## Deterministic global optimization for parameter estimation of dynamic systems (2006)

Venue: | Industrial and Engineering Chemistry Research |

Citations: | 6 - 4 self |

### BibTeX

@ARTICLE{Lin06deterministicglobal,

author = {Youdong Lin and Mark A. Stadtherr},

title = {Deterministic global optimization for parameter estimation of dynamic systems},

journal = {Industrial and Engineering Chemistry Research},

year = {2006},

pages = {8438--9448}

}

### OpenURL

### Abstract

A method is presented for deterministic global optimization in the estimation of parameters in models of dynamic systems. The method can be implemented as an ɛ-global algorithm, or, by use of the interval-Newton method, as an exact algorithm. In the latter case, the method provides a mathematically guaranteed and computationally validated global optimum in the goodness of fit function. A key feature of the method is the use of a new validated solver for parametric ODEs, which is used to produce guaranteed bounds on the solutions of dynamic systems with intervalvalued parameters, as well as on the first- and second-order sensitivities of the state variables with respect to the parameters. The computational efficiency of the method is demonstrated using several benchmark problems.

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Citation Context ...Rf ± Rg). (4) 7Thus a Taylor model of f ± g is given by Tf±g = (pf±g, Rf±g) = (pf ± pg, Rf ± Rg). (5) For the the product f × g, f × g ∈ (pf, Rf ) × (pg, Rg) ⊆ pf × pg + pf × Rg + pg × Rf + Rf × Rg. =-=(6)-=- Note that pf × pg is a polynomial of order 2q. In order to be consistent with the q-th order polynomial in a Taylor model, this term is split into the sum of a polynomial pf×g of up to q-th order, an... |

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Citation Context ...ed by Esposito and Floudas, 2 is based on the Lotka-Volterra predator-prey model from theoretical ecology. The system is described by two differential equations, ˙z1 = θ1z1(1 − z2) ˙z2 = θ2z2(z1 − 1) =-=(29)-=- z0 = [1.2, 1.1] T t ∈ [0, 10], where z1 represents the population of the prey and z2 the population of the predator. The measurement data was generated using the parameter values, θ = [3, 1] T , with... |

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Citation Context ...re less than ω, then no CPP will be applied on Xi. For the equality constraint c(x) = 0, the procedure is similar to but simpler than for the inequality case. If |ai| ≥ ω, eq 12 becomes aiU 2 i = Vi. =-=(16)-=- 11Then, with Wi = Vi/ai, it follows that ⎧ ∅ if Wi < 0 ⎪⎨ [ Ui = − √ Wi, √ ] Wi if Wi ≤ 0 ≤ Wi . (17) ⎪⎩ − √ Wi ∪ √ Wi if Wi > 0 ( Thus, the part of Xi retained is Xi = Xi ∩ Ui + xi0 − bi ) . If |ai... |

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Citation Context ... states (M ⊆ Z). In standard 3error-in-variables form, the parameter estimation problem for a dynamic model can be written as min φ = θ,zµ ∑ m∈M r∑ µ=1 (zµ,m − ¯zµ,m) 2 s.t. ˙zj = gj(z, θ, t); j ∈ J =-=(1)-=- zj(t0) = z0,j; j ∈ J 0 = h(z, θ, t) zµ,m = zm(tµ); m ∈ M; µ = 1, . . . , r t ∈ [t0, tf ]. Here θ is the vector of parameters (length p) which are to be estimated; z 0 = (z0,j; j ∈ J) is the vector (l... |

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Citation Context ...is similar to but simpler than for the inequality case. If |ai| ≥ ω, eq 12 becomes aiU 2 i = Vi. (16) 11Then, with Wi = Vi/ai, it follows that ⎧ ∅ if Wi < 0 ⎪⎨ [ Ui = − √ Wi, √ ] Wi if Wi ≤ 0 ≤ Wi . =-=(17)-=- ⎪⎩ − √ Wi ∪ √ Wi if Wi > 0 ( Thus, the part of Xi retained is Xi = Xi ∩ Ui + xi0 − bi ) . If |ai| < ω and |bi| ≥ ω, eq 14 becomes 2ai biUi = Vi, that is, Ui = Vi/bi. Thus, in this case, the part of X... |

