Some Results on the Complexity of Numerical Integration
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[1] Tino Ullrich,et al. Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square , 2013 .
[2] Erich Novak,et al. How Many Random Bits Do We Need for Monte Carlo Integration , 2004 .
[3] Vladimir N. Temlyakov,et al. Cubature formulas, discrepancy, and nonlinear approximation , 2003, J. Complex..
[4] Mario Ullrich,et al. On "Upper Error Bounds for Quadrature Formulas on Function Classes" by K.K. Frolov , 2014, MCQMC.
[5] Henryk Wozniakowski,et al. Tractability of Tensor Product Linear Operators , 1997, J. Complex..
[6] Leszek Plaskota,et al. Monte Carlo and Quasi-Monte Carlo methods 2010 , 2012 .
[7] Peter Kritzer,et al. Component-by-component construction of low-discrepancy point sets of small size , 2008, Monte Carlo Methods Appl..
[8] E. Novak,et al. Tractability of Multivariate Problems , 2008 .
[9] H. Wozniakowski,et al. On tractability of path integration , 1996 .
[10] E. Novak. Deterministic and Stochastic Error Bounds in Numerical Analysis , 1988 .
[11] H. Triebel. Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration , 2010 .
[12] K. Ritter,et al. The Curse of Dimension and a Universal Method For Numerical Integration , 1997 .
[13] Henryk Wozniakowski,et al. Multivariate integration of infinitely many times differentiable functions in weighted Korobov spaces , 2013, Math. Comput..
[14] Henryk Wozniakowski,et al. Explicit Cost Bounds of Algorithms for Multivariate Tensor Product Problems , 1995, J. Complex..
[15] Michael Gnewuch,et al. Lower Error Bounds for Randomized Multilevel and Changing Dimension Algorithms , 2012, ArXiv.
[16] Stefan Heinrich. Quantum integration in Sobolev classes , 2003, J. Complex..
[17] Henryk Wozniakowski,et al. When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals? , 1998, J. Complex..
[18] Stefan Heinrich,et al. Complexity of Banach Space Valued and Parametric Integration , 2013 .
[19] M. Skriganov,et al. Explicit constructions in the classical mean squares problem in irregularities of point distribution , 2002 .
[20] Henryk Wozniakowski,et al. Tractability of multivariate problems for standard and linear information in the worst case setting: Part I , 2015, J. Approx. Theory.
[21] Michael Gnewuch,et al. Infinite-dimensional integration on weighted Hilbert spaces , 2012, Math. Comput..
[22] H. Triebel,et al. Function Spaces in Lipschitz Domains and Optimal Rates of Convergence for Sampling , 2006 .
[23] Henryk Wozniakowski,et al. The curse of dimensionality for numerical integration of smooth functions , 2012, Math. Comput..
[24] Grzegorz W. Wasilkowski,et al. Tractability of infinite-dimensional integration in the worst case and randomized settings , 2011, J. Complex..
[25] K. Ritter,et al. Simple Cubature Formulas with High Polynomial Exactness , 1999 .
[26] E. Novak,et al. Tractability of Multivariate Problems Volume II: Standard Information for Functionals , 2010 .
[27] Vladimir Temlyakov,et al. ON A WAY OF OBTAINING LOWER ESTIMATES FOR THE ERRORS OF QUADRATURE FORMULAS , 1992 .
[28] Grzegorz W. Wasilkowski,et al. The power of adaptive algorithms for functions with singularities , 2009 .
[29] Stefan Heinrich,et al. Complexity of parametric integration in various smoothness classes , 2014, J. Complex..
[30] Vladimir N. Temlyakov,et al. On Approximate Recovery of Functions with Bounded Mixed Derivative , 1993, J. Complex..
[31] Fred J. Hickernell,et al. Multi-level Monte Carlo algorithms for infinite-dimensional integration on RN , 2010, J. Complex..
[32] N. S. Bakhvalov,et al. On the optimality of linear methods for operator approximation in convex classes of functions , 1971 .
[33] Hans-Joachim Bungartz,et al. Acta Numerica 2004: Sparse grids , 2004 .
[34] Van Kien Nguyen,et al. Change of Variable in Spaces of Mixed Smoothness and Numerical Integration of Multivariate Functions on the Unit Cube , 2015 .
[35] Michael Gnewuch,et al. On weighted Hilbert spaces and integration of functions of infinitely many variables , 2014, J. Complex..
[36] Carola Doerr,et al. Calculation of Discrepancy Measures and Applications , 2014, 1405.1653.
[37] Michael Gnewuch,et al. Entropy, Randomization, Derandomization, and Discrepancy , 2012 .
[38] Stefan Heinrich,et al. Lower bounds for the complexity of Monte Carlo function approximation , 1992, J. Complex..
[39] Aicke Hinrichs,et al. Covering numbers, Vapnik-ervonenkis classes and bounds for the star-discrepancy , 2004, J. Complex..
