6,366 research outputs found
Analysis of a parallelized nonlinear elliptic boundary value problem solver with application to reacting flows
A parallelized finite difference code based on the Newton method for systems of nonlinear elliptic boundary value problems in two dimensions is analyzed in terms of computational complexity and parallel efficiency. An approximate cost function depending on 15 dimensionless parameters is derived for algorithms based on stripwise and boxwise decompositions of the domain and a one-to-one assignment of the strip or box subdomains to processors. The sensitivity of the cost functions to the parameters is explored in regions of parameter space corresponding to model small-order systems with inexpensive function evaluations and also a coupled system of nineteen equations with very expensive function evaluations. The algorithm was implemented on the Intel Hypercube, and some experimental results for the model problems with stripwise decompositions are presented and compared with the theory. In the context of computational combustion problems, multiprocessors of either message-passing or shared-memory type may be employed with stripwise decompositions to realize speedup of O(n), where n is mesh resolution in one direction, for reasonable n
Automatic Frechet differentiation for the numerical solution of boundary-value problems
A new solver for nonlinear boundary-value problems (BVPs) in Matlab is presented, based on the Chebfun software system for representing functions and operators automatically as numerical objects. The solver implements Newton's method in function space, where instead of the usual Jacobian matrices, the derivatives involved are Frechet derivatives. A major novelty of this approach is the application of automatic differentiation (AD) techniques to compute the operator-valued Frechet derivatives in the continuous context. Other novelties include the use of anonymous functions and numbering of each variable to enable a recursive, delayed evaluation of derivatives with forward mode AD. The AD techniques are applied within a new Chebfun class called chebop which allows users to set up and solve nonlinear BVPs in a few lines of code, using the "nonlinear backslash" operator (\). This framework enables one to study the behaviour of Newton's method in function space
Discretization of the 3D Monge-Ampere operator, between Wide Stencils and Power Diagrams
We introduce a monotone (degenerate elliptic) discretization of the
Monge-Ampere operator, on domains discretized on cartesian grids. The scheme is
consistent provided the solution hessian condition number is uniformly bounded.
Our approach enjoys the simplicity of the Wide Stencil method, but
significantly improves its accuracy using ideas from discretizations of optimal
transport based on power diagrams. We establish the global convergence of a
damped Newton solver for the discrete system of equations. Numerical
experiments, in three dimensions, illustrate the scheme efficiency
On the convergence of a damped-secant method with modified right-hand side vector
[EN] We present a convergence analysis for a Damped Secant method with modified right-hand
side vector in order to approximate a locally unique solution of a nonlinear equation in a
Banach spaces setting. In the special case when the method is defined on Ri
, our method
provides computable error estimates based on the initial data. Such estimates were not
given in relevant studies such as (Herceg et al., 1996; Krejic´, 2002). Numerical examples
further validating the theoretical results are also presented in this study.The authors thank to the anonymous referee for his/her valuable comments and for the suggestions to improve the final version of the paper. This work is partially supported by UNIR Research Support Strategy 2013-2015, under the CYBERSE-CURITICS.es Research Group [http://research.unir.net].Argyros, IK.; Cordero Barbero, A.; Magreñán Ruiz, ÁA.; Torregrosa Sánchez, JR. (2015). On the convergence of a damped-secant method with modified right-hand side vector. Applied Mathematics and Computation. 252:315-323. doi:10.1016/j.amc.2014.12.029S31532325
A localized orthogonal decomposition method for semi-linear elliptic problems
In this paper we propose and analyze a new Multiscale Method for solving
semi-linear elliptic problems with heterogeneous and highly variable
coefficient functions. For this purpose we construct a generalized finite
element basis that spans a low dimensional multiscale space. The basis is
assembled by performing localized linear fine-scale computations in small
patches that have a diameter of order H |log H| where H is the coarse mesh
size. Without any assumptions on the type of the oscillations in the
coefficients, we give a rigorous proof for a linear convergence of the H1-error
with respect to the coarse mesh size. To solve the arising equations, we
propose an algorithm that is based on a damped Newton scheme in the multiscale
space
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