相关论文: Hierarchical Matrix Operations on GPUs: Matrix-Vec…
Matrix-vector multiplication forms the basis of many iterative solution algorithms and as such is an important algorithm also for hierarchical matrices which are used to represent dense data in an optimized form by applying low-rank…
Hierarchical matrices provide a highly memory-efficient way of storing dense linear operators arising, for example, from boundary element methods, particularly when stored in the H^2 format. In such data-sparse representations, iterative…
The multiplication of matrices is an important arithmetic operation in computational mathematics. In the context of hierarchical matrices, this operation can be realized by the multiplication of structured block-wise low-rank matrices,…
Hierarchical matrices approximate a given matrix by a decomposition into low-rank submatrices that can be handled efficiently in factorized form. $\mathcal{H}^2$-matrices refine this representation following the ideas of fast multipole…
Hierarchical matrices are data-sparse approximations of dense matrices that are widely used for fast matrix computations. Hierarchical matrices are built using a tree data structure, with low-rank blocks identified by various admissibility…
We present a distributed-memory library for computations with dense structured matrices. A matrix is considered structured if its off-diagonal blocks can be approximated by a rank-deficient matrix with low numerical rank. Here, we use…
In this work, we consider the reformulation of hierarchical ($\mathcal{H}$) matrix algorithms for many-core processors with a model implementation on graphics processing units (GPUs). $\mathcal{H}$ matrices approximate specific dense…
The hierarchical matrix framework partitions matrices into subblocks that are either small or of low numerical rank, enabling linear storage complexity and efficient matrix-vector multiplication. This work focuses on the $H^2$-matrix format…
Hierarchical Matrix (H-matrix) is an approximation technique which splits a target dense matrix into multiple submatrices, and where a selected portion of submatrices are low-rank approximated. The technique substantially reduces both time…
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…
Hierarchical $\mathcal{H}^2$-matrices are asymptotically optimal representations for the discretizations of non-local operators such as those arising in integral equations or from kernel functions. Their $O(N)$ complexity in both memory and…
Several researchers have developed a rich toolbox of matrix compression techniques that exploit structure and redundancy in large matrices. Classical methods such as the block low-rank format and the Fast Multipole Method make it possible…
Starting from the local structures to study hierarchical trees is a common research method. However, the cumbersome analysis and description make the naive method challenging to adapt to the increasingly complex hierarchical tree problems.…
Kernel matrices are ubiquitous in computational mathematics, often arising from applications in machine learning and scientific computing. In two or three spatial or feature dimensions, such problems can be approximated efficiently by a…
Boundary integral equations lead to dense system matrices when discretized, yet they are data-sparse. Using the $\mathcal{H}$-matrix format, this sparsity is exploited to achieve $\mathcal{O}(N\log N)$ complexity for storage and…
In this paper, a hierarchical Tucker low-rank (HTLR) matrix is proposed to approximate non-oscillatory kernel functions in linear complexity. The HTLR matrix is based on the hierarchical matrix, with the low-rank blocks replaced by Tucker…
We present high performance implementations of the QR and the singular value decomposition of a batch of small matrices hosted on the GPU with applications in the compression of hierarchical matrices. The one-sided Jacobi algorithm is used…
The discretization of non-local operators, e.g., solution operators of partial differential equations or integral operators, leads to large densely populated matrices. $\mathcal{H}^2$-matrices take advantage of local low-rank structures in…
Hierarchical matrix computations have attracted significant attention in the science and engineering community as exploiting data-sparse structures can significantly reduce the computational complexity of many important kernels. One…
We introduce a data distribution scheme for $\mathcal{H}$-matrices and a distributed-memory algorithm for $\mathcal{H}$-matrix-vector multiplication. Our data distribution scheme avoids an expensive $\Omega(P^2)$ scheduling procedure used…