[40] K. Ritter,et al. High dimensional integration of smooth functions over cubes , 1996 .
[41] Fred J. Hickernell,et al. Deterministic multi-level algorithms for infinite-dimensional integration on RN , 2011, J. Complex..
[42] Vladimir Temlyakov,et al. APPROXIMATE RECOVERY OF PERIODIC FUNCTIONS OF SEVERAL VARIABLES , 1987 .
[43] Henryk Wozniakowski,et al. Complexity of oscillatory integration for univariate Sobolev spaces , 2013, J. Complex..
[44] Henryk Wozniakowski,et al. Weighted Tensor Product Algorithms for Linear Multivariate Problems , 1999, J. Complex..
[45] S. Heinrich. Random Approximation in Numerical Analysis , 1994 .
[46] Josef Dick. A note on the existence of sequences with small star discrepancy , 2007, J. Complex..
[47] Harald Niederreiter,et al. Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.
[48] V. Babenko. Asymptotically sharp bounds for the remainder for the best quadrature formulas for several classes of functions , 1976 .
[49] Erich Novak,et al. Monte Carlo-Algorithmen , 2012, Springer-Lehrbuch.
[50] Steffen Dereich,et al. Quadrature for Self-affine Distributions on $${\mathbb R}^d$$Rd , 2015, Found. Comput. Math..
[51] Magnus Wahlström,et al. Algorithmic construction of low-discrepancy point sets via dependent randomized rounding , 2010, J. Complex..
[52] Henryk Wozniakowski,et al. Tractability of multivariate analytic problems , 2014, Uniform Distribution and Quasi-Monte Carlo Methods.
[53] Henryk Wozniakowski,et al. Liberating the dimension , 2010, J. Complex..
[54] Stefan Heinrich. Quantum Summation with an Application to Integration , 2002, J. Complex..
[55] Dirk Nuyens,et al. Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces , 2006, Math. Comput..
[56] Benjamin Doerr,et al. Construction of Low-Discrepancy Point Sets of Small Size by Bracketing Covers and Dependent Randomized Rounding , 2008 .
[57] Fred J. Hickernell,et al. Integration and approximation in arbitrary dimensions , 2000, Adv. Comput. Math..
[58] Erich Novak,et al. On weak tractability of the Clenshaw-Curtis Smolyak algorithm , 2014, J. Approx. Theory.
[59] György Elekes,et al. A geometric inequality and the complexity of computing volume , 1986, Discret. Comput. Geom..
[60] F. Pillichshammer,et al. Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration , 2010 .
[61] Jan Vybíral,et al. Weak and quasi-polynomial tractability of approximation of infinitely differentiable functions , 2013, J. Complex..
[62] F. Pillichshammer,et al. Discrepancy Theory and Quasi-Monte Carlo Integration , 2014 .
[63] Fred J. Hickernell,et al. On strong tractability of weighted multivariate integration , 2004, Math. Comput..
[64] Aicke Hinrichs. Optimal importance sampling for the approximation of integrals , 2010, J. Complex..
[65] Grzegorz W. Wasilkowski. Average case tractability of approximating ∞-variate functions , 2014, Math. Comput..
[66] Henryk Wozniakowski,et al. Information-based complexity , 1987, Nature.
[67] E. Novak,et al. Computation of Expectations by Markov Chain Monte Carlo Methods , 2013, 1311.1899.
[68] Ben Niu,et al. Deterministic Multi-level Algorithms for Infinite-dimensional Integration on {$\mathbb{R}^{\mathbb{N}}$} , 2010 .
[69] H. Woxniakowski. Information-Based Complexity , 1988 .
[70] Fred J. Hickernell,et al. The cost of deterministic, adaptive, automatic algorithms: Cones, not balls , 2013, J. Complex..
[71] Aicke Hinrichs,et al. Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions , 2016, Numerische Mathematik.
[72] E. Novak,et al. Tractability of Multivariate Problems, Volume III: Standard Information for Operators. , 2012 .
[73] Henryk Wozniakowski,et al. Lower bounds for the complexity of linear functionals in the randomized setting , 2011, J. Complex..
[74] Michael Gnewuch,et al. Infinite-Dimensional Integration in Weighted Hilbert Spaces: Anchored Decompositions, Optimal Deterministic Algorithms, and Higher-Order Convergence , 2012, Found. Comput. Math..
[75] P. L’Ecuyer,et al. Random Number Generation and Quasi-Monte Carlo† , 2015 .
[76] Henryk Wozniakowski,et al. Intractability Results for Integration and Discrepancy , 2001, J. Complex..
[77] Mario Ullrich,et al. The Role of Frolov's Cubature Formula for Functions with Bounded Mixed Derivative , 2015, SIAM J. Numer. Anal..
[78] E. Novak,et al. The inverse of the star-discrepancy depends linearly on the dimension , 2001 .
[79] Jan Vybíral,et al. Sampling numbers and function spaces , 2007, J. Complex..
