{"title":"Parameter certainty quantification in nonlinear models","authors":"","doi":"10.1016/j.ijengsci.2024.104163","DOIUrl":null,"url":null,"abstract":"<div><div>Estimating model parameters from experimental data is a common practice across various research fields. For nonlinear models, the parameters are estimated using an optimization algorithm that minimizes an objective function. Assessing the certainty of these parameter estimates is crucial to address questions such as “what is the probability the estimation error is smaller than 5%?”, “is our experiment sensitive enough to estimate all parameters?”, and “how much can we change each parameter while still fitting the data accurately?”. Typically, the certainty levels are quantified using a linear approximation of the model. However, we show that in models that are highly nonlinear in their parameters or in the presence of large experimental errors, this method fails to capture the certainty levels accurately. To address these limitations, we present an alternative method based on the Hessian approximation of the objective function. We show that this method captures the certainty levels more accurately and can be derived geometrically. We demonstrate the efficacy of our approach through a case study involving a nonlinear hyperelastic material constitutive model and an application on a nonlinear model for the conductivity of electrolyte solutions. Despite its higher computational cost, we recommend adopting the Hessian approximation when accurate certainty levels are required in highly nonlinear models.</div></div>","PeriodicalId":14053,"journal":{"name":"International Journal of Engineering Science","volume":null,"pages":null},"PeriodicalIF":5.7000,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Engineering Science","FirstCategoryId":"5","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0020722524001472","RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
Estimating model parameters from experimental data is a common practice across various research fields. For nonlinear models, the parameters are estimated using an optimization algorithm that minimizes an objective function. Assessing the certainty of these parameter estimates is crucial to address questions such as “what is the probability the estimation error is smaller than 5%?”, “is our experiment sensitive enough to estimate all parameters?”, and “how much can we change each parameter while still fitting the data accurately?”. Typically, the certainty levels are quantified using a linear approximation of the model. However, we show that in models that are highly nonlinear in their parameters or in the presence of large experimental errors, this method fails to capture the certainty levels accurately. To address these limitations, we present an alternative method based on the Hessian approximation of the objective function. We show that this method captures the certainty levels more accurately and can be derived geometrically. We demonstrate the efficacy of our approach through a case study involving a nonlinear hyperelastic material constitutive model and an application on a nonlinear model for the conductivity of electrolyte solutions. Despite its higher computational cost, we recommend adopting the Hessian approximation when accurate certainty levels are required in highly nonlinear models.
期刊介绍:
The International Journal of Engineering Science is not limited to a specific aspect of science and engineering but is instead devoted to a wide range of subfields in the engineering sciences. While it encourages a broad spectrum of contribution in the engineering sciences, its core interest lies in issues concerning material modeling and response. Articles of interdisciplinary nature are particularly welcome.
The primary goal of the new editors is to maintain high quality of publications. There will be a commitment to expediting the time taken for the publication of the papers. The articles that are sent for reviews will have names of the authors deleted with a view towards enhancing the objectivity and fairness of the review process.
Articles that are devoted to the purely mathematical aspects without a discussion of the physical implications of the results or the consideration of specific examples are discouraged. Articles concerning material science should not be limited merely to a description and recording of observations but should contain theoretical or quantitative discussion of the results.