Related papers: Estimates for generalized sparse grid hierarchical…
Coupled multiphysics problems often give rise to interface conditions naturally formulated in fractional Sobolev spaces. Here, both positive and negative fractionality are common. When designing efficient solvers for discretizations of such…
In this paper we study convergence estimates for a multigrid algorithm with smoothers of successive subspace correction (SSC) type, applied to symmetric elliptic PDEs. First, we revisit a general convergence analysis on a class of multigrid…
In this work, we extend the sparse iterative covariance-based estimator (SPICE), by generalizing the formulation to allow for different norm constraints on the signal and noise parameters in the covariance model. For a given norm, the…
This contribution presents a hierarchical multigrid approach for the solution of large-scale finite cell problems on both uniform grids and multi-level hp-discretizations. The proposed scheme leverages the hierarchical nature of the basis…
This paper concerns the preconditioning technique for discrete systems arising from time-harmonic Maxwell equations with absorptions, where the discrete systems are generated by N\'ed\'elec finite element methods of fixed order on meshes…
We describe a second-order accurate approach to sparsifying the off-diagonal blocks in the hierarchical approximate factorizations of sparse symmetric positive definite matrices. The norm of the error made by the new approach depends…
In this work we extend the shifted Laplacian approach to the elastic Helmholtz equation. The shifted Laplacian multigrid method is a common preconditioning approach for the discretized acoustic Helmholtz equation. In some cases, like…
In this paper we are concerned with restricted additive Schwarz with local impedance transformation conditions for a family of Helmholtz problems in two dimensions. These problems are discretized by the finite element method with conforming…
Common techniques for the spatial discretisation of PDEs on a macroscale grid include finite difference, finite elements and finite volume methods. Such methods typically impose assumed microscale structures on the subgrid fields, so…
For real matrices of full column-rank, we analyze the conditioning of several types of normal equations that are preconditioned by a randomized preconditioner computed in lower precision. These include symmetrically preconditioned normal…
An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into "local" subspaces and a global "coarse" space is developed. Particular applications of this…
For non-preconditioned Galerkin systems, the condition number grows with the number of elements as well as the quotient of the maximal and the minimal mesh-size. Therefore, reliable and effective numerical computations, in particular on…
Our main interest in this paper is to study some approximation problems for classes of functions with mixed smoothness. We use technique, based on a combination of results from hyperbolic cross approximation, which were obtained in 1980s --…
A well-known strategy for building effective preconditioners for higher-order discretizations of some PDEs, such as Poisson's equation, is to leverage effective preconditioners for their low-order analogs. In this work, we show that…
We present a unified framework for estimation and analysis of generalized additive models in high dimensions. The framework defines a large class of penalized regression estimators, encompassing many existing methods. An efficient…
We propose and analyze a robust BPX preconditioner for the integral fractional Laplacian on bounded Lipschitz domains. For either quasi-uniform grids or graded bisection grids, we show that the condition numbers of the resulting systems…
In this work, an extension of the generalized mixed Schwarz inequality is proved. A companion of the generalized mixed Schwarz inequality is established by merging both Cartesian and Polar decompositions of operators. Based on that some…
Coupled multiphysics problems often give rise to interface conditions naturally formulated in fractional Sobolev spaces. Here, both positive- and negative fractionality are common. When designing efficient solvers for discretizations of…
Two-level domain decomposition methods are preconditioned Krylov solvers. What separates one- and two-level domain decomposition methods is the presence of a coarse space in the latter. The abstract Schwarz framework is a formalism that…
Schmidt's subspace theorem in terms of Seshadri constants for closed subschemes in subgeneral position has been already developed sharply. We derive our theorem for numerically equivalent ample divisors by dint of the above theory step by…