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Fast error estimates for indirect measurements: applications to pavement engineering

Ьыстрое оценивание погрещностей при непрямых измерениях: приложения к проектированию мостовых

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Reliable Computing

Abstract

In many real-life situations, we have the following problem: we want to know the value of some characteristicy that is difficult to measure directly (e.g., lifetime of a pavement, efficiency of an engine, etc.). To estimatey, we must know the relationship betweeny and some directly measurable physical quantitiesx 1,...,x n . From this relationship, we extract an algorithmf that allows us, givenx i , to computey: y=f(x 1, ...,x n ). So, we measurex i , apply an algorithmf, and get the desired estimate.

Existing algorithms for error estimate (interval mathematics, Monte-Carlo methods, numerical differentiation, etc.) require computation time that is several times larger than the time necessary to computey=f(x 1, ...,x n ). So, if an algorithmf is already time-consuming, error estimates will take too long.

In many cases, this algorithmf consists of two parts: first, we usex i to determine the parametersz k of a model that describes the measured object, and second, we use these parameters to estimatey. The most time-consuming part is findingz k ; this is done by solving a system of non-linear equations; usually least squares method is used.

We show that for suchf, one can estimate errors repeating this time-consuming part off only once. So, we can compute bothy and an error estimate fory with practically no increase in total computation time. As an example of this methodology, we give pavement lifetime estimates.

Abstract

На практике часто встречается следующая задача: требуется определить значение некоей характеристической переменнойy, которую трудно измерить напрямую (например, срок жизни мостовой, эффективность двигателя и т.н.). Чтобы опенить величинуy, мы должны знать соотнощение междуy и некоторыми физическими величинамиx 1, ...,x n , которые подлаются прямому измерению. На основании зтого соотнощения строится алгоритмf, который позволяет исходя из заланныхx i вычислитьy: y=f(x 1, ...,x n ). После этого останется лищь измеритьx i , применить алгоритмf и nолучить требуемую оценку.

Существующие алгоритмы оценивания ногрещностей (интервальная математика, методы Монте-Карло, численное дифференцирование и т. п.) требуют затрат времени, в несколько раз иревыщающих те, которые нужны для вычисленияy=f(x 1, ...,x n ). Поэтому, если алгоритмf, и без того работает медленно, оценивание ногрещностей может занять неприемлемо больщое время.

Во многих случаях алгоритмf состоит из двух частей: сначала на основеx i онределяются нараметрыz k модели, описывающей измеряемый обьект, а затем зтн нараметры исиользуются для оценкиy. Наибольщих временных затрат требует нолучение значенийz k , для чего необходимо рещить систему нелинейных уравнений (как нравило, применяется метод наименыцих квадратов).

Показано, что для цодобных алгоритмовf можно нолучить оценки ногрещностей, нроизвеля эту трудоемкую часть вычислений только один раз. Таким образом, мы можем вычислить какy, так и оценку ногрещности дляy ирактически без увеличения общего времени вычислений. В качестве примера использования этой методики приводятся оценки срока жизни мостовой.

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Author notes

  1. C. -C. Chang

    Present address: , 8 Kan Lo St., Taichung, Taiwan Republic of China

Authors and Affiliations

  1. Department of Civil Engineering, University of Texas at El Paso, 79968, El Paso, TX, USA

    C. Ferregut, S. Nazarian & K. Vennalaganti

  2. Computer Science Department, University of Texas at El Paso, 79968, El Paso, TX, USA

    C. -C. Chang & V. Kreinovich

Authors
  1. C. Ferregut
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  2. S. Nazarian
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  3. K. Vennalaganti
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  4. C. -C. Chang
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  5. V. Kreinovich
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Additional information

C. Ferregut, S. Nazarian, K. Vennalaganti, C.-C. Chang, V. Kreinovich, 1996

This work was supported by NSF grants No. CDA-9015006 and IRI-92-11-662, NASA Research Grant No. 9-482, and a grant from the Materials Research Institute. We want to thank the United States Army Corps of Engineers, Waterways Experiment Station (USAE, WES), Vicksburg, MS, for supplying us with the Layered Elastic Programs, WELSEA and WESDEF. We are also greatly thankful to all the participants of the International Conference on Numerical Analysis with Automatic Result Verification, Lafayette, LA, February–March 1993, especially to Drs. S. Markov and W. Walster, for valuable discussions.

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Ferregut, C., Nazarian, S., Vennalaganti, K. et al. Fast error estimates for indirect measurements: applications to pavement engineering. Reliable Comput 2, 219–228 (1996). https://doi.org/10.1007/BF02391696

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  • DOI: https://doi.org/10.1007/BF02391696

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