, Volume 54, Issue 3, pp 963–1007 | Cite as

Data assimilation and sampling in Banach spaces

  • Ronald DeVore
  • Guergana PetrovaEmail author
  • Przemyslaw Wojtaszczyk


This paper studies the problem of approximating a function f in a Banach space \(\mathcal{X}\) from measurements \(l_j(f)\), \(j=1,\ldots ,m\), where the \(l_j\) are linear functionals from \(\mathcal{X}^*\). Quantitative results for such recovery problems require additional information about the sought after function f. These additional assumptions take the form of assuming that f is in a certain model class \(K\subset \mathcal{X}\). Since there are generally infinitely many functions in K which share these same measurements, the best approximation is the center of the smallest ball B, called the Chebyshev ball, which contains the set \(\bar{K}\) of all f in K with these measurements. Therefore, the problem is reduced to analytically or numerically approximating this Chebyshev ball. Most results study this problem for classical Banach spaces \(\mathcal{X}\) such as the \(L_p\) spaces, \(1\le p\le \infty \), and for K the unit ball of a smoothness space in \(\mathcal{X}\). Our interest in this paper is in the model classes \(K=\mathcal{K}(\varepsilon ,V)\), with \(\varepsilon >0\) and V a finite dimensional subspace of \(\mathcal{X}\), which consists of all \(f\in \mathcal{X}\) such that \(\mathrm{dist}(f,V)_\mathcal{X}\le \varepsilon \). These model classes, called approximation sets, arise naturally in application domains such as parametric partial differential equations, uncertainty quantification, and signal processing. A general theory for the recovery of approximation sets in a Banach space is given. This theory includes tight a priori bounds on optimal performance and algorithms for finding near optimal approximations. It builds on the initial analysis given in Maday et al. (Int J Numer Method Eng 102:933–965, 2015) for the case when \(\mathcal{X}\) is a Hilbert space, and further studied in Binev et al. (SIAM UQ, 2015). It is shown how the recovery problem for approximation sets is connected with well-studied concepts in Banach space theory such as liftings and the angle between spaces. Examples are given that show how this theory can be used to recover several recent results on sampling and data assimilation.


