| 1. |
- Arévalo, Carmen, et al.
(författare)
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A collocation formulation of multistep methods for variable step-size extensions
- 2002
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Ingår i: Applied Numerical Mathematics. - 0168-9274. ; 42:1-3, s. 5-16
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Tidskriftsartikel (refereegranskat)abstract
- Multistep methods are classically constructed by specially designed difference operators on an equidistant time grid. To make them practically useful, they have to be implemented by varying the step-size according to some error-control algorithm. It is well known how to extend Adams and BDF formulas to a variable step-size formulation. In this paper we present a collocation approach to construct variable step-size formulas. We make use of piecewise polynomials to show that every k-step method of order k + I has a variable step-size polynomial collocation formulation. (C) 2002 IMACS. Published by Elsevier Science B.V. All rights reserved.
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| 2. |
- Arévalo, Carmen
(författare)
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A note on numerically consistent initial values for high index differential-algebraic equations
- 2008
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Ingår i: Electronic Transactions on Numerical Analysis. - 1068-9613. ; 34, s. 14-19
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Tidskriftsartikel (refereegranskat)abstract
- When differential-algebraic equations of index 3 or higher are solved with backward differentiation formulas, the solution in the first few steps can have gross errors, the solution can have gross errors in the first few steps, even if the initial values are equal to the exact solution and even if the step size is kept constant. This raises the question of what are consistent initial values for the difference equations. Here we study how to change the exact initial values into what we call numerically consistent initial values for the implicit Euler method.
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| 3. |
- Arévalo, Carmen, et al.
(författare)
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A software platform for adaptive high order multistep methods
- 2020
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Ingår i: ACM Transactions on Mathematical Software. - : Association for Computing Machinery (ACM). - 0098-3500 .- 1557-7295. ; 46:1
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Tidskriftsartikel (refereegranskat)abstract
- We present a software package, Modes, offering h-adaptive and p-adaptive linear multistep methods for first order initial value problems in ordinary differential equations. The implementation is based on a new parametric, grid-independent representation of multistep methods [Arévalo and Söderlind 2017]. Parameters are supplied for over 60 methods. For nonstiff problems, all maximal order methods (p=k for explicit and p=k+1 for implicit methods) are supported. For stiff computation, implicit methods of order p=k are included. A collection of step-size controllers based on digital filters is provided, generating smooth step-size sequences offering improved computational stability. Controllers may be selected to match method and problem classes. A new system for automatic order control is also provided for designated families of multistep methods, offering simultaneous h- and p-adaptivity. Implemented as a Matlab toolbox, the software covers high order computations with linear multistep methods within a unified, generic framework. Computational experiments show that the new software is competitive and offers qualitative improvements. Modes is available for downloading and is primarily intended as a platform for developing a new generation of state-of-the-art multistep solvers, as well as for true ceteris paribus evaluation of algorithmic components. This also enables method comparisons within a single implementation environment.
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| 4. |
- Arévalo, Carmen, et al.
(författare)
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Constant coefficient linear multistep methods with step density control
- 2007
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier BV. - 0377-0427. ; 205:2, s. 891-900
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Tidskriftsartikel (refereegranskat)abstract
- In linear multistep methods with variable step size, the method's coefficients are functions of the step size ratios. The coefficients therefore need to be recomputed on every step to retain the method's proper order of convergence. An alternative approach is to use step density control to make the method adaptive. If the step size sequence is smooth, the method can use constant coefficients without losing its order of convergence. The paper introduces this new adaptive technique and demonstrates its feasibility with a few test problems. The technique works in perfect agreement with theory for a given step density function. For practical use, however, the density must be generated with data computed from the numerical solution. We introduce a local error tracking controller, which automatically adapts the density to computed data, and demonstrate in computational experiments that the technique works well at least up to fourth-order methods. (c) 2006 Elsevier B.V. All rights reserved.
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| 5. |
- Arévalo, Carmen, et al.
(författare)
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Convergence of multistep discretizations of DAEs
- 1995
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Ingår i: BIT. - 0006-3835. ; 35:2, s. 143-168
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Tidskriftsartikel (refereegranskat)abstract
- Standard ODE methods such as linear multistep methods encounter difficulties when applied to differential-algebraic equations (DAEs) of index greater than 1. In particular, previous results for index 2 DAEs have practically ruled out the use of all explicit methods and of implicit multistep methods other than backward difference formulas (BDFs) because of stability considerations. In this paper we embed known results for semi-explicit index 1 and 2 DAEs in a more comprehensive theory based on compound multistep and one-leg discretizations. This explains and characterizes the necessary requirements that a method must fulfill in order to be applicable to semi-explicit DAEs. Thus we conclude that the most useful discretizations are those that avoid discretization of the constraint. A freer use of e.g. explicit methods for the non-stiff differential part of the DAE is then possible.
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| 6. |
- Arévalo, Carmen, et al.
(författare)
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GRID-INDEPENDENT CONSTRUCTION OF MULTISTEP METHODS
- 2017
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Ingår i: Journal of Computational Mathematics. - : Global Science Press. - 0254-9409 .- 1991-7139. ; 35, s. 672-692
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Tidskriftsartikel (refereegranskat)abstract
- A new polynomial formulation of variable step size linear multistep methods is presented, where each k-step method is characterized by a fixed set of k-1 or k parameters. This construction includes all methods of maximal order (p=k for stiff, and p=k+1 for nonstiff problems). Supporting time step adaptivity by construction, the new formulation is not based on extending classical fixed step size methods; instead classical methods are obtained as fixed step size restrictions within a unified framework. The methods are implemented in Matlab, with local error estimation and a wide range of step size controllers. This provides a platform for investigating and comparing different multistep method in realistic operational conditions. Computational experiments show that the new multistep method construction and implementation compares favorably to existing software, although variable order has not yet been included.
