Academy of Sciences of the Czech Republic

We consider the compressible (barotropic) Navier-Stokes system on time-dependent domains, supplemented with slip boundary conditions. Our approach is based on penalization of the boundary behaviour, viscosity, and the pressure in the weak formulation. Global-in-time weak solutions are obtained. Secondly, we suppose that the characteristic speed of the fluid is domi- nated by the speed of sound and perform the low Mach number limit in the framework of weak solutions. The standard incompressible Navier-Stokes system is identified as the target problem. References:

[1] E. Feireisl, O. Kreml, S. Nečasová, J. Neustupa, and J. Stebel. Weak solutions to the barotropic NavierStokes system with slip boundary conditions in time dependent domains. J. Differential Equations, 254:125–140, 2013.

[2] E. Feireisl, O. Kreml, S. Nečasová, J. Neustupa, and J. Stebel. Incompressible limits of fluids excited by moving boundaries. Submitted

Regularization techniques based on the Golub-Kahan iterative bidiagonalization belong among popular approaches for solving large discrete ill-posed problems. First, the original problem is projected onto a lower dimensional subspace using the bidiagonalization algorithm, which by itself represents a form of regularization by projection. The projected problem, however, inherits a part of the ill-posedness of the original problem, and therefore some form of inner regularization must be applied. Stopping criteria for the whole process are then based on the regularization of the projected (small) problem.

We consider an ill-posed problem with a noisy right-hand side (observation vector), where the noise level is unknown. We show how the information from the Golub-Kahan iterative bidiagonalization can be used for estimating the noise level. Such information can be useful for constructing efficient stopping criteria in solving ill-posed problems.

This is joint work by Iveta Hn\v{e}tynkov\'{a}, Martin Ple\v{s}inger, and Zden\v{e}k Strako\v{s} (Faculty of Mathematics and Physics, Charles University, and Institute of Computer Science, Academy of Sciences, Prague)

For large--scale saddle point problems, the application of exact iterative schemes and preconditioners may be computationally expensive. In practical situations, only approximations to the inverses of the diagonal block or the related cross-product matrices are considered, giving rise to inexact versions of various solvers. Therefore, the approximation effects must be carefully studied. In this talk we study numerical behavior of several iterative Krylov subspace solvers applied to the solution of large-scale saddle point problems. Two main representatives of the segregated solution approach are analyzed: the Schur complement reduction method, based on an (iterative) elimination of primary variables and the null-space projection method which relies on a basis for the null-space for the constraints. We concentrate on the question what is the best accuracy we can get from inexact schemes solving either Schur complement system or the null-space projected system when implemented in finite precision arithmetic. The fact that the inner solution tolerance strongly influences the accuracy of computed iterates is known and was studied in several contexts.

In particular, for several mathematically equivalent implementations we study the influence of inexact solving the inner systems and estimate their maximum attainable accuracy. When considering the outer iteration process our rounding error analysis leads to results similar to ones which can be obtained assuming exact arithmetic. The situation is different when we look at the residuals in the original saddle point system. We can show that some implementations lead ultimately to residuals on the the roundoff unit level independently of the fact that the inner systems were solved inexactly on a much higher level than their level of limiting accuracy. Indeed, our results confirm that the generic and actually the cheapest implementations deliver the approximate solutions which satisfy either the second or the first block equation to the working accuracy. In addition, the schemes with a corrected direct substitution are also very attractive. We give a theoretical explanation for the behavior which was probably observed or it is already tacitly known. The implementations that we pointed out as optimal are actually those which are widely used and suggested in applications.

Consider a system of linear algebraic equations $Ax=b$ where $A$ is an $n$ by $n$ real matrix and $b$ a real vector of length $n$. Unlike in the linear iterative methods based on the idea of splitting of $A$, the Krylov subspace methods, which are used in computational kernels of various optimization techniques, look for some optimal approximate solution $x^n$ in the subspaces ${\cal K}_n (A, b) = \mbox{span} \{ b, Ab, \dots, A^{n-1}b\}, n = 1, 2, \dots$ (here we assume, with no loss of generality, $x^0 = 0$). As a consequence, though the problem $Ax = b$ is linear, Krylov subspace methods are not. Their convergence behaviour cannot be viewed as an (unimportant) initial transient stage followed by the subsequent convergence stage. Apart from very simple, and from the point of view of Krylov subspace methods uninteresting cases, it cannot be meaningfully characterized by an asymptotic rate of convergence. In Krylov subspace methods such as the conjugate gradient method (CG) or the generalized minimal residual method (GMRES), the optimality at each step over Krylov subspaces of increasing dimensionality makes any linearized description inadequate. CG applied to $Ax = b$ with a symmetric positive definite $A$ can be viewed as a method for numerical minimization the quadratic functional $1/2 (Ax, x) - (b,x)$. In order to reveal its nonlinear character, we consider CG a matrix formulation of the Gauss-Christoffel quadrature, and show that it essentially solves the classical Stieltjes moment problem. Moreover, though the CG behaviour is fully determined by the spectral decomposition of the problem, the relationship between convergence and spectral information is nothing but simple. We will explain several phenomena where an intuitive commonly used argumentation can lead to wrong conclusions, which can be found in the literature. We also show that rounding error analysis of CG brings fundamental understanding of seemingly unrelated problems in convergence analysis and in theory of the Gauss-Christoffel quadrature. In remaining time we demonstrate that in the unsymmetric case the spectral information is not generally sufficient for description of behaviour of Krylov subspace methods. In particular, given an arbitrary prescribed convergence history of GMRES and an arbitrary prescribed spectrum of the system matrix, there is always a system $Ax=b$ such that GMRES follows the prescribed convergence while $A$ has the prescribed spectrum.