Minimizing functional using FindMinimum

Question

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.

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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?

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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.

Answer 1

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.