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Citation Context ... respectively over the interval x ∈ X. For f ± g, f ± g ∈ Tf ± Tg = (pf, Rf ) ± (pg, Rg) = (pf ± pg, Rf ± Rg). (4) 7Thus a Taylor model of f ± g is given by Tf±g = (pf±g, Rf±g) = (pf ± pg, Rf ± Rg). =-=(5)-=- For the the product f × g, f × g ∈ (pf, Rf ) × (pg, Rg) ⊆ pf × pg + pf × Rg + pg × Rf + Rf × Rg. (6) Note that pf × pg is a polynomial of order 2q. In order to be consistent with the q-th order polyn... |

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Citation Context ...asic ideas of the method used. Additional details are given by Lin and Stadtherr. 13 Consider the following parametric ODE system, with state variables denoted by y: ˙y = f(y, θ), y(t0) = y 0, θ ∈ Θ, =-=(18)-=- where t ∈ [t0, tm] for some tm > t0. Note that a parameter interval Θ has been specified, and that it is desired to determine a validated enclosure of all possible solutions to this initial value pro... |

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Citation Context ...ng an interval Taylor series (ITS) with respect to time. That is, we determine hj and ˜Y j such that for Y j ⊆ ˜ Y 0 j, ∑k−1 ˜Y j = [0, hj] i F [i] (Y j, Θ) + [0, hj] k F [k] ( ˜ Y 0 j , Θ) ⊆ ˜ Y 0 . =-=(19)-=- i=0 Here k denotes the order of the Taylor expansion, and the coefficients F [i] are interval extensions of the Taylor coefficients f [i] of y(t) with respect to time, which can be obtained recursive... |

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Citation Context ... the first-order sensitivities zθi = ∂z/∂θi, i = 1, . . . , p, at each tµ, µ = 1, . . . , r. To do this, VSPODE is applied to integrate the first-order sensitivity equation, ˙zθi ∂g ∂g = zθi + ∂z ∂θi =-=(23)-=- zθi (t0) = 0, for each i = 1, . . . , p. Thus, the function range test is relatively expensive, and one must consider the tradeoff between the computational cost of the test and the reduction of Θ (k... |

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Citation Context ...sum φ determined only after integration to the final data time tr. Thus, after integration through the s-th data time ts (s < r), we have computed the partial sum φs = ∑ s∑ m∈M µ=1 (zµ,m − ¯zµ,m) 2 . =-=(22)-=- Since each term in the sum is positive, φs ≤ φs+1 ≤ φ, s = 1, . . . , r−1. Thus, for any s = 1, . . . , r−1, a lower bound on φs is a valid lower bound on φ, and the bounds improve (increase) as s in... |

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Citation Context ...erz. 18,20 Let Tf and Tg be the Taylor models (q-th order) of the functions f(x) and g(x) respectively over the interval x ∈ X. For f ± g, f ± g ∈ Tf ± Tg = (pf, Rf ) ± (pg, Rg) = (pf ± pg, Rf ± Rg). =-=(4)-=- 7Thus a Taylor model of f ± g is given by Tf±g = (pf±g, Rf±g) = (pf ± pg, Rf ± Rg). (5) For the the product f × g, f × g ∈ (pf, Rf ) × (pg, Rg) ⊆ pf × pg + pf × Rg + pg × Rf + Rf × Rg. (6) Note that... |

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Citation Context ...d by Tjoa and Biegler 34 and Esposito and Floudas. 2 The reaction system is A k1 ⇋ B k2 k3 ⇋ C. k4 26The differential equation model takes the form ˙z1 = −θ1z1 + θ2z2 ˙z2 = θ1z1 − (θ2 + θ3)z2 + θ4z3 =-=(27)-=- z3 = 1 − z1 − z2 z0 = [1, 0, 0] T t ∈ [0, 1], where the state vector, z, is defined as the concentration vector [A, B, C] T , and the parameter vector, θ, is defined as [k1, k2, k3, k4] T . In this p... |

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Citation Context ...ai > 0 and Vi < 0 √ Vi ai , [ ⎪⎩ −∞, − √ Vi ai ] [−∞, ∞] if ai < 0 and Vi ≥ 0 √ Vi ai ] [√ ] Vi ∪ , ∞ ai ( The part of Xi to be retained is then Xi = Xi ∩ Ui + xi0 − bi ) . 2ai if ai > 0 and Vi ≥ 0 . =-=(13)-=- if ai < 0 and Vi < 0 If |ai| < ω, then eq 9 cannot be used, but eq 8 can. Following a procedure similar to that used above, we now define Ui = Xi − xi0 and Vi = B(Tc) ⊖ bi(Xi − xi0). To identify boun... |

