• Corpus ID: 119645830

Adaptive compression of large vectors

@inproceedings{Borm2015AdaptiveCO,
  title={Adaptive compression of large vectors},
  author={Steffen Borm},
  year={2015}
}
Numerical algorithms for elliptic partial differential equations frequently employ error estimators and adaptive mesh refinement strategies in order to reduce the computational cost. We can extend these techniques to general vectors by splitting the vectors into a hierarchically organized partition of subsets and using appropriate bases to represent the corresponding parts of the vectors. This leads to the concept of hierarchical vectors. A hierarchical vector with m subsets and bases of rank k… 

References

SHOWING 1-10 OF 18 REFERENCES
Adaptive wavelet methods for elliptic operator equations: Convergence rates
TLDR
The main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error O(N -s ) in the energy norm, whenever such a rate is possible by N-term approximation.
Introduction to Hierarchical Matrices with Applications
Efficient arithmetic operations for rank-structured matrices based on hierarchical low-rank updates
TLDR
This work considers algorithms for performing algebraic operations with rank-structured matrices, i.e., for approximating the matrix product, inverse or factorizations in almost linear complexity, based on local low-rank updates that can be performed in linear complexity.
Data-sparse Approximation by Adaptive ℋ2-Matrices
TLDR
The basic ideas of ℋ- andℋ2-matrices are introduced and an algorithm that adaptively computes approximations of general matrices in the latter format is presented.
A Sparse ℋ-Matrix Arithmetic.
Abstract The preceding Part I of this paper has introduced a class of matrices (ℋ-matrices) which are data-sparse and allow an approximate matrix arithmetic of almost linear complexity. The matrices
A Sparse Matrix Arithmetic Based on H-Matrices. Part I: Introduction to H-Matrices
TLDR
This paper is the first of a series and is devoted to the first introduction of the H-matrix concept, which allows the exact inversion of tridiagonal matrices.
Approximation of Integral Operators by Variable-Order Interpolation
TLDR
A data-sparse, recursive matrix representation, so-called -matrices, is employed for the efficient treatment of discretized integral operators, showing that the optimal convergence is retained for the classical double layer potential discretization with piecewise constant functions.
Construction and Arithmetics of H-Matrices
TLDR
This paper presents a construction of the hierarchical matrix format for standard finite element and boundary element applications for which two criteria, the sparsity and idempotency, are sufficient to give the desired bounds.
A convergent adaptive algorithm for Poisson's equation
We construct a converging adaptive algorithm for linear elements applied to Poisson’s equation in two space dimensions. Starting from a macro triangulation, we describe how to construct an initial
Data Oscillation and Convergence of Adaptive FEM
TLDR
A simple and efficient adaptive FEM for elliptic partial differential equations (PDEs) with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants is constructed.
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