Sensitivity via the complex-step method for delay differential equations with non-smooth initial data

Banks, Harvey Thomas; Bekele-Maxwell, Kidist; Bociu, Lorena; Wang, Chuyue

doi:10.1090/qam/1458

Cited by 7 publications

(12 citation statements)

References 27 publications

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“…We fit seven mathematical models to three types of tumor data sets (breast, lung, and skin (HPV) tumors), and used γ values ranging from 0 to 1 for the statistical error model (2). Specifically, we take γ = 0, 0.25, 0.5, 0.75, 0.84, 1.…”

Section: Resultsmentioning

confidence: 99%

“…Here we use a method originally based on analyticity [15,16,17,18] of the functions involved. The method was recently developed further in [1,2]. This method, referred to as the complex-step method, as derived earlier relied on the Cauchy-Riemann equations in a crucial way.…”

Section: The Complex-step Methods For Computing Sensitivitiesmentioning

confidence: 99%

“…This method, referred to as the complex-step method, as derived earlier relied on the Cauchy-Riemann equations in a crucial way. More recently in [1,2] the method was shown to be much more widely applicable. Indeed one can argue for any C 2 function in the parameter θ, one has the 2nd order Taylor expansion using the complex step ih…”

Section: The Complex-step Methods For Computing Sensitivitiesmentioning

confidence: 99%

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The Effect of Statistical Error Model Formulation on the Fit and Selection of Mathematical Models of Tumor Growth for Small Sample Sizes

Bekele-Maxwell¹,

Noorman²,

Menda³

et al. 2018

Int. J. of Pure and Appl. Math.

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When fitting a mathematical model to a given data set using inverse problems, the correctness of both the mathematical model and the statistical error models are important since an incorrect statistical or observational model directly affects both the estimates and their corresponding standard errors. The effects of these models, among many other factors, are dependent on the sample size and the information content of the data set.In this article, we investigate how the choice of the statistical error model affects the mathematical model fit and accuracy of parameter estimates in small sample size tumor growth data sets. We specifically seek to determine the appropriate statistical error model for small sample size breast, lung and HPV tumor growth data sets obtained from studies on mice. T. Banks et alWe find that for small sample sizes the selection of the best statistical error model is not straightforward and requires the examination of multiple criteria for model fit and uncertainty.Therefore, selection of the best mathematical model is not an easy process for small sample size tumor data and selection of a model based on few data points may not prove accurate.We encourage further research on the optimal design of experiments (duration and number of observations) in order to best fit mathematical models to tumor growth data.

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Section: Resultsmentioning

confidence: 99%

Section: The Complex-step Methods For Computing Sensitivitiesmentioning

confidence: 99%

Section: The Complex-step Methods For Computing Sensitivitiesmentioning

confidence: 99%

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The Effect of Statistical Error Model Formulation on the Fit and Selection of Mathematical Models of Tumor Growth for Small Sample Sizes

Bekele-Maxwell¹,

Noorman²,

Menda³

et al. 2018

Int. J. of Pure and Appl. Math.

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“…In recent efforts [5,6], we demonstrated the efficiency of the complex-step method for computing sensitivities for biological models. There the method is applied to examples of various complexity, ranging from time delayed differential equations (DDEs) to Lamé systems, and the results are compared with solutions of traditional sensitivity equations.…”

Section: Sensitivity Analysis Via the Complex-step Methodmentioning

confidence: 99%

“…Further, we note that no boundary condition is needed to impose on ℘ since the total stress is already prescribed on Γ N in (9). In summary, we solve the balance equations (1a) and (1b) with constitutive equations (3), (4), (6), and 7along with initial condition (8) and boundary conditions (9)- (11). The principle unknowns are the solid displacement u and fluid pressure p.…”

Section: The Discharge Velocity Is Given Bymentioning

confidence: 99%

Local sensitivity via the complex-step derivative approximation for 1D Poro-Elastic and Poro-Visco-Elastic models

Banks¹,

Bekele-Maxwell²,

Bociu³

et al. 2019

Mathematical Control &Amp; Related Fields

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Poro-elastic systems have been used extensively in modeling fluid flow in porous media in petroleum and earthquake engineering. Nowadays, they are frequently used to model fluid flow through biological tissues, cartilages, and bones. In these biological applications, the fluid-solid mixture problems, which may also incorporate structural viscosity, are considered on bounded domains with appropriate non-homogeneous boundary conditions. The recent work in [12] provided a theoretical and numerical analysis of nonlinear poroelastic and poro-viscoelastic models on bounded domains with mixed boundary conditions, focusing on the role of visco-elasticity in the material. Their results show that higher time regularity of the sources is needed to guarantee bounded solution when visco-elasticity is not present. Inspired by their results, we have recently performed local sensitivity analysis on the solutions of these fluid-solid mixture problems with respect to the boundary source of traction associated with the elastic structure [3]. Our results show that the solution is more sensitive to boundary datum in the purely elastic case than when visco-elasticity is present in the solid matrix. In this article, we further extend this work in order to include local sensitivities of the solution of the coupled system to the boundary conditions imposed on the Darcy velocity. Sensitivity analysis is the first step in identifying important parameters to control or use as control terms in these poro-elastic and poro-visco-elastic models, which is our ultimate goal.

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Poro-Visco-Elasticity in Biomechanics: Optimal Control

Bociu

Strikwerda

2022

Association for Women in Mathematics Series

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Sensitivity via the complex-step method for delay differential equations with non-smooth initial data

Cited by 7 publications

References 27 publications

The Effect of Statistical Error Model Formulation on the Fit and Selection of Mathematical Models of Tumor Growth for Small Sample Sizes

The Effect of Statistical Error Model Formulation on the Fit and Selection of Mathematical Models of Tumor Growth for Small Sample Sizes

Local sensitivity via the complex-step derivative approximation for 1D Poro-Elastic and Poro-Visco-Elastic models

Poro-Visco-Elasticity in Biomechanics: Optimal Control

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