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Citation Context ...liminated in the objective range test or in the function range test, then the interval-Newton (IN) test is applied. The linear interval equation system F ′ (Θ (k) )(N (k) − ˜ θ (k) ) = −f( ˜ θ (k) ), =-=(24)-=- is solved for a new interval N (k) , where F ′ (Θ (k) ) is an interval extension of f ′ (θ), the Jacobian of f(θ) with respect to θ (i.e., F ′ (Θ (k) ) is an interval extension of the Hessian of φ(θ)... |

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Citation Context ...et x0 ∈ X. The Taylor theorem states that for each x ∈ X, there exists a ζ ∈ R with 0 < ζ < 1 such that f(x) = q∑ i=0 1 i! [(x − x0) · ▽] i 1 f (x0) + (q + 1)! [(x − x0) · ▽] q+1 f [x0 + (x − x0)ζ] , =-=(2)-=- where the partial differential operator [g · ▽] k is [g · ▽] k = ∑ j 1 +···+jm=k 0≤j 1 ,··· ,jm≤k k! j1! · · · jm! gj1 1 · · · gjm m ∂ k ∂x j1 1 · · · ∂xjm m . (3) The last (remainder) term in eq 2 c... |

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Citation Context ...closure Y j+1 = B(T y j+1 ). For the Taylor model computations, we begin by representing the parameters by the Taylor model T θ with components Tθi = (m(Θi) + (θi − m(Θi)), [0, 0]), i = 1, · · · , p, =-=(21)-=- 14where m(Θi) indicates the midpoint of the interval Θi. Then, we can determine Taylor models T f [i] of the interval Taylor series coefficients f [i] (y j, θ) by using RDA operations to compute T f... |

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Citation Context ...− x0) · ▽] q+1 f [x0 + (x − x0)ζ] , (2) where the partial differential operator [g · ▽] k is [g · ▽] k = ∑ j 1 +···+jm=k 0≤j 1 ,··· ,jm≤k k! j1! · · · jm! gj1 1 · · · gjm m ∂ k ∂x j1 1 · · · ∂xjm m . =-=(3)-=- The last (remainder) term in eq 2 can be quantitatively bounded over 0 < ζ < 1 using interval arithmetic or other methods to obtain an interval remainder bound. The Taylor model for f(x) then consist... |

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Citation Context ...a dependency problem. For |ai| ≥ ω, where ω is a small positive number, we can rearrange eq 8 such that each Xi occurs only once; that is, B(p) = m∑ [ i=1 ai ( Xi − xi0 + bi 2ai ) 2 − b2 i 4ai ] + Q. =-=(9)-=- In this way, the dependence problem in bounding the interval polynomial is alleviated so that a sharper bound can be obtained. If |ai| < ω, direction evaluation will be used instead. Taylor models fo... |

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7 |
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Citation Context ...0) 4ai where Si is determined by dependent subtraction (see section 3) using Si = B(Tc) ⊖ Now define the intervals Ui = Xi − xi0 + bi 2ai [ ai ( Xi − xi0 + bi 2ai and Vi = b2 i 4ai ) 2 − b2 i 4ai ] . =-=(11)-=- − Si, so that B(Tc) = aiU 2 i − Vi. The goal is to identify and retain only the part of Xi that contains values of xi for which it is possible 10to satisfy c(x) ≤ 0. Since B(Tc) bounds the range of ... |

6 | Advances in interval methods for deterministic global optimization in chemical engineering - Lin, Stadtherr |

5 |
Global Optimization with Nonlinear Ordinary Differential Equations
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Citation Context ...s the range of c(x) for x ∈ X, the constraint c(x) ≤ 0 will be satisfied if B(Tc) ≤ 0. Thus, to identify bounds on the part of Xi that satisfies the constraint, we can use the condition aiU 2 i ≤ Vi. =-=(12)-=- Then, the set Ui that satisfies eq 12, can be determined to be ⎧ [ ⎪⎨ − Ui = ∅ if ai > 0 and Vi < 0 √ Vi ai , [ ⎪⎩ −∞, − √ Vi ai ] [−∞, ∞] if ai < 0 and Vi ≥ 0 √ Vi ai ] [√ ] Vi ∪ , ∞ ai ( The part o... |

1 | Parameter estimation of Lotka-Volterra problem by direct search optimization - Luus - 1998 |