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| 7. |
- Arévalo, Carmen, et al.
(författare)
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Improving the accuracy of BDF methods for index 3 dierential algebraic equations
- 1995
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Ingår i: BIT. - 0006-3835 .- 1572-9125. ; 35:3, s. 297-308
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Tidskriftsartikel (refereegranskat)abstract
- Methods for solving index 3 DAEs based on BDFs suffer a loss of accuracy when there is a change of step size or a change of order of the method. A layer of nonuniform convergence is observed in these cases, and O(1) errors may appear in the algebraic variables. From the viewpoint of error control, it is beneficial to allow smooth changes of step size, and since most codes based on BDFs are of variable order, it is also of interest to avoid the inaccuracies caused by a change of order of the method. In the case of BDFs applied to index 3 DAEs in semi-explicit form, we present algorithms that correct to O(h) the inaccurate approximations to the algebraic variables when there are changes of step size in the backward Euler method. These algorithms can be included in an existing code at a very small cost. We have also described how to obtain formulas that correct the O(1) errors in the algebraic variables appearing after a change of order.
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| 8. |
- Arévalo, Carmen, et al.
(författare)
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Local error estimation and step size control in adaptive linear multistep methods
- 2021
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Ingår i: Numerical Algorithms. - : Springer Science and Business Media LLC. - 1017-1398 .- 1572-9265. ; 86:2, s. 537-563
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Tidskriftsartikel (refereegranskat)abstract
- In a k-step adaptive linear multistep methods the coefficients depend on the k − 1 most recent step size ratios. In a similar way, both the actual and the estimated local error will depend on these step ratios. The classical error model has been the asymptotic model, chp+ 1y(p+ 1)(t), based on the constant step size analysis, where all past step sizes simultaneously go to zero. This does not reflect actual computations with multistep methods, where the step size control selects the next step, based on error information from previously accepted steps and the recent step size history. In variable step size implementations the error model must therefore be dynamic and include past step ratios, even in the asymptotic regime. In this paper we derive dynamic asymptotic models of the local error and its estimator, and show how to use dynamically compensated step size controllers that keep the asymptotic local error near a prescribed tolerance tol. The new error models enable the use of controllers with enhanced stability, producing more regular step size sequences. Numerical examples illustrate the impact of dynamically compensated control, and that the proper choice of error estimator affects efficiency.
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| 9. |
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| 10. |
- Arévalo, Carmen, et al.
(författare)
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Program for the Promotion of Researchers (PPI) in Venezuela: acknowledgment or stimulus?
- 1996
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Ingår i: Interciencia. - 0378-1844. ; 21:2, s. 86-93
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Tidskriftsartikel (refereegranskat)abstract
- In 1990 Venezuela started its Program for the Promotion of Researchers (PPI), in order to foster research in science and technology A partial evaluation of the effects of the program on scientific and technological activities can be carried out using data from the System for :the Promotion of Researchers Foundation Fund (SPI). A complete and definitive evaluation is not possible, due to the fact that the Venezuelan science and technology community has obtained other incentives and has been negatively affected by situations not related to the PPI. Here we do not evaluate the influence of these other factors over scientific endeavor. After the initial pow at the start of the program in 1990, the number of membership requests remained at a fairly constant level until the 1995 convocation, when the number of requests was doubled. The increased interest in the program may be ascribed to the change of the Technical Secretary and his policies of dialogue and information. About 50% of all accredited researchers are classified in level I. The rates of growth of levels III and Candidate are lower than expected. The norms of access to level III may be over-valued, as indicated by the fact that in the area of physics, mathematics and chemistry (CMFQ), 48 years of production al the stipulated rate are needed to reach that level. The surprisingly low number of Candidates points to the need of forming human resources with post-graduate studies in science and technology. In Venezuela, a researcher produces approximately one paper every two years. This is the production rate established for the enhance and permanence at level I, and it has remained fairly invariant along the 5 years of activity of the PPI. The rate of scientific production in Venezuela is notably less than that of the other four countries with largest scientific production in Latin America. Thanks to the PPI the country has a data base of the scientific-technical sector. Through this characterization, the researcher has been acknowledged and valued in his productive activity; nevertheless, the PPI has not had the expected impact over the country's scientific and technological production. The composition of the Committees of Areas has been a polemical factor in the application of the program. Of the potential evaluators, only 15% have sewed in the committees for medical, biological and agricultural sciences, and for physics, mathematics and chemistry, 25% in the committee for the social sciences, and 51% in the committee for engineering, technology and earth sciences. Although these committees have been formed according to regulations, their composition are a cause of uneasiness in the community, and therefore a change of the rules should be analyzed. Similar uneasiness is caused by a lack of a Committee of Appeals. The financial contribution of the PPI is less than 20% of a researcher's salary. To maintain the program's economic incentives, the State should recapitalize the SPI. The institutions that employ 82% of all researchers in the program are IVIC, LUZ, UCV, ULA and USE. Those which implemented institutional policies of stimulus to research (LUZ and USB), were successful in raising efficiency We conclude it is necessary to evaluate the program integrally and to study some statute modifications. Furthermore, the country's scientific institutions must cooperate to stimulate the sector, creating additional incentives, the State must back the program financially, and the scientific community must remain vigilant to keep and perfect this effort to support and promote national research.
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