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Part of the book series: Scientific Computation ((SCIENTCOMP))

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Abstract

The question of high-dimension model representation (HDMR) is of increasing importance in computational mathematics and science.

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Notes

  1. 1.

    ‘Level’, in the context of Lagrange interpolation with the Chebyshev rule, implies the highest order polynomial that can be interpolated exactly.

References

  1. V. Barthelmann, E. Novak, K. Ritter, High dimensional polynomial interpolation on sparse grids. Adv. Comput. Math. 12, 273–288 (2000)

    Article  MathSciNet  Google Scholar 

  2. C. Cai, M. Lambert, Sparse grid-nested sampling for model selection in eddy-current testing, in Conference Paper, 20th International Workshop on Electromagnetic Nondestructive Evaluation (ENDE 2015), Sendai (2015)

    Google Scholar 

  3. J. Garcke, Sparse grid tutorial. http://en.wikipedia.org/wiki/Sparse-grid

  4. J. Garcke, M. Griebel, Data mining with sparse grids using simplicial basis functions. http://en.wikipedia.org/wiki/Sparse-grid

  5. M. Griebel, Sparse grids and related approximation schemes for higher dimensional problems, in Proceedings Foundations of Computational Mathematics 2005, (FoCM05), Santander, ed. by L. Pardo, A. Pinkus, E. Suli, M. Todd (Cambridge University Press, Cambridge, 2006), pp. 106–161

    Chapter  Google Scholar 

  6. A. Klimke, Sparse grid interpolation toolbox user’s guide, V. 5.1 (2008). http://www.ians.uni-stuttgart.de/spinterp

  7. A. Klimke, B. Wolhmuth, Algorithm 847: spinterp: piecewise multilinear hierarchical sparse grid interpolation in matlab. ACM Trans. Math. Softw. 31(4), 561–579 (2005)

    Article  MathSciNet  Google Scholar 

  8. E.K. Miller, Adaptive sparse sampling to estimate radiation and scattering patterns to a specified uncertainty with model-based parameter estimation. IEEE Antennas Propag. Mag. 103–113 (2015)

    Google Scholar 

  9. H.A. Sabbagh, R. Kim Murphy, E.H. Sabbagh, J.C. Aldrin, J.S. Knopp, Computational Electromagnetics and Model-Based Inversion: A Modern Paradigm for Eddy-Current Nondestructive Evaluation (Springer, New York, 2013)

    Book  Google Scholar 

  10. S.A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math. Dokl. 4, 240–243 (1963)

    MATH  Google Scholar 

  11. M. Stoyanov, User manual: TASMANIAN sparse grids. Oak Ridge National Laboratory (2013)

    Google Scholar 

  12. N. Zabaras, Solving stochastic inverse problems: a sparse grid collocation approach, in Computational Methods for Large-Scale Inverse Problems and Quantification of Uncertainty, ed. by People on Earth (Wiley, Hoboken, 2001)

    Google Scholar 

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Sabbagh, H.A., Kim Murphy, R., Sabbagh, E.H., Zhou, L., Wincheski, R. (2021). High-Dimension Model Representation via Sparse Grid Techniques. In: Advanced Electromagnetic Models for Materials Characterization and Nondestructive Evaluation. Scientific Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-67956-9_9

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