126 research outputs found
Numerical solutions of nonlinear elliptic problem using combined-block iterative methods
This thesis is concerned with iterative and monotone methods for numerical
solutions of nonlinear elliptic boundary value problems. The methods we study here
are called block iterative methods, which solve the nonlinear elliptic problems in twodimensional
domain in R2 or higher dimensional domain in Rn. In these methods the
nonlinear boundary value problem is discretized by the finite difference method. Two
iteration processes, block Jacobi and block Gauss-Seidel monotone iterations, are
investigated for computation of solutions of finite difference system using either an
upper solution or a lower solution as the initial iteration. The numerical examples are
presented for both linear and nonlinear problems, and for both block and pointwise
methods. The numerical results are compared and discussed
On supraconvergence phenomenon for second order centered finite differences on non-uniform grids
In the present study we consider an example of a boundary value problem for a
simple second order ordinary differential equation, which may exhibit a
boundary layer phenomenon. We show that usual central finite differences, which
are second order accurate on a uniform grid, can be substantially upgraded to
the fourth order by a suitable choice of the underlying non-uniform grid. This
example is quite pedagogical and may give some ideas for more complex problems.Comment: 26 pages, 2 figures, 2 tables, 37 references. Other author's papers
can be downloaded at http://www.denys-dutykh.com
Adaptive Algorithms
Overwhelming empirical evidence in computational science and engineering proved that self-adaptive mesh-generation is a must-do in real-life problem computational partial differential equations. The mathematical understanding of corresponding algorithms concerns the overlap of two traditional mathematical disciplines, numerical analysis and approximation theory, with computational sciences. The half workshop was devoted to the mathematics of optimal convergence rates and instance optimality of the Dörfler marking or the maximum strategy in various versions of space discretisations and time-evolution problems with all kind of applications in the efficient numerical treatment of partial differential equations
Numerical solution of the problems for plates on some complex partial internal supports
In the recent works, Dang and Truong proposed an iterative method for solving some problems of plates on one, two and three line partial internal supports (LPISs), and a cross internal support. In nature they are problems with strongly mixed boundary conditions for biharmonic equation. For this reason the method combines a domain decomposition technique with the reduction of the order of the equation from four to two. In this study, the method is developed for plates on internal supports of more complex configurations. Namely, we examine the cases of symmetric rectangular and H-shape supports, where the computational domain after reducing to the first quadrant of the plate is divided into three subdomains. Also, we consider the case of asymmetric rectangular support where the computational domain needs to be divided into 9 subdomains. The problems under consideration are reduced to sequences of weak mixed boundary value problems for the Poisson equation, which are solved by difference method. The performed numerical experiments show the effectiveness of the iterative method
A Multigrid Method for the Efficient Numerical Solution of Optimization Problems Constrained by Partial Differential Equations
We study the minimization of a quadratic functional subject to constraints given by a linear or semilinear elliptic partial differential equation with distributed control. Further, pointwise inequality constraints on the control are accounted for. In the linear-quadratic case, the discretized optimality conditions yield a large, sparse, and indefinite system with saddle point structure. One main contribution of this thesis consists in devising a coupled multigrid solver which avoids full constraint elimination. To this end, we define a smoothing iteration incorporating elements from constraint preconditioning. A local mode analysis shows that for discrete optimality systems, we can expect smoothing rates close to those obtained with respect to the underlying constraint PDE. Our numerical experiments include problems with constraints where standard pointwise smoothing is known to fail for the underlying PDE. In particular, we consider anisotropic diffusion and convection-diffusion problems. The framework of our method allows to include line smoothers or ILU-factorizations, which are suitable for such problems. In all cases, numerical experiments show that convergence rates do not depend on the mesh size of the finest level and discrete optimality systems can be solved with a small multiple of the computational cost which is required to solve the underlying constraint PDE. Employing the full multigrid approach, the computational cost is proportional to the number of unknowns on the finest grid level. We discuss the role of the regularization parameter in the cost functional and show that the convergence rates are robust with respect to both the fine grid mesh size and the regularization parameter under a mild restriction on the next to coarsest mesh size. Incorporating spectral filtering for the reduced Hessian in the control smoothing step allows us to weaken the mesh size restriction. As a result, problems with near-vanishing regularization parameter can be treated efficiently with a negligible amount of additional computational work. For fine discretizations, robust convergence is obtained with rates which are independent of the regularization parameter, the coarsest mesh size, and the number of levels. In order to treat linear-quadratic problems with pointwise inequality constraints on the control, the multigrid approach is modified to solve subproblems generated by a primal-dual active set strategy (PDAS). Numerical experiments demonstrate the high efficiency of this approach due to mesh-independent convergence of both the outer PDAS method and the inner multigrid solver. The PDAS-multigrid method is incorporated in the sequential quadratic programming (SQP) framework. Inexact Newton techniques further enhance the computational efficiency. Globalization is implemented with a line search based on the augmented Lagrangian merit function. Numerical experiments highlight the efficiency of the resulting SQP-multigrid approach. In all cases, locally superlinear convergence of the SQP method is observed. In combination with the mesh-independent convergence rate of the inner solver, a solution method with optimal efficiency is obtained
Stochastic Proximal Gradient Methods for Nonconvex Problems in Hilbert Spaces
For finite-dimensional problems, stochastic approximation methods have long
been used to solve stochastic optimization problems. Their application to
infinite-dimensional problems is less understood, particularly for nonconvex
objectives. This paper presents convergence results for the stochastic proximal
gradient method applied to Hilbert spaces, motivated by optimization problems
with partial differential equation (PDE) constraints with random inputs and
coefficients. We study stochastic algorithms for nonconvex and nonsmooth
problems, where the nonsmooth part is convex and the nonconvex part is the
expectation, which is assumed to have a Lipschitz continuous gradient. The
optimization variable is an element of a Hilbert space. We show almost sure
convergence of strong limit points of the random sequence generated by the
algorithm to stationary points. We demonstrate the stochastic proximal gradient
algorithm on a tracking-type functional with a -penalty term constrained
by a semilinear PDE and box constraints, where input terms and coefficients are
subject to uncertainty. We verify conditions for ensuring convergence of the
algorithm and show a simulation
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