Challenging the Curse of Dimensionality in Multidimensional Numerical Integration by Using a Low-Rank Tensor-Train Format
Numerical integration is a basic step in the implementation of more complex numerical algorithms suitable, for example, to solve ordinary and partial differential equations. The straightforward extension of a one-dimensional integration rule to a multidimensional grid by the tensor product of the sp...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/3/534 |
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| author | Boian Alexandrov Gianmarco Manzini Erik W. Skau Phan Minh Duc Truong Radoslav G. Vuchov |
| author_facet | Boian Alexandrov Gianmarco Manzini Erik W. Skau Phan Minh Duc Truong Radoslav G. Vuchov |
| author_sort | Boian Alexandrov |
| collection | DOAJ |
| description | Numerical integration is a basic step in the implementation of more complex numerical algorithms suitable, for example, to solve ordinary and partial differential equations. The straightforward extension of a one-dimensional integration rule to a multidimensional grid by the tensor product of the spatial directions is deemed to be practically infeasible beyond a relatively small number of dimensions, e.g., three or four. In fact, the computational burden in terms of storage and floating point operations scales exponentially with the number of dimensions. This phenomenon is known as the curse of dimensionality and motivated the development of alternative methods such as the Monte Carlo method. The tensor product approach can be very effective for high-dimensional numerical integration if we can resort to an accurate low-rank tensor-train representation of the integrand function. In this work, we discuss this approach and present numerical evidence showing that it is very competitive with the Monte Carlo method in terms of accuracy and computational costs up to several hundredths of dimensions if the integrand function is regular enough and a sufficiently accurate low-rank approximation is available. |
| first_indexed | 2024-03-11T09:35:08Z |
| format | Article |
| id | doaj.art-1983f35fc3704ec9aa0b3c4f9e5d097b |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T09:35:08Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-1983f35fc3704ec9aa0b3c4f9e5d097b2023-11-16T17:21:00ZengMDPI AGMathematics2227-73902023-01-0111353410.3390/math11030534Challenging the Curse of Dimensionality in Multidimensional Numerical Integration by Using a Low-Rank Tensor-Train FormatBoian Alexandrov0Gianmarco Manzini1Erik W. Skau2Phan Minh Duc Truong3Radoslav G. Vuchov4Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USATheoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USATheoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USATheoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USAComputational Mathematics Department, Sandia National Laboratories, Albuquerque, NM 87185, USANumerical integration is a basic step in the implementation of more complex numerical algorithms suitable, for example, to solve ordinary and partial differential equations. The straightforward extension of a one-dimensional integration rule to a multidimensional grid by the tensor product of the spatial directions is deemed to be practically infeasible beyond a relatively small number of dimensions, e.g., three or four. In fact, the computational burden in terms of storage and floating point operations scales exponentially with the number of dimensions. This phenomenon is known as the curse of dimensionality and motivated the development of alternative methods such as the Monte Carlo method. The tensor product approach can be very effective for high-dimensional numerical integration if we can resort to an accurate low-rank tensor-train representation of the integrand function. In this work, we discuss this approach and present numerical evidence showing that it is very competitive with the Monte Carlo method in terms of accuracy and computational costs up to several hundredths of dimensions if the integrand function is regular enough and a sufficiently accurate low-rank approximation is available.https://www.mdpi.com/2227-7390/11/3/534multidimensional numerical integrationhighly-dimensional cubaturesGauss-Legendre integration ruleClenshaw-Curtis integration rule |
| spellingShingle | Boian Alexandrov Gianmarco Manzini Erik W. Skau Phan Minh Duc Truong Radoslav G. Vuchov Challenging the Curse of Dimensionality in Multidimensional Numerical Integration by Using a Low-Rank Tensor-Train Format Mathematics multidimensional numerical integration highly-dimensional cubatures Gauss-Legendre integration rule Clenshaw-Curtis integration rule |
| title | Challenging the Curse of Dimensionality in Multidimensional Numerical Integration by Using a Low-Rank Tensor-Train Format |
| title_full | Challenging the Curse of Dimensionality in Multidimensional Numerical Integration by Using a Low-Rank Tensor-Train Format |
| title_fullStr | Challenging the Curse of Dimensionality in Multidimensional Numerical Integration by Using a Low-Rank Tensor-Train Format |
| title_full_unstemmed | Challenging the Curse of Dimensionality in Multidimensional Numerical Integration by Using a Low-Rank Tensor-Train Format |
| title_short | Challenging the Curse of Dimensionality in Multidimensional Numerical Integration by Using a Low-Rank Tensor-Train Format |
| title_sort | challenging the curse of dimensionality in multidimensional numerical integration by using a low rank tensor train format |
| topic | multidimensional numerical integration highly-dimensional cubatures Gauss-Legendre integration rule Clenshaw-Curtis integration rule |
| url | https://www.mdpi.com/2227-7390/11/3/534 |
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