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1

I recommend using a global optimization solver. You should be able to solve this, or at least try, using the BMIBNB global optimization solver https://yalmip.github.io/solver/bmibnb/ under YALMIP https://yalmip.github.io/tutorial/basics/ under MATLAB. In particular, note that YALMIP supports the exponential integral, expint, https://yalmip.github.io/command/...

1

I would also like to point at MatlabAutoDiff, which supports sparse Jacobians. Have tried it myself: it is possible to compute large Jacobians (tried with N=1e5) in a small amount of time.

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Julia has a whole ecosystem for generating sparsity patterns and doing sparse automatic differentiation in a way that mixes with scientific computing and machine learning (or scientific machine learning). Tools like SparseDiffTools.jl, ModelingToolkit.jl, and SparsityDetection.jl will do things like: Automatically find sparsity patterns from code Generate ...

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There are finite difference stencils specifically designed to have rotational symmetry. For example, instead of the standard second order stencil $$\frac{1}{h^2} \begin{bmatrix} & 1 & \\ 1 & -4 & 1\\ & 1 & \end{bmatrix}$$ you can use  \frac{1}{6 h^2} \begin{bmatrix} 1 & 4 & 1 \\ 4 & -20 & 4 \\ 1 & 4 & 1 \...

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