Dölz, J. ; Egger, H. ; Shashkov, V. (2021)
A fast and oblivious matrix compression algorithm for Volterra integral operators.
In: Advances in Computational Mathematics, 47 (6)
doi: 10.1007/s10444021099026
Article, Bibliographie
This is the latest version of this item.
Abstract
The numerical solution of dynamical systems with memory requires the efficient evaluation of Volterra integral operators in an evolutionary manner. After appropriate discretization, the basic problem can be represented as a matrixvector product with a lower diagonal but densely populated matrix. For typical applications, like fractional diffusion or largescale dynamical systems with delay, the memory cost for storing the matrix approximations and complete history of the data then becomes prohibitive for an accurate numerical approximation. For Volterra integral operators of convolution type, the fast and oblivious convolution quadrature method of Schädle, LopezFernandez, and Lubich resolves this issue and allows to compute the discretized evaluation with N time steps in O(N log N) complexity and only requires O(log N)active memory to store a compressed version of the complete history of the data. We will show that this algorithm can be interpreted as an Hmatrix approximation of the underlying integral operator. A further improvement can thus be achieved, in principle, by resorting to H2matrix compression techniques. Following this idea, we formulate a variant of the H2matrixvector product for discretized Volterra integral operators that can be performed in an evolutionary and oblivious manner and requires only O(N)operations and O(log N)active memory. In addition to the acceleration, more general asymptotically smooth kernels can be treated and the algorithm does not require a priori knowledge of the number of time steps. The efficiency of the proposed method is demonstrated by application to some typical test problems.
Item Type:  Article 

Erschienen:  2021 
Creators:  Dölz, J. ; Egger, H. ; Shashkov, V. 
Type of entry:  Bibliographie 
Title:  A fast and oblivious matrix compression algorithm for Volterra integral operators 
Language:  English 
Date:  26 October 2021 
Publisher:  Springer Science 
Journal or Publication Title:  Advances in Computational Mathematics 
Volume of the journal:  47 
Issue Number:  6 
Collation:  24 Seiten 
DOI:  10.1007/s10444021099026 
Corresponding Links:  
Abstract:  The numerical solution of dynamical systems with memory requires the efficient evaluation of Volterra integral operators in an evolutionary manner. After appropriate discretization, the basic problem can be represented as a matrixvector product with a lower diagonal but densely populated matrix. For typical applications, like fractional diffusion or largescale dynamical systems with delay, the memory cost for storing the matrix approximations and complete history of the data then becomes prohibitive for an accurate numerical approximation. For Volterra integral operators of convolution type, the fast and oblivious convolution quadrature method of Schädle, LopezFernandez, and Lubich resolves this issue and allows to compute the discretized evaluation with N time steps in O(N log N) complexity and only requires O(log N)active memory to store a compressed version of the complete history of the data. We will show that this algorithm can be interpreted as an Hmatrix approximation of the underlying integral operator. A further improvement can thus be achieved, in principle, by resorting to H2matrix compression techniques. Following this idea, we formulate a variant of the H2matrixvector product for discretized Volterra integral operators that can be performed in an evolutionary and oblivious manner and requires only O(N)operations and O(log N)active memory. In addition to the acceleration, more general asymptotically smooth kernels can be treated and the algorithm does not require a priori knowledge of the number of time steps. The efficiency of the proposed method is demonstrated by application to some typical test problems. 
Uncontrolled Keywords:  Volterra integral operators, Convolution quadrature, H2matrices, Matrix compression 
Additional Information:  Erstveröffentlichung; Art.No.: 81; Mathematics Subject Classification (2010): 65D20 · 45D05 
Classification DDC:  500 Science and mathematics > 510 Mathematics 
Divisions:  04 Department of Mathematics 04 Department of Mathematics > Numerical Analysis and Scientific Computing 
Date Deposited:  08 May 2024 11:49 
Last Modified:  25 Jun 2024 09:34 
PPN:  51936032X 
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A fast and oblivious matrix compression algorithm for Volterra integral operators. (deposited 30 Apr 2024 12:40)
 A fast and oblivious matrix compression algorithm for Volterra integral operators. (deposited 08 May 2024 11:49) [Currently Displayed]
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