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To simulate the physical process necessary to solve the arising systems of linear algebraic equations. The SLE matrix has a highly sparse form. There are a couple dozen non-zero elements in the string, including the diagonal. And all unknown a couple of million.

During the simulation, the matrix retains its portrait, but the coefficients change greatly. Therefore, it is more efficient (in terms of time) to use different solvers / preconditioners at different simulation times. It is necessary to come up with a criterion that uses some data about the matrix (perhaps its spectrum, but I think it is expensive to calculate), the characteristics of the computing system, and the estimate . According to this criterion, the decision will be made which solver / preconditioner to choose.

I can, of course, go the easy way. Run all solvers / preconditioners every N steps and select the fastest one.

But I would like to use something more difficult. Can you tell me what you can read? Or give your ideas that you used for something similar?

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The question of selection of solver and preconditioner is difficult and to a great degree depends on how much you actually know about where the matrix comes from and what kind of operator it represents. You might benefit from going through some of the considerations I lay out in videos 34 and following here: https://www.math.colostate.edu/~bangerth/videos.html

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