Optimal recovery Reduced modeling Data assimilation Sampling 

Mathematics Subject Classification

46B99 41A65 94A20 


  1. 1.
    Adcock, B., Hansen, A.C.: Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon. Appl. Comput. Harm. Anal. 32, 357–388 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adcock, B., Hansen, A.C., Poon, C.: Beyond consistent reconstructions: optimality and sharp bounds for generalized sampling, and application to the uniform resampling problem. SIAM J. Math. Anal. 45, 3132–3167 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Adcock, B., Hansen, A.C., Shadrin, A.: A stability barrier for reconstructions from Fourier samples. SIAM J. Numer. Anal. 52, 125–139 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed Point Theory for Lipschitzian-Type Mappings and Applications. Springer, Berlin (2009)Google Scholar
  5. 5.
    Bartle, R., Graves, L.: Mappings between function spaces. Trans. Am. Math. Soc. 72, 400–413 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis, vol. 48. American Mathematical Society Colloquium Publications, Providence, RI (2000)zbMATHGoogle Scholar
  7. 7.
    Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G., Wojtaszczyk, P.: Convergence rates for Greedy Algorithms in reduced basis methods. SIAM J. Math. Anal. 43, 1457–1472 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G., Wojtaszczyk, P.: Data Assimilation in Reduced Modeling, SIAM UQ, to appear; arXiv: 1506.04770v1 (2015)
  9. 9.
    , B.: Optimal Recovery of Functions and Integrals, First European Congress of Mathematics, Vol. I, pp. 371–390 (1992), Progr. Math., Birkhauser, Basel, 119 (1994)Google Scholar
  10. 10.
    Brown, A.: A rotund space have a subspace of codimension 2 with discontinuous metric projection. Mich. Math. J. 21, 145–151 (1974)CrossRefzbMATHGoogle Scholar
  11. 11.
    Cohen, A., DeVore, R.: Approximation of high dimensional parametric PDEs. Acta Numer. 24, 1–159 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Demanet, L., Townsend, A.: Stable extrapolation of analytic functions, preprintGoogle Scholar
  13. 13.
    DeVore, R., Lorentz, G.: Constructive Approximation, vol. 303. Springer, Grundlehren (1993)zbMATHGoogle Scholar
  14. 14.
    Figiel, T.: On the moduli of convexity and smoothness. Stud. Math. 56, 121–155 (1976)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Figiel, T., Lindenstrauss, J., Milman, V.: The dimension of almost spherical sections of convex bodies. Acta Math. 139, 53–94 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fournier, J.: An interpolation problem for coefficients of \(H^\infty \) functions. Proc. Am. Math. Soc. 42, 402–407 (1974)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Garkavi, A.: The Best Possible Net and the Best Possible Cross Section of a Set in a Normed Space, Vol. 39, pp. 111–132. Translations Series 2. American Mathematical Society, Providence, RI (1964)Google Scholar
  18. 18.
    Hanner, O.: On the uniform convexity of \(L^p\) and \(\ell ^p\). Ark. Matematik 3, 239–244 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Henrion, D., Tarbouriech, S., Arzelier, D.: LMI approximations for the radius of the intersection of ellipsoids: survey. J. Optim. Theory Appl. 108, 1–28 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kadec, M., Snobar, M.: Certain functionals on the Minkowski compactum. Mat. Zamet. 10, 453–457 (1971). (Russian)MathSciNetGoogle Scholar
  21. 21.
    Konyagin, S.: A remark on renormings of nonreflexive spaces and the existence of a Chebyshev center. Mosc. Univ. Math. Bull. 43, 55–56 (1988)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Lewis, J., Lakshmivarahan, S., Dhall, S.: Dynamic Data Assimilation: A Least Squares Approach, Encyclopedia of Mathematics and its Applications, vol. 104. Cambridge University Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  23. 23.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)CrossRefzbMATHGoogle Scholar
  24. 24.
    Maday, Y., Patera, A., Penn, J., Yano, M.: A parametrized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics. Int. J. Numer. Method Eng. 102, 933–965 (2015)CrossRefzbMATHGoogle Scholar
  25. 25.
    Micchelli, C., Rivlin, T.: Lectures on optimal recovery, numerical analysis, Lancaster 1984 (Lancaster, 1984), 21–93. Lecture Notes in Math, vol. 1129. Springer, Berlin (1985)Google Scholar
  26. 26.
    Micchelli, C., Rivlin, T., Winograd, S.: The optimal recovery of smooth functions. Numerische Mathematik 26, 191–200 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Milman, V., Schechtman, G.: Asymptotic Theory of Finite Dimensional Normed Spaces. Lecture Notes in Mathematics, vol. 1200. Springer, Berlin (1986)Google Scholar
  28. 28.
    Powell, M.: Approximation Theory and Methods. Cambridge University Press, Cambridge (1981)zbMATHGoogle Scholar
  29. 29.
    Platte, R., Trefethen, L., Kuijlaars, A.: Impossibility of fast stable approximation of analytic functions from equispaced samples. SIAM Rev. 53, 308–318 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Repovš, D., Semenov, P.: Continuous Selections of Multivalued Mappings. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  31. 31.
    Singer, I.: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Die Grundlehren der mathematischen Wissenschaften, vol. 171. Springer, Berlin (1970)CrossRefGoogle Scholar
  32. 32.
    Schönhage, A.: Fehlerfort pflantzung bei interpolation. Numer. Math. 3, 62–71 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Smith, P., Ward, J.: Restricted centers in \(C(\Omega )\). Proc. Am. Math. Soc. 48, 165–172 (1975)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Szarek, S.: On the best constants in the Khinchin inequality. Stud. Math. 58, 197–208 (1976)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Traub, J., Wozniakowski, H.: A General Theory of Optimal Algorithms. Academic Press, London (1980)zbMATHGoogle Scholar
  36. 36.
    Trefethen, L.N., Weideman, J.A.C.: Two results on polynomial interpolation in equally spaced points. J. Approx. Theory 65, 247–260 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wojtaszczyk, P.: Banach Spaces for Analysts. Cambridge University Press, Cambridge (1991)CrossRefzbMATHGoogle Scholar
  38. 38.
    Zygmund, A.: Trigonometric Series. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia 2017

Authors and Affiliations

  • Ronald DeVore
    • 1
  • Guergana Petrova
    • 1
    Email author
  • Przemyslaw Wojtaszczyk
    • 2
  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.Interdisciplinary Center for Mathematical and Computational ModellingUniversity of WarsawWarsawPoland

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