Thirty Years of Virtual Substitution

Sturm, Thomas

doi:10.1145/3208976.3209030

Cited by 14 publications

(10 citation statements)

References 51 publications

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“…It would be indeed interesting to work on such parametric data. In the presence of parameters, one would consider effective quantifier elimination over real closed fields [15,20,39,69,70,80] as a generalization of SMT solving. Robust implementations are freely available [9,19] and well supported.…”

Section: Discussionmentioning

confidence: 99%

Algorithmic Reduction of Biological Networks with Multiple Time Scales

et al. 2021

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We present a symbolic algorithmic approach that allows to compute invariant manifolds and corresponding reduced systems for differential equations modeling biological networks which comprise chemical reaction networks for cellular biochemistry, and compartmental models for pharmacology, epidemiology and ecology. Multiple time scales of a given network are obtained by scaling, based on tropical geometry. Our reduction is mathematically justified within a singular perturbation setting. The existence of invariant manifolds is subject to hyperbolicity conditions, for which we propose an algorithmic test based on Hurwitz criteria. We finally obtain a sequence of nested invariant manifolds and respective reduced systems on those manifolds. Our theoretical results are generally accompanied by rigorous algorithmic descriptions suitable for direct implementation based on existing off-the-shelf software systems, specifically symbolic computation libraries and Satisfiability Modulo Theories solvers. We present computational examples taken from the well-known BioModels database using our own prototypical implementations.

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Section: Discussionmentioning

confidence: 99%

Algorithmic Reduction of Biological Networks with Multiple Time Scales

et al. 2021

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“…Over the years, several algorithms have been developed to perform a quantifier elimination. In the opinion of the author, the most common algorithms are cylindrical algebraic decomposition [5] and virtual substitution [6]. To carry out quantifier elimination we employed the REDUCE package REDLOG [7], which support both algorithms.…”

Section: Ofmentioning

confidence: 99%

Formal Calculation of Positive Invariant Sets for the Lorenz Family Combining Lyapunov Approaches and Quantifier Elimination

Röbenack

2021

Proc Appl Math and Mech

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The Lorenz family is a class of dynamic systems which is parameterized in one variable and includes both the classic Lorenz system and the Chen system. Both systems (and therefore the Lorenz family) can show chaotic behaviour. For different questions of system analysis, e.g. for the estimation of various types of dimensions, it is helpful to be able to determine bounds on the attractor. One typical approach ist the usage of Lyapunov functions to describe positive invariant sets. However, this approach usually requires a good knowledge of the system to perform the necessary calculations. For polynomial systems, these calculations can be carried out using a computation technique known as quantifier elimination. In this contribution we describe this approach and employ it to calculate bounds on the Lorenz family.

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“…Furthermore, virtual substitution adds the guarding conditions γ in a suitable way. For examples and surveys of the virtual substitution method see the work of Sturm (2017Sturm ( , 2018.…”

Section: Virtual Substitutionmentioning

confidence: 99%

Identifying the parametric occurrence of multiple steady states for some biological networks

Bradford

Davenport

England

et al. 2020

Journal of Symbolic Computation

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We consider a problem from biological network analysis of determining regions in a parameter space over which there are multiple steady states for positive real values of variables and parameters. We describe multiple approaches to address the problem using tools from Symbolic Computation. We describe how progress was made to achieve semi-algebraic descriptions of the multistationarity regions of parameter space, and compare symbolic results to numerical methods.The biological networks studied are models of the mitogen-activated protein kinases (MAPK) network which has already consumed considerable effort using special insights into its structure of corresponding models. Our main example is a model with 11 equations in 11 variables and 19 parameters, 3 of which are of interest for symbolic treatment. The model also imposes positivity conditions on all variables and parameters.We apply combinations of symbolic computation methods designed for mixed equality/inequality systems, specifically virtual substitution, lazy real triangularization and cylindrical algebraic decomposition, as well as a simplification technique adapted from Gaussian elimination and graph theory.We are able to determine multistationarity of our main example over a 2-dimensional parameter space. We also study a second MAPK model and a symbolic grid sampling technique which can locate such regions in 3-dimensional parameter space.

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Thirty Years of Virtual Substitution

Cited by 14 publications

References 51 publications

Algorithmic Reduction of Biological Networks with Multiple Time Scales

Algorithmic Reduction of Biological Networks with Multiple Time Scales

Formal Calculation of Positive Invariant Sets for the Lorenz Family Combining Lyapunov Approaches and Quantifier Elimination

Identifying the parametric occurrence of multiple steady states for some biological networks

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