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Currently, I am studying a 2D nonlinear Schroedinger equation and searching for the fastest method. $$ \begin{equation} i \frac{\partial \psi}{\partial t} = [-\frac{1}{2} \nabla^2 + V_0(r) - i\gamma + g|\psi|^2]\psi + \Phi(r,t), \end{equation} $$ Studying the amplitude of the solution to this time-dependent equation I decided to split the $\psi$ function into two independent functions of time and coordinate: $$ \begin{equation} \psi = u(r)e^{-i\varepsilon t}, \Phi(r,t) = A(r)e^{-i\varepsilon t} \end{equation} $$ This simplification speeds up the solution program because I no longer need to simulate the evolution of the system and just observe the final result at the stationary regime. Then I obtain such equation: $$ \begin{equation} [-\frac{1}{2} \nabla^2 + V_0(r) - i\gamma - \varepsilon + g|\psi|^2]\psi + A(r) = 0 \end{equation} $$

I use Newton's method to find the solution (note that $r$ stands for $(x,y)$). Because of the 2D problem, I discretize the domain in 1024x1024 vertices. This leads to a problem of inversing (10241024)x(10241024) matrix. This task doesn't allow me to use GPU because I have low memory resources and I can't parallelise it with CPU.

At this moment a MATLAB code I have created takes way too much time (up to $2500$ seconds to converge to accuracy $10^{-9}$). So I wonder if there is a faster method I could implement to achieve the same accuracy within a smaller time.

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  • $\begingroup$ I split my u(x,y) function on real and complex parts and using multivariate Newton's method $\endgroup$
    – Andrew
    Commented Apr 10, 2023 at 15:55
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    $\begingroup$ Are you storing a dense or a sparse matrix? $\endgroup$ Commented Apr 10, 2023 at 21:10
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    $\begingroup$ Newton-GMRES with a proper matrix-free implementation would likely perform the best on a problem of this size $\endgroup$
    – whpowell96
    Commented Apr 10, 2023 at 21:26
  • $\begingroup$ How does the error develop as you do more Newton iterations? How do you compute the initial guess? Can you solve the problem on a coarser grid, extrapolate the solution to the fine grid and then polish it off with a single Newton step? Why is a parallel linear solve not possible in this context? $\endgroup$ Commented Apr 11, 2023 at 8:28
  • $\begingroup$ @DanielShapero I store sparse matrix $\endgroup$
    – Andrew
    Commented Apr 11, 2023 at 15:01

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Rather than looking for a better method it's probably worth trying a faster programming language. If you do this in Julia (or C++/Fortran) you would likely see a significant speedup. Julia would be especially nice since there is a lot of libraries for doing matrix free solves.

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    $\begingroup$ Speed up depends on if OP is using a MATLAB function that is really just a wrapper to a lower level language library or if OP is running pure MATLAB code with for loops. $\endgroup$
    – user7257
    Commented Apr 20, 2023 at 20:40

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