Efficient Numerical Methods for Non-local Operators: by Steffen Borm

By Steffen Borm

Hierarchical matrices current an effective method of treating dense matrices that come up within the context of quintessential equations, elliptic partial differential equations, and keep an eye on idea. whereas a dense $n\times n$ matrix in usual illustration calls for $n^2$ devices of garage, a hierarchical matrix can approximate the matrix in a compact illustration requiring basically $O(n okay \log n)$ devices of garage, the place $k$ is a parameter controlling the accuracy. Hierarchical matrices were effectively utilized to approximate matrices coming up within the context of boundary critical equipment, to build preconditioners for partial differential equations, to judge matrix features, and to unravel matrix equations utilized in keep an eye on conception. $\mathcal{H}^2$-matrices provide a refinement of hierarchical matrices: utilizing a multilevel illustration of submatrices, the potency should be considerably stronger, relatively for big difficulties. This booklet provides an advent to the elemental innovations and offers a basic framework that may be used to research the complexity and accuracy of $\mathcal{H}^2$-matrix concepts. ranging from easy principles of numerical linear algebra and numerical research, the idea is built in a simple and systematic method, obtainable to complex scholars and researchers in numerical arithmetic and medical computing. targeted recommendations are required basically in remoted sections, e.g., for convinced periods of version difficulties. A ebook of the eu Mathematical Society (EMS). disbursed in the Americas by means of the yankee Mathematical Society.

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Additional resources for Efficient Numerical Methods for Non-local Operators: $\mathcal{h}^2$-matrix Compression, Algorithms and Analysis

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22 (Number of clusters). Let T« be a cluster tree constructed by Algorithm 2. 11. Proof. Due to « ¤ ;, Algorithm 2 ensures that tO ¤ ; holds for all t 2 T« . 9, the index sets tO of all leaf clusters t 2 L« are disjoint, so there cannot be more than n« leaf clusters. 52. 23 (Improved bound). , we have #tO m for all leaf clusters t 2 L« . 9, all leaves correspond to disjoint index sets, and since all of these sets contain at least m indices, there cannot be more than n« =m leaves. 52 yields the bound n« 1 c« Ä 2 m for the number of clusters.

62], [68], [67], [52], [63]). 5 introduces general cluster bases and the basic concepts for estimating the complexity of algorithms for H 2 -matrices. 6 is devoted to the general definition of H 2 -matrices and H 2 -matrix spaces and to proving bounds for the storage complexity (cf. [70]). 7 presents the most important algorithm in the context of H 2 -matrices: the evaluation of the product of an H 2 -matrix and an arbitrary vector. 8 contains a number of definitions that allow us to express complexity estimates in terms of matrix dimensions instead of numbers of clusters.

T/ with i 2 tOi . t/ since the i -th row of V t 0 can only differ from zero if i 2 tO0 , and we have already established that this only happens for t 0 D ti . , the matrix V t corresponding to any non-leaf cluster t can be expressed in terms of the matrices V t 0 corresponding to the sons of t . Cluster bases with this property are called nested. V t / t2T« : we store V t only for leaf clusters and use the expansion matrices E t to express all other cluster basis matrices. Since the expansion matrices require only m2 units of storage, this representation is far more efficient than the original one.

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