#### DMCA

## Efficient Syntax-driven Lumping of Differential Equations

### Citations

534 | The university of florida sparse matrix collection
- Davis, Hu
- 2011
(Show Context)
Citation Context ...rry over. For these, we report considerable aggregations for real-world applications in atmospheric modeling, structural mechanics, and electrical engineering, taken from the Sparse Matrix Collection =-=[15]-=-. Further related work. FB is a special case of the theory of ODE lumping [16,17], which is more general because it considers an arbitrary linear transformation of the state space, as opposed to a sum... |

526 | Bisimulation through probabilistic testing - Larsen, Skou - 1991 |

455 |
Three partition refinement algorithms
- Paige, Tarjan
- 1996
(Show Context)
Citation Context ...ion systems, the notion of bisimilarity has played a key role for model reduction, with efficient algorithms [5,6,7] based on Paige and Tarjan’s celebrated solution to the coarsest refinement problem =-=[8]-=-. The main contribution of this paper is to lift this approach to ODE systems. In particular we focus on a class of polynomial systems, where the time derivatives are multivariate polynomials of degre... |

315 |
Iterative Methods for Sparse Linear Systems (2nd edn
- Saad
- 2003
(Show Context)
Citation Context ...te the effectiveness on affine systems, we propose an application of the bisimulations beyond ODEs: we consider linear systems of equationsAx = b. Stationary iterative methods such as Jacobi’s (e.g., =-=[14]-=-) can be interpreted as an affine dynamical system in discrete time, to which case the bisimulations carry over. For these, we report considerable aggregations for real-world applications in atmospher... |

157 | Chemical reaction network structure and the stability of complex isothermal reactors I. the deficiency zero and deficiency one theorems. Chemical Engineering Science 42:2229–2268
- Feinberg
- 1987
(Show Context)
Citation Context ...l ODE system (with degree at most two), we slightly extend the CRN syntax by allowing negative rates. This has important repercussions on the applicability of established results of CRN theory (e.g., =-=[10]-=-). Hence, to disambiguate, we refer to this extension as Reaction Networks (RN). Instead, all the results for exact quantitative bisimulations for CRNs, recently proposed by these authors in [11] (cf.... |

116 | Exact and ordinary lumpability in finite Markov chains.
- Buchholz
- 1994
(Show Context)
Citation Context ...f computing the largest bisimulations by developing a variant of Paige and Tarjan’s algorithm, along the lines of the efficient partition refinement algorithms of [6] and [7] for Markov chain lumping =-=[13]-=-, and of [5] for probabilistic transition systems. In particular, for technical reasons that will be clarified later, we build on the Markov chain lumping algorithm of [7]. Our algorithm, presented in... |

87 |
BioNetGen: software for rule-based modeling of signal transduction based on the interactions of molecular domains
- Blinov, Faeder, et al.
(Show Context)
Citation Context ...he system under consideration, the state space size (intended as the number of ODE variables) is so large that it makes the numerical solution intractable (e.g., in protein-based interaction networks =-=[1,2]-=-). Formal kinds of analyses such as reachability computation suffer from the curse of dimensionality, particularly for nonlinear systems (e.g., [3,4]). It is therefore an important goal to be able to ... |

68 | A Markov reward model checker.
- Katoen, Khattri, et al.
- 2002
(Show Context)
Citation Context ...e our algorithm using i) the biochemical networks evaluated in [11] as case studies for degree-two polynomial systems; ii) Ax = b systems from [15]; and iii) selected CTMCs from the MRMC distribution =-=[33]-=-. Comparing against the reductions of [11] and MRMC also allowed us to validate the implementation of our algorithm. 13 Original model FB reduction BB reduction Id Ref. |R| |S| Red.(s) [11] Red.(s) |R... |

44 | Symmetry Reduction for Probabilistic Model Checking
- Kwiatkowska, Norman, et al.
(Show Context)
Citation Context ...kov chains C1 [30] 22 871 849 3 101 445 4.00E+4 2.01E+3 1 069 777 135 752 — 1.34E+3 1 166 931 148 092 C2 [31] 11 583 520 2 373 652 1.73E+2 9.78E+1 5 792 531 1 187 597 — 3.07E+2 5 814 622 1 187 597 C3 =-=[32]-=- 10 485 761 1 048 576 1.48E+1 1.76E+1 3301 792 — 1.23E+1 5083 792 Table 1. FB and BB reductions. Entries labeled with “—” indicate that the reduction algorithm did not terminate within 24 hours. Greye... |

