I figured it out. I needed to make sure that Mathematica understood that action takes vector arguments by defining it explicitly as

action[f_ /; (VectorQ[f])] := derList[f, 1] . derList[f, 1];

That solves the Mathematica question.

But the answer it gives me is 'too good' and takes advantage of the discretization to oscillate every time step rather than being a linear function which is the proper continuum answer. This can be fixed by using a forward difference rather than a symmetric difference. i.e. replace derList by derForward defined as

derForward[f_, delta_] := 
  Module[{v, l}, l = Length[f]; v = ListConvolve[{1, -1}, f]; 
   AppendTo[v, (0 - f[[l]])]; PrependTo[v, (f[[1]] - \[Pi])]; 
   1/delta v];

Then FindMinimum[action[f], {f, f0}] returns the expected linear solution.

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  • 2
    $\begingroup$ I think action[f_List] := ... would be simpler. $\endgroup$
    – Carl Woll
    Sep 21, 2021 at 17:51
  • $\begingroup$ @CarlWoll, thanks, I wasn't aware you could do that $\endgroup$
    – octonion
    Sep 21, 2021 at 17:55
  • $\begingroup$ @octonion How you know that your solution minimize your Lagrangian? $\endgroup$ Sep 23, 2021 at 3:19

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