[80] Erich Novak. Quantum Complexity of Integration , 2001, J. Complex..
[81] Martin E. Dyer,et al. Random Volumes in the n-Cube , 1990, Polyhedral Combinatorics.
[82] Markus Hofer,et al. Probabilistic discrepancy bound for Monte Carlo point sets , 2014, Math. Comput..
[83] Faithful asymptotics of remainders optimal for some glasses of functions with cubic weight formulas , 1976 .
[84] S. Heinrich,et al. Optimal Summation and Integration by Deterministic, Randomized, and Quantum Algorithms , 2002 .
[85] V V Dubinin. Cubature formulae for Besov classes , 1997 .
[86] D. Rudolf,et al. Explicit error bounds for Markov chain Monte Carlo , 2011, 1108.3201.
[87] Michael Gnewuch,et al. Optimal Randomized Multilevel Algorithms for Infinite-Dimensional Integration on Function Spaces with ANOVA-Type Decomposition , 2012, SIAM J. Numer. Anal..
[88] K. F. Roth,et al. On irregularities of distribution IV , 1979 .
[89] Ian H. Sloan,et al. Component-by-component construction of good lattice rules , 2002, Math. Comput..
[90] K. Ritter,et al. Multi-Level {M}onte {C}arlo Algorithms for Infinite-Dimensional Integration on $\mathbb{R}^\mathbb{N}$ , 2010 .
[91] G. Leobacher,et al. Introduction to Quasi-Monte Carlo Integration and Applications , 2014 .
[92] Frances Y. Kuo,et al. High-dimensional integration: The quasi-Monte Carlo way*† , 2013, Acta Numerica.
[93] Henryk Wozniakowski,et al. Finite-order weights imply tractability of multivariate integration , 2004, J. Complex..
[94] Josef Dick,et al. Numerical integration of Hölder continuous, absolutely convergent Fourier, Fourier cosine, and Walsh series , 2013, J. Approx. Theory.
[95] Henryk Wozniakowski,et al. The curse of dimensionality for numerical integration of smooth functions II , 2012, J. Complex..
[96] Henryk Wozniakowski,et al. Good Lattice Rules in Weighted Korobov Spaces with General Weights , 2006, Numerische Mathematik.
[97] Dirk Nuyens,et al. Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points , 2006, J. Complex..
[98] Dirk Nuyens,et al. Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules , 2011, Numerical Algorithms.
[99] Henryk Wozniakowski,et al. Liberating the weights , 2004, J. Complex..
[100] W. Sickel,et al. The Smolyak agorithm, sampling on sparse grids and functions spaces of dominated mixed smoothness , 2007 .
[101] Daan Huybrechs,et al. Highly Oscillatory Problems: Highly oscillatory quadrature , 2009 .
[102] Erich Novak,et al. On the Power of Adaption , 1996, J. Complex..
[103] Peter Mathé,et al. The Optimal Error of Monte Carlo Integration , 1995, J. Complex..
[104] W. Sickel,et al. Spline interpolation on sparse grids , 2011 .
[105] Henryk Wozniakowski,et al. Exponential convergence and tractability of multivariate integration for Korobov spaces , 2011, Math. Comput..
[106] F. J. Hickernell,et al. Tractability of Multivariate Integration for Periodic Functions , 2001, J. Complex..
[107] K. F. Roth. On irregularities of distribution , 1954 .
[108] Przemyslaw Wojtaszczyk,et al. Linear vs. nonlinear algorithms for linear problems , 2004, J. Complex..
[109] Henryk Wozniakowski,et al. Path Integration on a Quantum Computer , 2002, Quantum Inf. Process..
[110] Josef Dick,et al. The weighted star discrepancy of Korobov's $p$-sets , 2014, 1404.0114.
[111] Aicke Hinrichs,et al. Discrepancy, Integration and Tractability , 2013 .
[112] W. Sickel,et al. Smolyak’s Algorithm, Sampling on Sparse Grids and Function Spaces of Dominating Mixed Smoothness , 2006 .
[113] A. Stroud. Approximate calculation of multiple integrals , 1973 .
[114] Grzegorz W. Wasilkowski,et al. Randomly shifted lattice rules for unbounded integrands , 2006, J. Complex..
[115] Frances Y. Kuo,et al. Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces , 2003, J. Complex..
[116] Steffen Dereich,et al. Infinite-Dimensional Quadrature and Approximation of Distributions , 2009, Found. Comput. Math..
[117] Frances Y. Kuo,et al. On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces , 2002, Math. Comput..
[118] Henryk Wozniakowski,et al. A general theory of optimal algorithms , 1980, ACM monograph series.
[119] Benjamin Doerr,et al. A lower bound for the discrepancy of a random point set , 2012, J. Complex..
[120] Michael Gnewuch,et al. Optimal randomized changing dimension algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition , 2014, J. Approx. Theory.
[121] Christoph Aistleitner,et al. Covering numbers, dyadic chaining and discrepancy , 2011, J. Complex..