Linear preconditioning conjugate method

In terms of the computational work per outer Newton step k), TN methods based on preconditioned conjugate gradient require a Hessian-vector product (Hd) at each inner loop iteration, and one solution of a linear system Mz = r where M is the preconditioner. Since M may be sparse, this linear solution often takes a very small percentage of the total CPU time (e.g., <3% ). The benefits of faster convergence generally far outweigh these costs. 

In the work of Lindborg et al [119], the resulting linear equation systems were solved with preconditioned Krylov subspace projection methods [166]. The Poisson equation was solved by a conjugate gradient (CG)-solver, while the other transport equations were solved using a bi-conjugate gradient (BCG)-solver which can handle also non-symmetric equations systems. The solvers were preconditioned with a Jacobi preconditioner. 

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