41 | Deciding bisimilarity and similarity for probabilistic processes.
- Baier, Engelen, et al.
- 2000
(Show Context)
Citation Context ...erve the original dynamics. For discrete-state quantitative models based on labeled transition systems, the notion of bisimilarity has played a key role for model reduction, with efficient algorithms =-=[5,6,7]-=- based on Paige and Tarjan’s celebrated solution to the coarsest refinement problem [8]. The main contribution of this paper is to lift this approach to ODE systems. In particular we focus on a class ... |

39 | Bisimulation Minimisation Mostly Speeds Up Probabilistic Model Checking
- Katoen, Kemna, et al.
- 2007
(Show Context)
Citation Context ...red to compute, at each iteration, the equivalence classes for multi-sets of species according to Definition 4. CTMCs (C1–C3). These are the three largest CTMCs of the MRMC distribution [33], used in =-=[34]-=- to study the impact of ordinary CTMC lumpability in model checking. In particular, these are: a protocol for wireless group communication (C1, original model name FDT3E3 PE16E4 S4OD40 ); a cluster mo... |

37 |
J.-P.: On the use of model checking techniques for dependability evaluation
- Haverkort, Hermanns, et al.
- 2000
(Show Context)
Citation Context ...15] 706 577 116 836 3.23E+0 3.11E+0 609 459 73 423 — 2.86E+0 609 307 73 348 Continuous-time Markov chains C1 [30] 22 871 849 3 101 445 4.00E+4 2.01E+3 1 069 777 135 752 — 1.34E+3 1 166 931 148 092 C2 =-=[31]-=- 11 583 520 2 373 652 1.73E+2 9.78E+1 5 792 531 1 187 597 — 3.07E+2 5 814 622 1 187 597 C3 [32] 10 485 761 1 048 576 1.48E+1 1.76E+1 3301 792 — 1.23E+1 5083 792 Table 1. FB and BB reductions. Entries ... |

28 | Optimal state-space lumping in Markov chains.
- Derisavi, Hermanns, et al.
- 2003
(Show Context)
Citation Context ...erve the original dynamics. For discrete-state quantitative models based on labeled transition systems, the notion of bisimilarity has played a key role for model reduction, with efficient algorithms =-=[5,6,7]-=- based on Paige and Tarjan’s celebrated solution to the coarsest refinement problem [8]. The main contribution of this paper is to lift this approach to ODE systems. In particular we focus on a class ... |

25 |
The effect of lumping and expanding on kinetic differential equations.
- Tomlin, Li, et al.
- 1997
(Show Context)
Citation Context ...ations in atmospheric modeling, structural mechanics, and electrical engineering, taken from the Sparse Matrix Collection [15]. Further related work. FB is a special case of the theory of ODE lumping =-=[16,17]-=-, which is more general because it considers an arbitrary linear transformation of the state space, as opposed to a sum of variables for FB. While the theory is established, 2 no algorithm is availabl... |

24 | Abstracting the differential semantics of rule-based models: Exact and automated model reduction
- Danos, Feret, et al.
- 2010
(Show Context)
Citation Context ...he system under consideration, the state space size (intended as the number of ODE variables) is so large that it makes the numerical solution intractable (e.g., in protein-based interaction networks =-=[1,2]-=-). Formal kinds of analyses such as reachability computation suffer from the curse of dimensionality, particularly for nonlinear systems (e.g., [3,4]). It is therefore an important goal to be able to ... |

19 |
A general analysis of exact lumping in chemical kinetics.
- Li, Rabitz
- 1989
(Show Context)
Citation Context ...ations in atmospheric modeling, structural mechanics, and electrical engineering, taken from the Sparse Matrix Collection [15]. Further related work. FB is a special case of the theory of ODE lumping =-=[16,17]-=-, which is more general because it considers an arbitrary linear transformation of the state space, as opposed to a sum of variables for FB. While the theory is established, 2 no algorithm is availabl... |

