The Monte Carlo method

Granovskii, B.L.; Ermakov, S. M.

doi:10.1007/bf01084250

Cited by 10 publications

(5 citation statements)

References 150 publications

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“…Here, L i,ω can be from the class Λ std or Λ all , the random mapping φ ω can be linear or nonlinear, and we may use adaption; see e.g., [4,6,7,12,13,15,18,21,26]. Moreover, the number n ω = n ω (f ) of functional evaluations may be random and may depend adaptively on f ; in such a case card(A) is defined by sup f H ≤1 E ω n ω (f ).…”

Section: Randomized Settingmentioning

confidence: 99%

The power of standard information for multivariate approximation in the randomized setting

Wasilkowski

Woźniakowski

2006

Math. Comp.

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Abstract. We study approximating multivariate functions from a reproducing kernel Hilbert space with the error between the function and its approximation measured in a weighted L 2 -norm. We consider functions with an arbitrarily large number of variables, d, and we focus on the randomized setting with algorithms using standard information consisting of function values at randomly chosen points.We prove that standard information in the randomized setting is as powerful as linear information in the worst case setting. Linear information means that algorithms may use arbitrary continuous linear functionals, and by the power of information we mean the speed of convergence of the nth minimal errors, i.e., of the minimal errors among all algorithms using n function evaluations. Previously, it was only known that standard information in the randomized setting is no more powerful than the linear information in the worst case setting.We also study (strong) tractability of multivariate approximation in the randomized setting. That is, we study when the minimal number of function evaluations needed to reduce the initial error by a factor ε is polynomial in ε −1 (strong tractability), and polynomial in d and ε −1 (tractability). We prove that these notions in the randomized setting for standard information are equivalent to the same notions in the worst case setting for linear information. This result is useful since for a number of important applications only standard information can be used and verifying (strong) tractability for standard information is in general difficult, whereas (strong) tractability in the worst case setting for linear information is known for many spaces and is relatively easy to check.We illustrate the tractability results for weighted Korobov spaces. In particular, we present necessary and sufficient conditions for strong tractability and tractability. For product weights independent of d, we prove that strong tractability is equivalent to tractability.We stress that all proofs are constructive. That is, we provide randomized algorithms that enjoy the maximal speed of convergence. We also exhibit randomized algorithms which achieve strong tractability and tractability error bounds.

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Section: Randomized Settingmentioning

confidence: 99%

The power of standard information for multivariate approximation in the randomized setting

Wasilkowski

Woźniakowski

2006

Math. Comp.

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“…We also recommend the survey article [100], the recent bibliography [84], and the elementary introduction in [306]. The main reason for the popularity of the Monte Carlo method is its applicability to a never-ending variety of problems in numerical analysis, statistics, applied mathematics, particle physics, engineering, systems analysis, and so on.…”

Section: [25] [28] [66] [106] [309])mentioning

confidence: 99%

Quasi-Monte Carlo methods and pseudo-random numbers

Niederreiter¹

1978

Bull. Amer. Math. Soc.

722

359

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“…This resulting framework shortens calculation time by grouping vehicles of the same nature together. Errors from this new method will be calculated using Monte Carlo simulation techniques [18]. This classic method is applied by many studies around the world to calculate model errors.…”

Section: Emission Inventory Methodsmentioning

confidence: 99%

Characteristics and distribution of greenhouse gas emissions from motorcycles in Hanoi, Vietnam

Tran,

Nghiem,

Hoang Vu

et al. 2024

IOP Conf. Ser.: Earth Environ. Sci.

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Motorcycles contribute to the notable emission of greenhouse gases (GHG) and air pollutants in urban areas. These vehicles are the most widely used means of road transportation in Vietnam. This study is the first to use the EMISENS model in combination with the IVE modelling system to calculate greenhouse gas emissions from motorcycles at tier 3 in IPCC guideline. There are more than 8700 motorcycles circulating on the roads in Hanoi per hour. With the characteristics of motorcycles compiled, the average age of motorcycles is around 11 years, and each vehicle travels an average of 16.4 km per day with an average start-up frequency of 3.46. The emission factors of greenhouse gases CO2 and CH4 when running (erunning) were 72.55 and 0.24 grams per km, respectively. The emission factors for CO2, CH4 and N2O during vehicle start-up (estart-up) were 6.96, 0.53 and 0.0009 grams per start-up, respectively. Inventory emission data from 2021-2023 reveal that that motorcycles in Hanoi emitted nearly 4.6 mil tons of CO2e per year. Emission distribution is concentrated in inner-city districts and some residential areas at the intersections with provincial roads, national highways, and industrial parks.

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The Monte Carlo method

Cited by 10 publications

References 150 publications

The power of standard information for multivariate approximation in the randomized setting

The power of standard information for multivariate approximation in the randomized setting

Quasi-Monte Carlo methods and pseudo-random numbers

Characteristics and distribution of greenhouse gas emissions from motorcycles in Hanoi, Vietnam

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