Picard iteration algorithm combined with Gauss–Seidel technique for initial value problems

Youssef, I. K.; El-Arabawy, H. A.

doi:10.1016/j.amc.2007.01.058

Cited by 22 publications

(34 citation statements)

References 11 publications

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“…The idea in [21,26] is to set up an iteration between a linear least squares optimization and a forward solution, which is partly based on a Picard iteration [34]. This approach is therefore distinctively different from other integral formulations like the modulating function approach [35] which does not iterate and is therefore not directly suitable for the discrete data and high non-linearities present in this application.…”

Section: Unique Parameter Identificationmentioning

confidence: 99%

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Unique parameter identification for cardiac diagnosis in critical care using minimal data sets

Hann

Chase

Desaive

et al. 2010

Computer Methods and Programs in Biomedicine

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Lumped parameter approaches for modelling the cardiovascular system typically have many parameters of which a significant percentage are often not identifiable from limited data sets. Hence, significant parts of the model are required to be simulated with little overall effect on the accuracy of data fitting, as well as dramatically increasing the complexity of parameter identification. This separates sub-structures of more complex cardiovascular system models to create uniquely identifiable simplified models that are one to one with the measurements. In addition, a new concept of parameter identification is presented where the changes in the parameters are treated as an actuation force into a feed back control system, and the reference output is taken to be steady state values of measured volume and pressure. The major advantage of the method is that when it converges, it must be at the global minimum so that the solution that best fits the data is always found.By utilizing continuous information from the arterial/pulmonary pressure waveforms and the end-diastolic time, it is shown that potentially, the ventricle volume is not required in the data set, which was a requirement in earlier published work. The simplified models can also act as a bridge to identifying more sophisticated cardiac models, by providing an initial set of patient specific parameters that can reveal trends and interactions in the data over time. The goal is to apply the simplified models to retrospective data on groups of patients to help characterize population trends or un-modelled dynamics within known bounds. These trends can assist in improved prediction of patient responses to cardiac disturbance and therapy intervention with potentially smaller and less invasive data sets. In this way a more complex model that takes into account individual patient variation can be developed, and applied to the improvement of cardiovascular management in critical care. Keywords :Model-based cardiac diagnosis ; Cardiovascular system ; Integral-based parameter identification ; Pressure waveform ; ECG ; Intensive care unit IntroductionIn critical care, cardiovascular dysfunction can be easily misdiagnosed due to incomplete information and the complexities involved, leading to premature discharge or non-optimal treatment [1][2][3]. It is also a major cause of increased length of stay and death [4,5]. Demand for critical care is growing dramatically severely affecting healthcare delivery [6][7][8]. The overall goal of this research is to use computational cardiac models to better aggregate available clinical data in an intensive care unit (ICU) into a more readily understood physiological context for clinicians. The computational models can be used to reveal non-linear dynamics and interactions that are not readily apparent in the data.A major difficulty faced with cardiovascular modelling in general, is the level of detail these models typically include. For example multi-scale modelling approaches utilizing finite elements have successfully expl...

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Section: Unique Parameter Identificationmentioning

confidence: 99%

“…Step 5 If the maximum volumes and aortic pressures are matched within a given tolerance to data in Equation (34), go to Step 6, otherwise go back to Step 3.…”

Section: Fig 3 -Parameter Identification Algorithm Formentioning

confidence: 99%

Unique parameter identification for cardiac diagnosis in critical care using minimal data sets

Hann

Chase

Desaive

et al. 2010

Computer Methods and Programs in Biomedicine

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“…A Picard iteration is capable of solving non-linear systems using discretised transforms of the governing ODEs [28,29]. In this case, the iteration uses updating predictions of the Y(t) profile to converge to a highly accurate Y(t).…”

Section: (T) and A Picard Iteration To Find Y(t)mentioning

confidence: 99%

“…These equations can be readily solved by linear least squares to give an initial estimate for the value of n L . For this given value of n L , Equations (3) and (4) are then numerically solved using a Picard iteration [28,29] …”

