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Change of variable in spaces of mixed smoothness and numerical integration of multivariate functions on the unit cube. (English) Zbl 1376.65028
This paper deals with the optimal approximation of multivariate integrals \(\int_{[0,1]^d} f(x)\,{\mathrm d}x\) of nonperiodic/periodic integrands \(f\) via Frolov cubature formulae. The authors show that the asymptotic order of the minimal worst-case integration error is not affected by boundary conditions in Besov spaces and Triebel-Lizorkin spaces of functions with dominating mixed smoothness. As essential tool of this research, the Besov spaces and Triebel-Lizorkin spaces of dominating mixed smoothness are characterized by mixed iterated differences. The authors present two modified Frolov cubature formulae that yield the same order of convergence up to a constant. This constant involves the norms of a change of variable mapping and a pointwise multiplication mapping between function spaces of dominating mixed smoothness. In the main part of this paper, the authors present new results on boundedness of change of variable mapping and pointwise multiplication mapping between function spaces of dominating mixed smoothness.

65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47B38 Linear operators on function spaces (general)
Full Text: DOI
[1] Amanov, T.I.: Spaces of Differentiable Functions with Dominating Mixed Derivatives. Nauka Kaz. SSR, Alma-Ata (1976)
[2] Bakhvalov, NS, Optimal convergence bounds for quadrature processes and integration methods of Monte Carlo type for classes of functions, Zh. Vychisl. Mat. i Mat. Fiz., 4, 5-63, (1963)
[3] Bykovskii, V.A.: On the correct order of the error of optimal cubature formulas in spaces with dominant derivative, and on quadratic deviations of grids. Akad. Sci. USSR, Vladivostok, Computing Center Far-Eastern Scientific Center (Preprint, 1985)
[4] Christ, M; Seeger, A, Necessary conditions for vector-valued operator inequalities in harmonic analysis., Proc. Lond. Math. Soc. (3), 93, 447-473, (2006) · Zbl 1099.42020
[5] DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, New York (1993) · Zbl 0797.41016
[6] Dick, J., Pillichshammer, F.: Discrepancy theory and quasi-Monte Carlo integration. In: A panorama of discrepancy theory. Lecture Notes in Math, vol. 2107, pp. 539-619. Springer, Cham (2014) · Zbl 1358.11086
[7] Dubinin, V.V.: Cubature formulas for classes of functions with bounded mixed difference. Matem. Sbornik 183(7), (1992) (Math. USSR Sbornik )76(283-292), (1993) · Zbl 0786.41025
[8] Dubinin, VV, Cubature formulae for Besov classes, Izv. Math., 61, 259-283, (1997) · Zbl 0882.41019
[9] Dũng, D., Temlyakov, V.N., Ullrich, T.: Hyperbolic cross approximation. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser/Springer, to appear. arXiv:1601.03978v2 [math.NA]
[10] Dũng, D; Ullrich, T, Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square, Math. Nachr., 288, 743-762, (2015) · Zbl 1317.65067
[11] Fefferman, C; Stein, EM, Some maximal inequalities, Am. J. Math., 93, 107-115, (1971) · Zbl 0222.26019
[12] Frolov, KK, Upper error bounds for quadrature formulas on function classes, Dokl. Akad. Nauk SSSR, 231, 818-821, (1976) · Zbl 0358.65014
[13] Glasserman, P.: Monte Carlo Methods in Financial Engineering. Applications of Mathematics : Stochastic Modelling and Applied Probability. Springer, Berlin (2004) · Zbl 1038.91045
[14] Goda, T; Suzuki, K; Yoshiki, T, Optimal order quasi-Monte Carlo integration in weighted Sobolev spaces of arbitrary smoothness, IMA J. Numer. Anal., 37, 505-518, (2017) · Zbl 1433.65031
[15] Goda, T; Suzuki, K; Yoshiki, T, An explicit construction of optimal order quasi-Monte Carlo rules for smooth integrands, SIAM J. Numer. Anal., 54, 2664-2683, (2016) · Zbl 1357.65006
[16] Hinrichs, A, Discrepancy of hammersley points in Besov spaces of dominating mixed smoothness, Math. Nachr., 283, 478-488, (2010) · Zbl 1198.11073
[17] Hinrichs, A; Markhasin, L; Oettershagen, J; Ullrich, T, Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions, Numer. Math., 134, 163-196, (2016) · Zbl 1358.65004
[18] Hlawka, E, Zur angenäherten berechnung mehrfacher integrale, Monatsh. Math., 66, 140-151, (1962) · Zbl 0105.04603
[19] Kacwin, C., Oettershagen, J., Ullrich, T.: On the orthogonality of the Chebyshev-Frolov lattice and applications. ArXiv e-prints (2016) arXiv:1606.00492 [math.NA] · Zbl 1417.11135
[20] Korobov, NM, Approximate evaluation of repeated integrals, Dokl. Akad. Nauk SSSR, 124, 1207-1210, (1959) · Zbl 0089.04201
[21] Krieg, D., Novak, E.: A universal algorithm for multivariate integration. Found. Comput. Math. (2016). doi:10.1007/s10208-016-9307-y · Zbl 1384.65003
