Koksma–Hlawka inequality

E824092

mathematical inequality result in discrepancy theory result in numerical analysis

The Koksma–Hlawka inequality is a fundamental result in numerical analysis and discrepancy theory that bounds the error of quasi-Monte Carlo integration by the product of a function’s variation and the discrepancy of the sampling points.

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Observed surface forms (2)

Surface form	Occurrences
Koksma inequality	1
Koksma–Hlawka inequality in numerical integration	1

Statements (48)

Predicate	Object
instanceOf	mathematical inequality ⓘ result in discrepancy theory ⓘ result in numerical analysis ⓘ
appliesTo	deterministic numerical integration ⓘ integration over the unit cube [0,1]^s ⓘ
assumes	function of bounded variation in the sense of Hardy–Krause ⓘ point set in the s-dimensional unit cube ⓘ
comparedWith	Monte Carlo error bounds based on variance ⓘ
compares	exact integral ⓘ quasi-Monte Carlo integration error ⓘ
errorBoundDependsOn	discrepancy of the sampling points ⓘ variation of the integrand ⓘ
errorType	worst-case error bound ⓘ
field	discrepancy theory ⓘ numerical analysis ⓘ quasi-Monte Carlo methods ⓘ
generalizedBy	various weighted discrepancy inequalities ⓘ
givesUpperBoundOn	absolute integration error ⓘ
hasVariant	Koksma–Hlawka inequality for anchored variation NERFINISHED ⓘ weighted Koksma–Hlawka inequality NERFINISHED ⓘ
holdsFor	any dimension s ≥ 1 ⓘ
implies	quasi-Monte Carlo can achieve faster convergence than Monte Carlo for smooth low-variation functions ⓘ
influencedDevelopmentOf	quasi-Monte Carlo theory ⓘ
involves	supremum over axis-aligned boxes for discrepancy ⓘ
isNonProbabilistic	true ⓘ
languageOfOriginalWork	German ⓘ
mathematicalDomain	analysis ⓘ number theory ⓘ
motivates	construction of low-discrepancy sequences ⓘ
namedAfter	Edmund Hlawka NERFINISHED ⓘ J. F. Koksma NERFINISHED ⓘ
relatedTo	Erdős–Turán–Koksma inequality NERFINISHED ⓘ Faure sequence ⓘ Halton sequence NERFINISHED ⓘ Sobol sequence NERFINISHED ⓘ low-discrepancy sequences ⓘ
relatesConcept	discrepancy of point sets ⓘ quasi-Monte Carlo integration ⓘ variation of a function ⓘ
statesThat	integration error is bounded by the product of variation and discrepancy ⓘ
timePeriod	20th century ⓘ
typicalFormulation	\| (1/N) Σ_{n=1}^N f(x_n) − ∫_[0,1]^s f(u) du \| ≤ V_HK(f) D_N^*(x_1,…,x_N) ⓘ
usedIn	analysis of quasi-Monte Carlo algorithms ⓘ computational finance ⓘ high-dimensional numerical integration ⓘ uncertainty quantification ⓘ
usesConcept	Hardy–Krause variation NERFINISHED ⓘ star discrepancy ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Johan Frederik Koksma → notableWork → Koksma–Hlawka inequality ⓘ

Johan Frederik Koksma → notableConcept → Koksma–Hlawka inequality ⓘ

this entity surface form: Koksma inequality

Johan Frederik Koksma → notableConcept → Koksma–Hlawka inequality ⓘ

this entity surface form: Koksma–Hlawka inequality in numerical integration