Allow block-diagonal linear operators with diagonalize=True

[hippalectryon-0]

Use case: we have a lot of degrees of freedom (e.g. N=30000, for example a 3D grid 20x20x20 with 4 fields), but the linear step is diagonal by block with small blocks.

Example equation: homogeneous Rayleigh-Bénard system with variables $u_x,u_y,u_z,\theta$:
$\partial_t u_i + u_j\partial_ju_i + \nabla P = Pr\Delta u_i + RaPr \theta\delta_{i=z}$
$\partial\theta+u_j\partial_j\theta=\Delta \theta+u_z$.

In this system, the linear term for $\theta$ is extremely simple $\mathcal{L}(k_x,k_y,k_z)\theta=-k^2\theta+u_z$. It is block-diagonal with blocks of size 4 (the number of equations).

However, if we use (for instance) EDT35, we have to compute the NxN full matrix of the operator, which is way too big (1e10 elements !!).

Moreover, looking at the code of _ETD35_Diagonalized, it seems to me that all the quantities involved are stable by block diagonalization, so I expect that we could handle gracefully those cases (maybe using scipy.sparse ?).

[whalenpt]

Hi, Things may have changed but at the time I studied ETD with adaptive step, nondiagonal linear operators were typically handled with other types of exponential integrators, usually involving krylov subspace approximations to the system thereby "shrinking" the sparse nondiagonal system under investigation with a close approximation. If I knew of a good python library for that I would point you to it, but I'm not aware of one, though you may want to look into it. I see the temptation in the block diagonal structure of your system to look for ways to fit it into ETD35 but I think you would be getting into an area that is or at least was not well studied or understood. If you would like to try modifying the code to see if something works using scipy sparse please let me know how that goes. Best regards, Patrick

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On Wed, May 10, 2023, 5:54 AM Hippalectryon ***@***.***> wrote: Use case: we have a lot of degrees of freedom (e.g. N=30000, for example a 3D grid 20x20x20 with 4 fields), but the linear step is diagonal by block with small blocks. Example equation: homogeneous Rayleigh-Bénard system with variables $u_x,u_y,u_z,\theta$: $\partial_t u_i + u_j\partial_ju_i + \nabla P = Pr\Delta u_i + RaPr \theta\delta_{i=z}$ $\partial\theta+u_j\partial_j\theta=\Delta \theta+u_z$. In this system, the linear term for $\theta$ is extremely simple $\mathcal{L}(k_x,k_y,k_z)\theta=-k^2\theta+u_z$. It is block-diagonal with blocks of size 4 (the number of equations). However, if we use (for instance) EDT35, we have to compute the NxN full matrix of the operator, which is way too big (1e10 elements !!). Moreover, looking at the code of _ETD35_Diagonalized, it seems to me that all the quantities involved are stable by block diagonalization, so I expect that we could handle gracefully those cases (maybe using scipy.sparse ?). — Reply to this email directly, view it on GitHub <#4>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AGS5CMCTMBUZMAP7Z4LRSEDXFOFXNANCNFSM6AAAAAAX4WOEFM> . You are receiving this because you are subscribed to this thread.Message ID: ***@***.***>