17 |
Mathematical biology. I. An introduction, 3rd edn
- Murray
- 2002
(Show Context)
Citation Context ... where the time derivatives are multivariate polynomials of degree at most two in the ODE variables. This class is quite general because it incorporates models frequently used in (bio-)chemistry (cf. =-=[9]-=-) as well as the ubiquitous linear ODEs. We reconcile the established approaches based on discrete-state models with the continuous-state semantics of ODEs by reasoning at the level of a discrete-stat... |

15 |
Efficient modeling, simulation and coarse-graining of biological complexity with NFsim.
- Sneddon, Faeder, et al.
- 2011
(Show Context)
Citation Context ...to validate the implementation of our algorithm. 13 Original model FB reduction BB reduction Id Ref. |R| |S| Red.(s) [11] Red.(s) |R| |S| Red.(s) [11] Red.(s) |R| |S| Biochemical reaction networks M1 =-=[22]-=- 3 538 944 262 146 4.61E+4 7.49E+0 990 222 7.65E+4 1.21E+1 2 614 222 M2 [22] 786 432 65 538 1.92E+3 1.58E+0 720 167 3.68E+3 2.51E+0 1 873 167 M3 [22] 172 032 16 386 8.15E+1 2.89E–1 504 122 1.77E+2 6.0... |

14 | Computing reachable states for nonlinear biological models, in:
- Dang, Guernic, et al.
- 2009
(Show Context)
Citation Context ...table (e.g., in protein-based interaction networks [1,2]). Formal kinds of analyses such as reachability computation suffer from the curse of dimensionality, particularly for nonlinear systems (e.g., =-=[3,4]-=-). It is therefore an important goal to be able to obtain reduced size models that appropriately preserve the original dynamics. For discrete-state quantitative models based on labeled transition syst... |

13 |
Investigation of early events in FcεRI-mediated signaling using a detailed mathematical model
- Faeder, Hlavacek, et al.
(Show Context)
Citation Context ...6 386 8.15E+1 2.89E–1 504 122 1.77E+2 6.03E–1 1 305 122 M4 [22] 48 18 1.00E–3 1.00E–3 24 12 2.00E–3 2.00E–3 44 12 M5 [23] 194 054 14 531 3.72E+1 3.88E–1 142 165 10 855 1.32E+3 6.00E–1 91 001 6 634 M6 =-=[24]-=- 187 468 10 734 3.07E+1 6.09E–1 57 508 3 744 2.71E+2 1.40E+0 145 650 5 575 M7 [24] 32 776 2 506 1.26E+0 1.19E–1 16 481 1 281 1.66E+1 2.14E–1 32 776 2 506 M8 [25] 41 233 2 562 1.12E+0 2.69E–1 33 075 1 ... |

12 |
On the inverse problem of reaction kinetics
- Hárs, Tóth
- 1981
(Show Context)
Citation Context ...of linear equations using stationary iterative methods. Finally, we relate RN bisimulations to CTMC lumpability [13]. It is easy to see that the encoding of polynomials ODEs to RNs is not unique (cf. =-=[21]-=- for CRNs). Here, we propose one for which the algorithmic complexity can be directly related to the number of monomials appearing in the ODE system, leaving the question of investigating minimality i... |

12 | Morphisms of reaction networks that couple structure to function. BMC systems biology
- Cardelli
- 2014
(Show Context)
Citation Context ...[26] 5 797 796 1.61E–1 1.90E–2 4 210 503 7.37E–1 3.30E–2 5 797 796 M11 [27] 5832 730 3.89E–1 1.50E–2 1296 217 6.00E–1 2.40E–2 237 217 M12 [28] 487 85 2.00E–3 2.00E–3 264 56 6.00E–3 3.00E–3 431 56 M13 =-=[29]-=- 24 18 1.20E–2 4.00E–3 24 18 7.00E–3 4.00E–3 7 3 Affine systems F1 [15] 10 319 760 1 489 753 9.74E+3 8.70E+2 1 295 514 188 101 — 2.23E+2 10 319 760 1 489 753 F2 [15] 8 814 880 1 270 433 8.86E+2 5.58E+... |