[22] Kuo, F; Sloan, IH; Woźniakowski, H, Periodization strategy may fail in high dimensions, Numer. Algorithms, 46, 369-391, (2007) · Zbl 1140.65011
[23] Markhasin, L, Discrepancy and integration in function spaces with dominating mixed smoothness, Diss. Math., 494, 1-81, (2013) · Zbl 1284.46030
[24] Nguyen, V.K., Sickel, W.: Pointwise multipliers for Sobolev and Besov spaces of dominating mixed smoothness. J. Math. Anal. Appl. (2017). doi:10.1016/j.jmaa.2017.02.046 · Zbl 1382.46023
[25] Nikol’skij, S.M.: Approximation of Functions of Several Variables and Embedding Theorems. Nauka Moskva, Moscow (1977)
[26] Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems: Standard Information for Functionals, vol. II. EMS Tracts in Mathematics, 12. European Mathematical Society (EMS), Zürich, (2010) · Zbl 1357.65006
[27] Nuyens, D; Cools, R, Higher order quasi Monte Carlo methods: a comparison, AIP Conf. Proc., 1281, 553-557, (2010)
[28] Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. de Gruyter Series in Nonlinear Analysis and Applications, vol. 3. Walter de Gruyter & Co, Berlin (1996) · Zbl 0873.35001
[29] Schmeisser, H.-J., Triebel, H.: Topics in Fourier Analysis and Function Spaces. Geest & Portig, Leipzig (1987) and Wiley, Chichester (1987) · Zbl 0661.46024
[30] Sickel, W, On pointwise multipliers for \(F^s_{p, q}\). the case \(σ _{p, q} < s < n/p\), Annali di Matematica pura ed applicata, 174, 209-250, (1999) · Zbl 0956.46027
[31] Skriganov, MM, Constructions of uniform distributions in terms of geometry of numbers, Algebra i Analiz, 6, 200-230, (1994) · Zbl 0840.11041
[32] Stöckert, B, Ungleichungen vom Plancherel-polya-nikolskij-typ in gewichteten \(L_p^{Ω }\)-Räumen mit gemischten normen, Math. Nachr., 86, 19-32, (1978) · Zbl 0412.46021
[33] Temlyakov, VN, On reconstruction of multivariate periodic functions based on their values at the knots of number-theoretical nets, Anal. Math., 12, 287-305, (1986) · Zbl 0621.41004
[34] Temlyakov, VN, On a way of obtaining lower estimates for the errors of quadrature formulas, Mat. Sb., 181, 1403-1413, (1990)
[35] Temlyakov, VN, Error estimates for Fibonacci quadrature formulas for classes of functions with a bounded mixed derivative, Trudy Mat. Inst. Steklov., 200, 327-335, (1991) · Zbl 0813.41023
[36] Temlyakov, V.N.: Approximation of Periodic Functions, Computational Mathematics and Analysis Series. Nova Science Publishers Inc, Commack, NY (1993) · Zbl 0899.41001
[37] Temlyakov, V.N.: On error estimates for cubature formulas. Trudy Mat. Inst. Steklov 207, 326-338 (1994); Proc. Steklov Inst. Math. 207(6), 299-302 (1995) · Zbl 1126.41303
[38] Temlyakov, VN, Cubature formulas, discrepancy, and nonlinear approximation, J. Complex., 19, 352-391, (2003) · Zbl 1031.41016
[39] Temlyakov, V, Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness, Mat. Sb., 206, 131-160, (2015) · Zbl 1362.41009
[40] Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983) · Zbl 0546.46028
[41] Triebel, H.: Theory of Function Spaces II. Birkhäuser, Basel (1992) · Zbl 0763.46025
[42] Triebel, H.: Function Spaces and Wavelets on Domains. EMS Publ. House, Zürich (2008) · Zbl 1158.46002
[43] Triebel, H.: Bases in Function Spaces, Sampling Discrepancy, Numerical Integration. EMS Tracts in Mathematics. European Mathematical Society (EMS), Zürich (2010) · Zbl 1202.46002
[44] Ullrich, M.: On “Upper error bounds for quadrature formulas on function classes” by Frolov, K. K. In: Cools, R., Nuyens, D. (eds.) Monte Carlo and Quasi-Monte Carlo Methods, Proceedings in Mathematics & Statistics, vol. 163, pp. 571-582. Springer, Cham (2016) · Zbl 1356.65089
[45] Ullrich, M.: A Monte Carlo method for integration of multivariate smooth functions. SIAM J. Numer. Anal. (to appear) · Zbl 1365.65060
[46] Ullrich, M; Ullrich, T, The role of frolov’s cubature formula for functions with bounded mixed derivative, SIAM J. Numer. Anal., 54, 969-993, (2016) · Zbl 1336.65023
[47] Ullrich, T.: Function spaces with dominating mixed smoothness; characterization by differences. Jenaer Schriften zur Mathematik und Informatik, Math/Inf/05/06 (2006)
[48] Ullrich, T.: Smolyaks algorithm, sparse grid approximation and periodic function spaces with dominating mixed smoothness. Ph.D Thesis, Friedrich-Schiller-Universität Jena, Jena (2007)
[49] Ullrich, T.: Local mean characterization of Besov-Triebel-Lizorkin type spaces with dominating mixed smoothness on rectangular domains. Preprint, Uni Bonn (2008) · Zbl 1140.65011
[50] Vybíral, J, Function spaces with dominating mixed smoothness, Diss. Math., 436, 73, (2006) · Zbl 1101.46023
[51] Yserentant, H.: Regularity and Approximability of Electronic Wave Functions. In: Lecture Notes in Mathematics. Springer, Berlin (2010) · Zbl 1204.35003
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