Local Multi-Trace formulation

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Table of contents

1 Introduction 
1.1 Context of this work
1.2 Multi-Trace formulations
1.3 Matrix-compression and low-rank approximations
1.4 Summary and Contributions
I Muti-trace formulations 
2 Local Multi-Trace formulation 
2.1 Preliminaries
2.2 Functional and trace spaces
2.2.1 Trace spaces
2.3 Local Multi-Trace operator
2.4 Inverse of the Local Multi-Trace Operator
2.5 Numerical Experiments
2.5.1 Verifying the inversion formula
2.5.2 Preconditioner efficiency
2.6 Conclusions of the chapter
3 Stability of Local-MTF for Maxwell equation 
3.1 Preliminaries
3.2 Problem setting
3.3 Local multi-trace operator for Maxwell equation
3.4 Separation of variables
3.5 Computation of accumulation points
3.6 Numerical results
3.7 Stability of local MTF
3.8 Conclusions of the chapter
II Low-rank approximations 
4 Introduction to Low-Rank approximations 
4.1 Preliminaries
4.2 Best Low-rank Approximation
4.3 Low-Rank Approximation using Pivoted QR Factorization
4.4 Low-rank Approximation using Subspace Iteration
4.5 Conclusions of the chapter
5 Affine low-rank approximations 
5.1 Preliminaries
5.2 Affine Low-rank Approximation
5.2.1 Low-Rank Approximation as Projection of Rows and Columns
5.2.2 Getting an Affine Low-Rank Approximation
5.3 Correlation of Matrices Using their Gravity Center
5.3.1 Matrices with Exponentially Decreasing Singular Values
5.3.2 Characterization of Matrices using their Gravity Center
5.3.3 Measuring the Correlation of Matrices
5.3.4 Matrices with High Correlation
5.4 Numerical Experiments
5.4.1 Low-rank Approximation of Challenging Matrices
5.4.2 Approximation of the Matrix Norm
5.4.3 Analyzing the Correlation Coefficient
5.5 Conclusions of the chapter
6 Liner-time CUR approximations for BEM matrices 
6.1 Preliminaries
6.2 CUR approximations
6.3 Linear-time CUR approximation via Geometric Sampling
6.3.1 Geometrical sampling
6.3.2 Bound on the error of CUR approximation with geometric sampling
6.3.3 Discussion on geometric sampling technique
6.4 Numerical Experiments
6.4.1 BEM matrix from Laplacian kernel
6.4.2 BEM matrix from Exponential kernel
6.4.3 BEM matrix from Gravity kernel
6.4.4 When ACA with partial pivoting fails
6.4.5 Approximating a Hierarchical matrix
6.5 Conclusions of the chapter
7 Conclusion
Bibliography 
A CALRQR: Communication avoiding low-rank QR approximation 
A.1 Communication avoiding algorithm low-rank QR
A.1.1 TSQR: Tall-Skinny QR factorization
A.1.2 Tournament pivoting
A.1.3 CALRQR: Communication avoiding low-rank QR factorization
A.2 Numerical Results
B Extra proofs and algorithms 
B.1 Best Fitting Line Analysis
B.2 Proof of Lemma 4.1
B.3 Algorithms
B.3.1 CUR via Geometric sampling