9 |
Model checking dependability attributes of wireless group communication
- Massink, Katoen, et al.
- 2004
(Show Context)
Citation Context ...6 F4 [15] 4 706 074 143 572 6.72E+0 6.70E+0 565 288 47 858 — 1.47E+1 2 739 188 112 444 F5 [15] 706 577 116 836 3.23E+0 3.11E+0 609 459 73 423 — 2.86E+0 609 307 73 348 Continuous-time Markov chains C1 =-=[30]-=- 22 871 849 3 101 445 4.00E+4 2.01E+3 1 069 777 135 752 — 1.34E+3 1 166 931 148 092 C2 [31] 11 583 520 2 373 652 1.73E+2 9.78E+1 5 792 531 1 187 597 — 3.07E+2 5 814 622 1 187 597 C3 [32] 10 485 761 1 ... |

6 | Simple O(m logn) time Markov chain lumping - Valmari, Franceschinis - 2010 |

6 | Exact fluid lumpability for Markovian process algebra
- Tschaikowski, Tribastone
- 2012
(Show Context)
Citation Context ... for FB. While the theory is established, 2 no algorithm is available to compute such aggregations. BB is a generalization of a behavioral equivalence originally defined for Markovian process algebra =-=[18]-=-. FB and BB have been recently put in a unifying algorithmic context in [19], using the notions of forward and backward differential equivalences for a low-level syntax describing a more general class... |

4 | Reachability analysis of polynomial systems using linear programming relaxations
- Sassi, Testylier, et al.
- 2012
(Show Context)
Citation Context ...table (e.g., in protein-based interaction networks [1,2]). Formal kinds of analyses such as reachability computation suffer from the curse of dimensionality, particularly for nonlinear systems (e.g., =-=[3,4]-=-). It is therefore an important goal to be able to obtain reduced size models that appropriately preserve the original dynamics. For discrete-state quantitative models based on labeled transition syst... |

4 | A bipolar clamp mechanism for activation of jak-family protein tyrosine kinases
- Barua, Faeder, et al.
(Show Context)
Citation Context ...10 855 1.32E+3 6.00E–1 91 001 6 634 M6 [24] 187 468 10 734 3.07E+1 6.09E–1 57 508 3 744 2.71E+2 1.40E+0 145 650 5 575 M7 [24] 32 776 2 506 1.26E+0 1.19E–1 16 481 1 281 1.66E+1 2.14E–1 32 776 2 506 M8 =-=[25]-=- 41 233 2 562 1.12E+0 2.69E–1 33 075 1 897 1.89E+1 3.97E–1 41 233 2 562 M9 [25] 5 033 471 1.91E–1 1.60E–2 4 068 345 4.35E–1 2.40E–2 5 033 471 M10 [26] 5 797 796 1.61E–1 1.90E–2 4 210 503 7.37E–1 3.30E... |

3 | Machines vs. ensembles: Effective MAPK signaling through heterogeneous sets of protein complexes
- Suderman, Deeds
(Show Context)
Citation Context ... 786 432 65 538 1.92E+3 1.58E+0 720 167 3.68E+3 2.51E+0 1 873 167 M3 [22] 172 032 16 386 8.15E+1 2.89E–1 504 122 1.77E+2 6.03E–1 1 305 122 M4 [22] 48 18 1.00E–3 1.00E–3 24 12 2.00E–3 2.00E–3 44 12 M5 =-=[23]-=- 194 054 14 531 3.72E+1 3.88E–1 142 165 10 855 1.32E+3 6.00E–1 91 001 6 634 M6 [24] 187 468 10 734 3.07E+1 6.09E–1 57 508 3 744 2.71E+2 1.40E+0 145 650 5 575 M7 [24] 32 776 2 506 1.26E+0 1.19E–1 16 48... |

