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  1. Original Research ARTICLE
  2. A localized orthogonal decomposition method for semi-linear elliptic problems
  3. Computational Mathematics II

To browse Academia. Skip to main content. You're using an out-of-date version of Internet Explorer. By using our site, you agree to our collection of information through the use of cookies. To learn more, view our Privacy Policy. Log In Sign Up. A new iterative method to compute nonlinear equations Applied Mathematics and Computation, Francisco Calheiros.

Viriato Semiao. A new iterative method to compute nonlinear equations. Applied Mathematics and Computation — www. The new method is based on the proposals of Abbasbandy on improving the order of accuracy of Newton—Raphson method [S. Babolian, J. Biazar, On the order of convergence of Ado- mian method, Applied Mathematics and Computation —]. The con- vergence of the new scheme is proved and at least the cubic order of convergence is established. Several examples are presented and compared to other methods, showing the accuracy and fast convergence of this new method.

Original Research ARTICLE

E-mail address: mbasto ipca. All rights reserved. Basto et al. Introduction In the beginning of the 80s, a new method for exactly solving nonlinear functional equations of various kinds has been proposed by Adomian, the so called Adomian decomposition method ADM [1—5]. Over the last 20 years, such method has been applied to solve a wide range of problems, both deter- ministic and stochastic, linear and nonlinear, arising from physics, chemistry, biology, engineering, etc. The technique uses a decomposition of the nonlinear operator as a series function. This series usually exhibits fast convergence to the exact solution, if there is a single one possible, and to one of the possible solu- tions, if several exist [5] and [6].

For each n, the polynomial An de- pends only on x0, x1,. Adomian introduced formulas to generate these polynomials for all kind of nonlinearities [1,2,4,5]. To improve the order of convergence of the sequence Sn n as given in [19], instead of solving Eq. Also, Eqs. Besides, if one solves Eqs. Writing Eq. The truncated series solution of Eq. As it has been mentioned above, to obtain the real solution one should mod- ify Eq. A new iterative method 2. Applying 10 to Eq. The Rosenbrock method is over 8 times faster than the standard Runge-Kutta 4th order method, which is a non-adaptive time step method.

While Dormand—Prince is an adaptive time step method that is even faster than Rosenbrock, as will be shown below, it is an explicit method that does not always converge to the correct solution.

Numerical Solutions of Linear Systems - Jacobi and Gauss-Seidel Matrix forms

The fastest time of execution for the Dormand-Prince method is not surprising since an explicit method is easier to evaluate computationally than an implicit method, in particular because there is no need for matrix inversions. Second, a comparison of the methods behavior over 14 days of simulation is presented in Figure 4. The advantage of the Rosenbrock method in terms of stability and convergence to the correct solution is clearly noticed.

Figure 4. Finally, we also report in Table 7 the sensitivity of the Rosenbrock method. We notice that most of the perturbations are symmetrical. The results Table 7 indicate that the Rosenbrock method is sufficiently robust for our needs. The parameters file input is shown in Figure 6. For illustration, two example simulations are provided, one over 2 days Figure 7 and one over 28 days Figure 8. In Figure 7 we show how different parameter ensembles can be displayed simultaneously. Combining both factors shows a multiphasic viral decline which was observed under treatment with HCV NS5A inhibitors [ 25 ].

In Figure 8 we present the difference between the long-term approximation and the Rosenbrock method for those three sets of parameters over longer than 2 days treatment durations e. Thus, in all cases the trends from the first 2 days continue in the next 4 weeks and the error grows minimally. Figure 5. Graphical user interface GUI. The displayed values are the default options and can be modified. At the bottom are the options to add experimental data and parameters values from a file.

An option to export the parameters is also available. Figure 6. An example of a parameters file that can be automatically uploaded. Lines starting with a are comments ignored. This file can also be exported from our software, allowing users to share them easily. Figure 7. Several simulations can be displayed simultaneously. Three treatment scenarios are plotted with similar results as previously shown in Rong et al. Figure 8. One can notice how even at the longer treatment duration the difference remains minimal and mostly constant.

Modeling intracellular viral RNA dynamics within infected cells is becoming an improtant mean for considering various curative treatment options. A viral dynamic model that considers intracellular viral RNA replication, namely an age-structured PDE multiscale model, has been recently put forth to study viral hepatitis dynamics during antiviral therapy [ 25 , 41 , 43 ]. This type of model is more complicated to solve than previous ODEs models.

The seminal works that introduce the age-based multiscale model predominantly use analytical approximations, with some numerical solutions that are either based on simple first-order methods or canned solvers [ 25 , 41 , 43 ]. Neither of these numerical solutions are satisfactory for realistic simulations of several days of infection and therapy in terms of accuracy, efficiency and stability. We first showed that the long-term approximation is an underestimate of the PDE model solution, which was anticipated in Rong and Perelson [ 43 ] because some infection events are being ignored in this analytical approximation.

