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The initial idea is usually attributed to the work of Hardy with Srinivasa Ramanujan a few years earlier, in 1916 and 1917, on the asymptotics of the partition function.It was taken up by many other researchers, including Harold Davenport and I. M. Vinogradov, who modified the formulation slightly (moving from complex analysis to exponential sums), without changing the broad lines.
Matrix-free conjugate gradient method has been applied in the non-linear elasto-plastic finite element solver. [7] Solving these equations requires the calculation of the Jacobian which is costly in terms of CPU time and storage. To avoid this expense, matrix-free methods are employed.
The finite point method (FPM) is a meshfree method for solving partial differential equations (PDEs) on scattered distributions of points. The FPM was proposed in the mid-nineties in (Oñate, Idelsohn, Zienkiewicz & Taylor, 1996a), [1] (Oñate, Idelsohn, Zienkiewicz, Taylor & Sacco, 1996b) [2] and (Oñate & Idelsohn, 1998a) [3] with the purpose to facilitate the solution of problems involving ...
That is, in Benders decomposition, the variables of the original problem are divided into two subsets so that a first-stage master problem is solved over the first set of variables, and the values for the second set of variables are determined in a second-stage subproblem for a given first-stage solution.
MFEM is a free, lightweight, scalable C++ library for finite element methods that features arbitrary high-order finite element meshes and spaces, support for a wide variety of discretizations, and emphasis on usability, generality, and high-performance computing efficiency.
For two clusters, we can assign a binary variable to the point corresponding to the -th row in , indicating whether it belongs to the first (=) or second cluster (=). Consequently, we have 20 binary variables, which form a binary vector x ∈ B 20 {\displaystyle x\in \mathbb {B} ^{20}} that corresponds to a cluster assignment of all points (see ...
The state variables obtained after Step 2 are averaged over each cell defining a new piecewise constant approximation resulting from the wave propagation during the time interval . To be consistent, the time interval Δ t {\displaystyle {\Delta t}\,} should be limited such that the waves emanating from an interface do not interact with waves ...
The Jacobian itself might be too difficult to compute, but the GMRES method does not require the Jacobian itself, only the result of multiplying given vectors by the Jacobian. Often this can be computed efficiently via difference formulae. Solving the Newton iteration formula in this manner, the result is a Jacobian-Free Newton-Krylov (JFNK ...