Robust preconditioning and error estimates for optimal control of the convection-diffusion-reaction equation with limited observation in Isogeometric analysis
In this paper we analyze an optimization problem with limited observation governed by a convection--diffusion--reaction equation. Motivated by a Schur complement approach, we arrive at continuous norms that enable analysis of well-posedness and subsequent derivation of error analysis and a preconditioner that is robust with respect to the parameters of the problem. We provide conditions for inf-sup stable discretizations and present one such discretization for box domains with constant convection. We also provide a priori error estimates for this discretization. The preconditioner requires a fourth order problem to be solved. For this reason, we use Isogeometric Analysis as a method of discretization. To efficiently realize the preconditioner, we consider geometric multigrid with a standard Gauss-Seidel smoother as well as a new macro Gauss-Seidel smoother. The latter smoother provides good results with respect to both the geometry mapping and the polynomial degree.
This paper deals with the fast solution of linear systems associated with the mass matrix, in the context of isogeometric analysis. We propose a preconditioner that is both efficient and easy to implement, based on a diagonal-scaled Kronecker product of univariate parametric mass matrices. Its application is faster than a matrix-vector product involving the mass matrix itself. We prove that the condition number of the preconditioned matrix converges to 1 as the mesh size is reduced, that is, the preconditioner is asymptotically equivalent to the exact inverse. Moreover, we give numerical evide…
Robust Preconditioning for Space-Time Isogeometric Analysis of Parabolic Evolution Problems
Johannes Kepler University（约翰内斯·开普勒大学）；Institute of Computational Mathematics, Johannes Kepler University（约翰内斯·开普勒大学计算数学研究所）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉顿计算与应用数学研究所）
In this work we focus on the preconditioning of a Galerkin space-time isogeometric discretization of the heat equation. Exploiting the tensor product structure of the basis functions in the parametric domain, we propose a preconditioner that is the sum of Kronecker products of matrices and that can be efficiently applied thanks to an extension of the classical Fast Diagonalization method. The preconditioner is robust w.r.t. the polynomial degree of the spline space and the time required for the application is almost proportional to the number of degrees-of-freedom, for a serial execution. By i…
Scalable multigrid methods for immersed finite element methods and immersed isogeometric analysis
Eindhoven University of Technology（埃因霍温理工大学）；University of Colorado Boulder（科罗拉多大学博尔德分校）；German Aerospace Center (DLR)（德国航空航天中心）；Walt Disney Animation Studios（华特迪士尼动画工作室）
We derive novel, fast, and parameter-robust preconditioned iterative methods for steady and time-dependent Navier--Stokes control problems. Our approach may be applied to time-dependent problems which are discretized using backward Euler or Crank--Nicolson, and is also a valuable candidate for Stokes control problems discretized using Crank--Nicolson. The key ingredients of the solver are a saddle-point type approximation for the linear systems, an inner iteration for the $(1,1)$-block accelerated by a preconditioner for convection--diffusion control, and an approximation to the Schur compleme…
Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. II: The incompressible Navier-Stokes equations
Motivated by a wide range of real-world problems whose solutions exhibit boundary and interior layers, the numerical analysis of discretizations of singularly perturbed differential equations is an established sub-discipline within the study of the numerical approximation of solutions to differential equations. Consequently, much is known about how to accurately and stably discretize such equations on \textit{a priori} adapted meshes, in order to properly resolve the layer structure present in their continuum solutions. However, despite being a key step in the numerical simulation process, muc…
Analysis of Block Jacobi/Gauss-Seidel and additive/multiplicative Schwarz preconditioning through the theory of GLT sequences, with applications to domain decomposition discretizations
University of Rome Tor Vergata（罗马托尔维加塔大学）；University Mohammed VI Polytechnic（穆罕默德六世polytechnic大学）；University of Insubria（因苏布里亚大学）；Uppsala University（乌普萨拉大学）
Monolithic and Block Overlapping Schwarz Preconditioners for the Incompressible Navier-Stokes Equations
Delft Institute of Applied Mathematics, Delft University of Technology（代尔夫特理工大学应用数学研究所）；Department of Mathematics and Computer Science, University of Cologne（科隆大学数学与计算机科学系）；Center for Data and Simulation Science, University of Cologne（科隆大学数据与模拟科学中心）
This work addresses optimal control problems governed by a linear time-dependent partial differential equation (PDE) as well as integer constraints on the control. Moreover, partial observations are assumed in the objective function. The resulting problem poses several numerical challenges due to the mixture of combinatorial aspects, induced by integer variables, and large scale linear algebra issues, arising from the PDE discretization. Since classical solution approaches such as the branch-and-bound framework are typically overwhelmed by such large-scale problems, this work extends an improv…
In this article, we derive a new, fast, and robust preconditioned iterative solution strategy for the all-at-once solution of optimal control problems with time-dependent PDEs as constraints, including the heat equation and the non-steady convection--diffusion equation. After applying an optimize-then-discretize approach, one is faced with continuous first-order optimality conditions consisting of a coupled system of PDEs. As opposed to most work in preconditioning the resulting discretized systems, where a (first-order accurate) backward Euler method is used for the discretization of the time…
