Sparse solvers

An interesting special case, mentioned in Chap. 3, is that of the second derivative on four points, u"(4). For arbitrarily (unequally) spaced points, this is a second-order accurate approximation and, as described in Chap. 9, it has some advantages. It allows the use of an efficient extended Thomas algorithm, rather than a pentadiagonal solver or a sparse solver required if... 

In order to measure the performance of the new algorithm, two solvers are being used, namely, the full scheme and the advanced sparse solver. The distillation model is excited with an external input. Three cases were generated ... 

Furthermore it is worth mentioning, that the apiphcation of the Ekill / Ealive technique requires a sensible adjustment of the ANSYS calculation option in order to guarantee convergence. In particular option for an adaptive time integration scheme, non-Unear geometry, and the SPARSE solver is recommended. 

Bieniasz LK, Britz D (2001) Chronopotentiometry at a microband electrode simulation study using a Rosenbrock time integration scheme for differentiai-aigebraic equations and a direct sparse solver. J Electroanal Chem 503 141-152... 

Whether the simulation is on a direct discretisation of the equations in cylindrical or transformed coordinates, the discretisation process results in a (usually) linear system of ordinary differential equations, that must be solved. In two dimensions, the number of these will often be large and the equation system is banded. One approach is to ignore the sparse nature of the system and simply to solve it, using lower-upper decomposition (LUD) [212]. The method is very simple to apply and has been used [133,213,214]—it is especially appropriate in curvilinear coordinates and multipoint derivative approximations, where the system is of minimal size [214], and can outperform the more obvious method, using a sparse solver such as MA2 8 (see later). However, many simulators tend to prefer other methods, that avoid using implicit solution in two dimensions simultaneously but still are implicit. Of these, two stand out. 

Unequal intervals Chap. 7. These are essential for most programs. The second spatial derivative requires four points if second-order is wanted (and is recommended). With four-point discretisation, an efficient extended Thomas algorithm can be used, obviating the need for a sparse solver. Very few points can then be used across the concentration profile. For two-dimensional simulations, direct three-point discretisation on the unequally spaced grid was shown to be comparable with using transformation and discretisation in transformed space. 

The highest level of integration would be to establish one large set of equations and to apply one solution process to both thermal and airflow-related variables. Nevertheless, a very sparse matrix must be solved, and one cannot use the reliable and well-proven solvers of the present codes anymore. Therefore, a separate solution process for thermal and airflow parameters respectively remains the most promising approach. This seems to be appropriate also for the coupling of computational fluid dynamics (CFD) with a thermal model. ... 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

In order to exploit the sparseness of the matrix, iterative solvers can be applied. The iterative procedure is initialized with a guess for the solution vector O. In... 

Modem LP solvers can solve very large LPs very quickly and reliably on a PC or workstation. LP size is measured by several parameters (1) the number of variables n, (2) the number of constraints m, and (3) the number of nonzero entries nz in the constraint matrix A. The best measure is the number of nonzero elements nz because it directly determines the required storage and has a greater effect on computation time than n or m. For almost all LPs encountered in practice, nz is much less than mn, because each constraint involves only a few of the variables jc. The problem density 100(nz/mn) is usually less than 1%, and it almost always decreases as m and n increase. Problems with small densities are called sparse, and real world LPs are always sparse. Roughly speaking, a problem with under 1000 nonzeros is small, between 1000 and 50,000 is medium-size, and over 50,000 is large. A small problem probably has m and n in the hundreds, a medium-size problem in the low to mid thousands, and a large problem above 10,000. 

Warner [176] has given a comprehensive discussion of the principal approaches to the solution of stiff differential equations, including a hundred references among the most pertinent books, papers and application packages directed at simulating kinetic models. Emphasis has been put not only on numerical and software problems such as robustness, improving the linear equation solvers, using sparse matrix techniques, etc., but also on the availability of a chemical compiler, i.e. a powerful interface between kineticist and computer. 

In sum, the greatest virtues of CG methods are their modest storage and computational requirements (both order n) and their better convergence than the SD method. These properties have made them popular linear solvers and minimization choices in many applications18-20 84-88 and perhaps the only candidates for very large problems. The linear CG is often applied to systems arising from discretizations of partial differential equations,81 89 90 where the matrices are frequently positive-definite, sparse, and structured. 

Finite Element Methods Applied to Many-body Perturbation Theory. - Over the past ten years, the finite element method, which is a classical tool in classical science and engineering applications, has been developed into a technique for the accurate solution of the atomic243 and molecular244,245 electronic structure problem. The piece-wise definition of the form functions employed in the finite element method prevents the computational linear dependencies which occur in the finite basis set expansion method and, moreover, leads to sparse, band structured matrices for which efficient solvers are available. 

It is well known that solvers that exploit the sparse structure of its Jacobian are more efficient in terms of CPU computation time than solvers that do not Unfortunately, those solvers are not always available or applicable to the specific studied case. 

An approach that solves the unstructured part separately (e.g., quadrature of the integral), while the remaining sparse structure is efficiently calculated with an appropriate solver. 

A novel approach was recently introduced to the BzzMath library (Manenti et al., 2009). A new DAE solver was created for very sparse systems corrupted by a few unstructured elements, as they typically occur in the process control field. The implementation of this algorithm is now part of... 

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