4,492 research outputs found
A robust multigrid method for the time-dependent Stokes problem
In the present paper we propose an all-at-once multigrid method for generalized Stokes flow problems. Such problems occur as subproblems in implicit time-stepping approaches for time-dependent Stokes problems. The discretized optimality system is a large scale linear system whose condition number depends on the grid size of the spacial discretization and of the length of the time step. Recently, for this problem an all-at-once multigrid method has been proposed, where in each smoothing step the Poisson problem has to be solved (approximatively) for the pressure field. In the present paper, we propose an all-at-once multigrid method where the solution of such subproblems is not needed. We prove that the proposed method shows robust convergence behavior in the grid size of the spacial discretization and of the length of the time-step
A Primal-Dual Algorithmic Framework for Constrained Convex Minimization
We present a primal-dual algorithmic framework to obtain approximate
solutions to a prototypical constrained convex optimization problem, and
rigorously characterize how common structural assumptions affect the numerical
efficiency. Our main analysis technique provides a fresh perspective on
Nesterov's excessive gap technique in a structured fashion and unifies it with
smoothing and primal-dual methods. For instance, through the choices of a dual
smoothing strategy and a center point, our framework subsumes decomposition
algorithms, augmented Lagrangian as well as the alternating direction
method-of-multipliers methods as its special cases, and provides optimal
convergence rates on the primal objective residual as well as the primal
feasibility gap of the iterates for all.Comment: This paper consists of 54 pages with 7 tables and 12 figure
Numerical Analysis of Sparse Initial Data Identification for Parabolic Problems
In this paper we consider a problem of initial data identification from the
final time observation for homogeneous parabolic problems. It is well-known
that such problems are exponentially ill-posed due to the strong smoothing
property of parabolic equations. We are interested in a situation when the
initial data we intend to recover is known to be sparse, i.e. its support has
Lebesgue measure zero. We formulate the problem as an optimal control problem
and incorporate the information on the sparsity of the unknown initial data
into the structure of the objective functional. In particular, we are looking
for the control variable in the space of regular Borel measures and use the
corresponding norm as a regularization term in the objective functional. This
leads to a convex but non-smooth optimization problem. For the discretization
we use continuous piecewise linear finite elements in space and discontinuous
Galerkin finite elements of arbitrary degree in time. For the general case we
establish error estimates for the state variable. Under a certain structural
assumption, we show that the control variable consists of a finite linear
combination of Dirac measures. For this case we obtain error estimates for the
locations of Dirac measures as well as for the corresponding coefficients. The
key to the numerical analysis are the sharp smoothing type pointwise finite
element error estimates for homogeneous parabolic problems, which are of
independent interest. Moreover, we discuss an efficient algorithmic approach to
the problem and show several numerical experiments illustrating our theoretical
results.Comment: 43 pages, 10 figure
On the finite termination of an entropy function based smoothing Newton method for vertical linear complementarity problems
By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that the proposed algorithm finds an exact solution of VLCP in a finite number of iterations, under some conditions milder than those assumed in literature. Some computational results are included to illustrate the potential of this approach.Newton method;Finite termination;Entropy function;Smoothing approximation;Vertical linear complementarity problems
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