3 | Modeling the effect of APC truncation on destruction complex function in colorectal cancer cells. PLoS Comput Biol (2013
- Barua, WS
(Show Context)
Citation Context ...9E–1 16 481 1 281 1.66E+1 2.14E–1 32 776 2 506 M8 [25] 41 233 2 562 1.12E+0 2.69E–1 33 075 1 897 1.89E+1 3.97E–1 41 233 2 562 M9 [25] 5 033 471 1.91E–1 1.60E–2 4 068 345 4.35E–1 2.40E–2 5 033 471 M10 =-=[26]-=- 5 797 796 1.61E–1 1.90E–2 4 210 503 7.37E–1 3.30E–2 5 797 796 M11 [27] 5832 730 3.89E–1 1.50E–2 1296 217 6.00E–1 2.40E–2 237 217 M12 [28] 487 85 2.00E–3 2.00E–3 264 56 6.00E–3 3.00E–3 431 56 M13 [29]... |

3 |
Simulation of large-scale rule-based models, Bioinformatics 25
- Colvin, Monine, et al.
- 2009
(Show Context)
Citation Context ...2E+0 2.69E–1 33 075 1 897 1.89E+1 3.97E–1 41 233 2 562 M9 [25] 5 033 471 1.91E–1 1.60E–2 4 068 345 4.35E–1 2.40E–2 5 033 471 M10 [26] 5 797 796 1.61E–1 1.90E–2 4 210 503 7.37E–1 3.30E–2 5 797 796 M11 =-=[27]-=- 5832 730 3.89E–1 1.50E–2 1296 217 6.00E–1 2.40E–2 237 217 M12 [28] 487 85 2.00E–3 2.00E–3 264 56 6.00E–3 3.00E–3 431 56 M13 [29] 24 18 1.20E–2 4.00E–3 24 18 7.00E–3 4.00E–3 7 3 Affine systems F1 [15]... |

3 |
The interplay of double phosphorylation and scaffolding in MAPK pathways
- Kocieniewski, Faeder, et al.
(Show Context)
Citation Context ...3 471 1.91E–1 1.60E–2 4 068 345 4.35E–1 2.40E–2 5 033 471 M10 [26] 5 797 796 1.61E–1 1.90E–2 4 210 503 7.37E–1 3.30E–2 5 797 796 M11 [27] 5832 730 3.89E–1 1.50E–2 1296 217 6.00E–1 2.40E–2 237 217 M12 =-=[28]-=- 487 85 2.00E–3 2.00E–3 264 56 6.00E–3 3.00E–3 431 56 M13 [29] 24 18 1.20E–2 4.00E–3 24 18 7.00E–3 4.00E–3 7 3 Affine systems F1 [15] 10 319 760 1 489 753 9.74E+3 8.70E+2 1 295 514 188 101 — 2.23E+2 1... |

2 | Forward and backward bisimulations for chemical reaction networks
- Cardelli, Tribastone, et al.
- 2015
(Show Context)
Citation Context ....g., [10]). Hence, to disambiguate, we refer to this extension as Reaction Networks (RN). Instead, all the results for exact quantitative bisimulations for CRNs, recently proposed by these authors in =-=[11]-=- (cf. Section 2), do carry over to RN. The forward bisimulation (FB) gives a partition of the ODE variables such that the sum of the ODEs can be written as an explicit function of the sum of the varia... |

2 | Symbolic Computation of Differential Equivalences.
- Cardelli, Tribastone, et al.
- 2016
(Show Context)
Citation Context ...pute such aggregations. BB is a generalization of a behavioral equivalence originally defined for Markovian process algebra [18]. FB and BB have been recently put in a unifying algorithmic context in =-=[19]-=-, using the notions of forward and backward differential equivalences for a low-level syntax describing a more general class of nonlinear ODE systems. A symbolic partition-refinement algorithm to comp... |

2 |
Differential bisimulation for a Markovian process algebra
- Iacobelli, Tribastone, et al.
- 2015
(Show Context)
Citation Context ... ODE reduction algorithms are available for domain specific languages such as rule-based models of biochemical networks [1] and Markovian process algebra using FB-like (though not BB-like) conditions =-=[20]-=-. 2 Background Reaction Networks. An RN (S,R) is a pair of a finite set of species S and a finite set of reactions R. A reaction is a triple written in the form ρ α−→ pi, where ρ and pi are multisets ... |