A localized orthogonal decomposition method for semi-linear elliptic problems

We then observed that the governing differential equations are stiff and therefore advanced numerical methods are needed. Methods with fixed stepsize or alternatively, the use of canned routines, do not offer a comprehensive solution to the model and are limited in scope. For example, in the multiscale model, there is an integral term that needs to be computed at each time step depending on previous iterations and this is inadequately done by canned solvers. Methods with adaptive stepsize offer a considerable improvement for realistic simulations.

Having previously investigated these methods, we found the Rosebrock method to be the most accurate and efficient among adaptive stepsize methods for this model. Future work from the numerical standpoint could potentially include a more comprehensive treatment of the associated integral. In particular the adaptive stepsize techniques rely on the evaluation of the error produced at each evaluation of the ODE equations. Therefore, the error produced by the integral itself is never taken into account.

Including that term would allow to relax the restrictions on the stepsize increase and potentially decrease further, beyond what is observed in Table 5 as least number of iterations, the number of iterations that are required to achieve a sufficient accuracy. The Rosenbrock method implementation provided here can be generalized to potentially assist in understating treatment failure due to drug resistance by the expansion of the age-based model to include viral strains [ 57 ]. Finally, since the current age-based HCV multiscale model is a successful milestone but still fails to predict cure in some DAA-treated patients [ 59 — 61 ], further model modification will be needed [ 38 , 39 ] and future developments would necessitate some updates in the numerical method.

While such models become quickly overwhelming to solve analytically, an extension of the presented method is straightforward. The simulator provided here based on the Rosenbrock method can be modified accordingly and be developed further to accommodate the modelers needs. DB conceived research. VR implemented method and developed the corresponding equations with assistance of DB. The research was supported in part by the U. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors thank Susan Uprichard for careful reading of the manuscript and helpful comments. They would also like to thank Libin Rong and Daniel Coffield for providing the initial code of the multiscale model and the reviewers for their comprehensive comments. Global prevalence and genotype distribution of hepatitis C virus infection in a modelling study. Lancet Gastroenterol Hepatol. World Health Organization. Geneva: World Health Organization Google Scholar. Hepatology 64 — Hepatology 62 — CrossRef Full Text.

Rosen HR. Hepatology 65 —9. Perelson AS. Modelling viral and immune system dynamics. Nat Rev Immunol. Nature HIV-1 dynamics in vivo : virion clearance rate, infected cell life-span, and viral generation time. Science Mathematical modeling of viral kinetics under immune control during primary HIV-1 infection. The role of cells refractory to productive infection in acute hepatitis B viral dynamics.

Modeling complex decay profiles of hepatitis B virus during antiviral therapy. Hepatology 49 —8. Viral dynamics in hepatitis B virus infection. Oral prenylation inhibition with lonafarnib in chronic hepatitis D infection: a proof-of-concept randomised, double-blind, placebo-controlled phase 2A trial. Lancet Infect Dis. Understanding early serum hepatitis D virus and hepatitis B surface antigen kinetics during pegylated interferon-alpha therapy via mathematical modeling. Hepatology 60 — Pharmacokinetics and pharmacodynamics modeling of lonafarnib in patients with chronic hepatitis delta virus infection.

Hepatol Commun. Modeling the acute and chronic phases of Theiler murine encephalomyelitis virus infection. J virol.

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Frequent release of low amounts of herpes simplex virus from neurons: results of a mathematical model. Sci Trans Med. A mathematical model of hepatitis C virus dynamics in patients with high baseline viral loads or advanced liver disease.

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    PLoS Comput Biol. Beauchemin CA, Handel A. A review of mathematical models of influenza A infections within a host or cell culture: lessons learned and challenges ahead. BMC Public Health. Ebola virus dynamics in mice treated with favipiravir. Modeling shows that the NS5A inhibitor daclatasvir has two modes of action and yields a shorter estimate of the hepatitis C virus half-life. Modeling subgenomic hepatitis C virus RNA kinetics during treatment with alpha interferon. Mathematical modeling of subgenomic hepatitis C virus replication in Huh-7 cells.

    Novel mechanism of antibodies to hepatitis B virus in blocking viral particle release from cells. Hepatology 52 — Mathematical modeling of HCV infection and treatment. Methods Mol Biol. Hepatitis C viral kinetics in the era of direct acting antiviral agents and interleukinB. Through a diverse and skilled staff, we are constantly improving quality, integrity and the reliability standards that define us.

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    Computational Mathematics II

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