Section: (T) and A Picard Iteration To Find Y(t)mentioning

confidence: 99%

DISTq: An Iterative Analysis of Glucose Data for Low-Cost, Real-Time and Accurate Estimation of Insulin Sensitivity

Docherty¹

2010

TOMINFOJ

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Insulin sensitivity (SI) estimation has numerous uses in medical and clinical situations. However, highresolution tests that are useful for clinical diagnosis and monitoring are often too intensive, long and costly for regular use. Simpler tests that mitigate these issues are not accurate enough for many clinical diagnostic or monitoring scenarios. The gap between these tests presents an opportunity for new approaches.The quick dynamic insulin sensitivity test (DISTq) utilises the model-based DIST test protocol and a series of population estimates to eliminate the need for insulin or C-peptide assays to enable a high resolution, low-intensity, real-time evaluation of SI. The method predicts patient specific insulin responses to the DIST test protocol with enough accuracy to yield a useful clinical insulin sensitivity metric for monitoring of diabetes therapy.The DISTq method replicated the findings of the fully sampled DIST test without the use of insulin or C-peptide assays. Correlations of the resulting SI values was R=0.91. The method was also compared to the euglycaemic hyperinsulinaemic clamp (EIC) in an in-silico Monte-Carlo analysis and showed a good ability to re-evaluate SI EIC (R=0.89), compared to the fully sampled DIST (R=0.98) Population-derived parameter estimates using a-posteriori population-based functions derived from DIST test data enables the simulation of insulin profiles that are sufficiently accurate to estimate SI to a relatively high precision. Thus, costly insulin and C-peptide assays are not necessary to obtain an accurate, but inexpensive, real-time estimate of insulin sensitivity. This estimate has enough resolution for SI prediction and monitoring of response to therapy. In borderline cases, re-evaluation of stored (frozen) blood samples for insulin and C-peptide would enable greater accuracy where necessary, enabling a hierarchy of tests in an economical fashion.

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“…The first three are linear, and the rests are nonlinear. The second, fourth and sixth systems are also used in [5,6] as the test problems for comparing different numerical methods. The exact solutions to the six ODEs are shown in fig.…”

Section: Numerical Examplesmentioning

confidence: 99%

Hybrid dynamic iterations for the solution of initial value problems

Yu¹,

Srinivasan²

2011

Journal of Parallel and Distributed Computing

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Many scientific problems are posed as Ordinary Differential Equations (ODEs). A large subset of these are initial value problems, which are typically solved numerically. The solution starts by using a known state-space of the ODE system to determine the state at a subsequent point in time. This process is repeated several times. When the computational demand is high due to large state space, parallel computers can be used efficiently to reduce the time to solution. Conventional parallelization strategies distribute the state space of the problem amongst cores and distribute the task of computing for a single time step amongst the cores. They are not effective when the computational problems have fine granularity, for example, when the state space is relatively small and the computational effort arises largely from the long time span of the initial value problem. We propose a hybrid dynamic iterations method 1 which combines conventional sequential ODE solvers with dynamic iterations to parallelize the time domain. Empirical results demonstrate a factor of two to four improvement in performance of the hybrid dynamic iterations method over a conventional ODE solver on an 8 core processor. Compared to Picard iterations (also parallelized in the time domain), the proposed method shows better convergence and speedup results when high accuracy is required. Keywords: time parallelization, hybrid dynamic iterations, ODE solverEmail address: {yu,asriniva}@cs.fsu.edu (Yanan Yu and Ashok Srinivasan) 1 This paper is an extended version of a conference paper [1]. In this paper, we have considered an additional underlying ODE solver for the hybrid method, empirically evaluated with more ODE systems, and also evaluated the relative performance with low and high accuracy requirements.

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Picard iteration algorithm combined with Gauss–Seidel technique for initial value problems

Cited by 22 publications

References 11 publications

Unique parameter identification for cardiac diagnosis in critical care using minimal data sets

Unique parameter identification for cardiac diagnosis in critical care using minimal data sets

DISTq: An Iterative Analysis of Glucose Data for Low-Cost, Real-Time and Accurate Estimation of Insulin Sensitivity

Hybrid dynamic iterations for the solution of initial value problems

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