# Which analogs of Newton's multivariate method are faster?

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

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.

• I split my u(x,y) function on real and complex parts and using multivariate Newton's method Commented Apr 10, 2023 at 15:55
• Are you storing a dense or a sparse matrix? Commented Apr 10, 2023 at 21:10
• Newton-GMRES with a proper matrix-free implementation would likely perform the best on a problem of this size Commented Apr 10, 2023 at 21:26
• 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? Commented Apr 11, 2023 at 8:28
• @DanielShapero I store sparse matrix Commented Apr 11, 2023 at 15:01