Koksma–Hlawka Inequality

Abstract

The Koksma–Hlawka inequality is a tight error bound on the approximation of an integral by the sample average of integrand values. The integration error is bounded by a product of two terms, the discrepancy of the sample points, \documentclass{minimal}\usepackage{amsmath}\begin{document}$D(\{\vec{x}_{i}\})$\end{document} , and the variation of the integrand, V ( g ). These two quantities measure the quality of the sample points and the roughness of the integrand, respectively. The Koksma–Hlawka inequality plays a key role in the development of quasi–Monte Carlo methods. Such methods replace simple random sample points by low discrepancy points. The Koksma–Hlawka inequality has also influenced the study of experimental design and led to the creation of UNIFORM DESIGNS.

Improving Approximate Bayesian Computation via Quasi-Monte Carlo

“…The importance of the notion of discrepancy and in particular the star discrepancy is highlighted by the Koksma-Hlawka inequality (Hickernell, 2006), which relates the error of the integration to the coverage of the space and the variation of the function that is integrated:…”

A Computational Spectral Approach to Interest Rate Models

“…Note that, although the increased accuracy, the error displays again a stationary behavior, but actually no improvement on numerical methods can be implemented, hence we can once again conclude that this is the only source of error. 3.2060e-10 4.1608e-08 3.1112e-12 6.0758e-11 7 3.2058e-10 4.5733e-09 3.1111e-12 6.6780e-12 8 3.2057e-10 4.2648e-09 3.1109e-12 6.2276e-12 9 3.2058e-10 4.2643e-09 3.1111e-12 6.2268e-12 10 3.2055e-10 4.2613e-09 3.1108e-12 6.2225e-12 11 3.2048e-10 4.2659e-09 3.1101e-12 6.2291e-12 12 3.2108e-10 4.2641e-09 3.1159e-12 6.2265e-12 13 3.2077e-10 4.2605e-09 3.1129e-12 6.2213e-12 14 3.2074e-10 4.2667e-09 3.1126e-12 6.2303e-12 15 3.2070e-10 4.2598e-09 3.1122e-12 6.2203e-12 T for p = 15, achieved with standard Monte Carlo technique, which is compared with the probability density function of the analytical solution to eq. (16) at time T , whose distribution is lognormal, namely…”

“…Note that, although the increased accuracy, the error displays again a stationary behavior, but actually no improvement on numerical methods can be implemented, which allows to say that this is the only source of error. 3.2065e-10 1.1972e-08 3.1118e-12 1.1972e-11 8 3.2063e-10 6.2851e-09 3.1115e-12 6.2852e-12 9 3.2064e-10 6.2283e-09 3.1116e-12 6.2285e-12 10 3.2063e-10 6.2219e-09 3.1115e-12 6.2220e-12 11 3.2057e-10 6.2321e-09 3.1109e-12 6.2322e-12 12 3.2115e-10 6.2226e-09 3.1166e-12 6.2228e-12 13 3.2085e-10 6.2201e-09 3.1137e-12 6.2203e-12 14 3.2078e-10 6.2329e-09 3.1130e-12 6.2330e-12 15 3.2075e-10 6.2165e-09 3.1127e-12 6.2167e-12 Table 9: Absolute error of the average and the variance of PCE approximation of gBm at time T = 1 ( whose parameters are r = 3%, σ = 30% and starting value S 0 = 100) for higher precision at computing…”

“…5.2 for further details. 2.6330e-01 3.5319e-01 5 2.6330e-01 3.5319e-01 6 2.6330e-01 3.5319e-01 7 2.6330e-01 3.5319e-01 8 2.6330e-01 3.5319e-01 9 2.6330e-01 3.5319e-01 10 2.6330e-01 3.5319e-01 11 2.6330e-01 3.5319e-01 12 2.6330e-01 3.5319e-01 13 2.6330e-01 3.5319e-01 14 2.6330e-01 3.5319e-01 15 2.6330e-01 3.5319e-01 The quantiles computed by means of a standard Monte Carlo sampling of the analytical solution, whose size is equal to the one used for PCE, give the following standard error…”

“…The former is based on pseudo-random sampling and its convergence properties basically rely on the Law of large numbers and the Central limit theorem. The latter uses low-discrepancy sequences to simulate the process and it is theoretically based on the Koksma-Hlawka-inequality, see [9], which provides error bounds for computations involving QMC methods, see, e.g., [21,Chapter 5].…”