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This is a header-only library providing a fast matrix-vector product and linear system solver for tensor product matrices with a downward-closed index set restriction. This can especially be applied to compute interpolation coefficients for a sparse grid basis or evaluate a sparse interpolant at sparse grid points. The library code is located in the src folder. This is a header-only library providing a fast matrix-vector product and linear system solver for tensor product matrices with a downward-closed index set restriction. This can especially be applied to compute interpolation coefficients for a sparse grid basis or evaluate a sparse interpolant at sparse grid points. The library code is located in the src folder.
The fast_sparse_interpolation library is published under an Apache 2.0 license. If you use this project for research purposes, please cite the following publication which describes the mathematical background: The fast_sparse_interpolation library is published under an Apache 2.0 license. If you use this project for research purposes, please cite the following publication which describes the mathematical background:
- David Holzmüller, Dirk Pflüger: Fast Sparse Grid Interpolation Coefficients Via LU Decomposition (to be published in 2019). - David Holzmüller, Dirk Pflüger: Fast Sparse Grid Operations: A Unified Framework using Sparsified Generalized Triangular Tensor Products (currently submitted).
The algorithm was first proposed in: The essential ideas behind the algorithm were first proposed in:
- Gustavo Avila and Tucker Carrington Jr.: A multi-dimensional Smolyak collocation method in curvilinear coordinates for computing vibrational spectra (2015). - Gustavo Avila and Tucker Carrington Jr.: A multi-dimensional Smolyak collocation method in curvilinear coordinates for computing vibrational spectra (2015).
Example code for usage can be found in test/main.cpp. This file can be compiled by executing the top-level Makefile (make debug / make release). This Makefile executes cmake on CMakeLists.txt to generate a Makefile in the build folder, which is then automatically executed using make to compile the executable test code. The test folder also contains code for performance measurement and plotting of performance data, which we used for our 2019 paper. Example code for usage can be found in test/main.cpp. This file can be compiled by executing the top-level Makefile (make debug / make release). This Makefile executes cmake on CMakeLists.txt to generate a Makefile in the build folder, which is then automatically executed using make to compile the executable test code. The test folder also contains code for performance measurement and plotting of performance data, which we used for our 2019 paper.
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- The class BoundedSumIterator provides an iterator that allows to iterate over a downward closed index set, specifically a set of the form {(i_1, ..., i_d) | i_1, ..., i_d >= 0, i_1 + ... + i_d <= bound}. It is possible to provide other iterator classes with the same interface in order to use different downward closed index sets. - The class BoundedSumIterator provides an iterator that allows to iterate over a downward closed index set, specifically a set of the form {(i_1, ..., i_d) | i_1, ..., i_d >= 0, i_1 + ... + i_d <= bound}. It is possible to provide other iterator classes with the same interface in order to use different downward closed index sets.
- The function evaluateFunction() evaluates a function at grid points and returns a MultiDimVector with the function values. - The function evaluateFunction() evaluates a function at grid points and returns a MultiDimVector with the function values.
- The function createInterpolationOperator() creates a SparseTPOperator whose matrix corresponds to a sparse grid interpolation problem with given basis functions and grid points. - The function createInterpolationOperator() creates a SparseTPOperator whose matrix corresponds to a sparse grid interpolation problem with given basis functions and grid points.
- For custom extensions (such as evaluating derivatives), helper functions for multiplication with matrices of the form ($I \otimes \hdots \otimes I \otimes M$) are available (for arbitrary M, for lower triangular M, for upper triangular M and for M = I). Each single multiplication permutes the indices in the MultiDimVector, after d multiplications (where d is the number of dimensions), the indices are ordered as in the beginning. - For custom extensions (such as evaluating derivatives), helper functions for multiplication with matrices of the form ($\widehat{I \otimes \hdots \otimes I \otimes M}$) are available (for arbitrary M, for lower triangular M, for upper triangular M and for M = I). Each single multiplication permutes the indices in the MultiDimVector, after d multiplications (where d is the number of dimensions), the indices are ordered as in the beginning.
The basis functions and grid points used in the examples are only easy-to-implement examples. For practical purposes, you should probably use other basis functions and grid points, for example ones provided in the SG++ library. Recommendations: The basis functions and grid points used in the examples are only easy-to-implement examples. For practical purposes, you should probably use other basis functions and grid points, for example ones provided in the SG++ library. Recommendations:
- Legendre polynomials and Leja points (or L2-Leja points, see SG++) - Legendre polynomials and Leja points (or L2-Leja points, see SG++)
- Splines and uniform points - Splines and uniform points
- Maybe also kernel functions and uniform points - Maybe also kernel functions or trigonometric functions and uniform points
Note for developers:
To run the script ./prepend_header.sh, modify it to only prepend the header in files that do not contain one and run
shopt -s globstar
beforehand