Publikační činnost Katedry aplikované matematiky / Publications of Department of Applied Mathematics (470)
http://hdl.handle.net/10084/64748
Kolekce obsahuje bibliografické záznamy publikační činnosti (článků) akademických pracovníků Katedry aplikované matematiky (470) v časopisech registrovaných ve Web of Science od roku 2003 po současnost.2021-12-02T23:42:51Z2-Dimensional primal domain decomposition theory in detail
http://hdl.handle.net/10084/110518
2-Dimensional primal domain decomposition theory in detail
Lukáš, Dalibor; Bouchala, Jiří; Vodstrčil, Petr; Malý, Lukáš
We give details of the theory of primal domain decomposition (DD) methods for a 2-dimensional second order elliptic equation with homogeneous Dirichlet boundary conditions and jumping coefficients. The problem is discretized by the finite element method. The computational domain is decomposed into triangular subdomains that align with the coefficients jumps. We prove that the condition number of the vertex-based DD preconditioner is O((1 + log(H/h))2), independently of the coefficient jumps, where H and h denote the discretization parameters of the coarse and fine triangulations, respectively. Although this preconditioner and its analysis date back to the pioneering work J.H.Bramble, J. E.Pasciak, A.H. Schatz (1986), and it was revisited and extended by many authors including M.Dryja, O.B.Widlund (1990) and A.Toselli, O.B.Widlund (2005), the theory is hard to understand and some details, to our best knowledge, have never been published. In this paper we present all the proofs in detail by means of fundamental calculus.
2015-01-01T00:00:00ZA boundary element method for homogenization of periodic structures
http://hdl.handle.net/10084/139059
A boundary element method for homogenization of periodic structures
Lukáš, Dalibor; Of, Günther; Zapletal, Jan; Bouchala, Jiří
Homogenized coefficients of periodic structures are calculated via an auxiliary partial differential equation in the periodic cell. Typically, a volume finite element discretization is employed for the numerical solution. In this paper, we reformulate the problem as a boundary integral equation using Steklov-Poincare operators. The resulting boundary element method only discretizes the boundary of the periodic cell and the interface between the materials within the cell. We prove that the homogenized coefficients converge super-linearly with the mesh size, and we support the theory with examples in two and three dimensions.
2019-01-01T00:00:00ZA comparison of approaches for the construction of reduced basis for stochastic Galerkin matrix equations
http://hdl.handle.net/10084/139459
A comparison of approaches for the construction of reduced basis for stochastic Galerkin matrix equations
Béreš, Michal
We examine different approaches to an efficient solution of the stochastic Galerkin (SG) matrix equations coming from the Darcy flow problem with different, uncertain coefficients in apriori known subdomains. The solution of the SG system of equations is usually a very challenging task. A relatively new approach to the solution of the SG matrix equations is the reduced basis (RB) solver, which looks for a low-rank representation of the solution. The construction of the RB is usually done iteratively and consists of multiple solutions of systems of equations. We examine multiple approaches and their modifications to the construction of the RB, namely the reduced rational Krylov subspace method and Monte Carlo sampling approach. We also aim at speeding up the process using the deflated conjugate gradients (DCG). We test and compare these methods on a set of problems with a varying random behavior of the material on subdomains as well as different geometries of subdomains.
2020-01-01T00:00:00ZA comparison of deterministic and Bayesian inverse with application in micromechanics
http://hdl.handle.net/10084/133590
A comparison of deterministic and Bayesian inverse with application in micromechanics
Blaheta, Radim; Béreš, Michal; Domesová, Simona; Pan, Pengzhi
The paper deals with formulation and numerical solution of problems of identification of material parameters for continuum mechanics problems in domains with heterogeneous microstructure. Due to a restricted number of measurements of quantities related to physical processes, we assume additional information about the microstructure geometry provided by CT scan or similar analysis. The inverse problems use output least squares cost functionals with values obtained from averages of state problem quantities over parts of the boundary and Tikhonov regularization. To include uncertainties in observed values, Bayesian inversion is also considered in order to obtain a statistical description of unknown material parameters from sampling provided by the Metropolis-Hastings algorithm accelerated by using the stochastic Galerkin method. The connection between Bayesian inversion and Tikhonov regularization and advantages of each approach are also discussed.
2018-01-01T00:00:00Z