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ACM41000:  Uncertainty Quantication

Assignment 3

1. Why do zebras have stripes and cheetahs have spots? A Turing pattern is a concept introduced by English mathematician Alan Turing in a 1952 that describes how patterns that emerge in nature, such as stripes and spots, may arise naturally and autonomously from a homogeneous, uniform state. It is often described in coupled models of reacting and diﬀusing chemicals. Consider such a model below for the dynamics of Ⅹl (t．s) and Ⅹ2 (t．s)

= Dl   + 1 + l(2)2  - 4l

θs   = D2   θt2    + 3l  - l(2)2

(a)  Solve the coupled system numerically in R using the ReacTran package.

Solve in t e [0．1] and s e [0．10]. Take the diﬀusion coeﬃcients to be

Dl  = D2  = 0．02．

boundary conditions to be

Ⅹl (0．s) = Ⅹl (1．s) = 1 and initial conditions

Ⅹl (t．0) = 1 + sin(2πt)

2 (0s) = 2 (1s) = 3

Ⅹ2 (t．0) = 3

The initial setup is provided to you in an accompanying ﬁle on Brightspace.

(b) Derive the sensitivity equations for Ⅹl  and Ⅹ2  with respect to the diﬀusivities Dl  and D2 :

5ll  =

θl

θDl

52l  =

θ2

θDl

5l2  =

θl

θD2

522  =

θ2

θD2

(c)  Solve the sensitivity equations taking initial conditions of all 5ij  to be zero and boundary conditions on 5ij  also to be zero.

(d) Brieﬂy summarise which variables are most sensitive to which param- eters.

2. Based on the research article,

Goulet, Dave.  “Modeling, simulating, and parameter ﬁtting of biochemical  kinetic  experiments.”    SIAM  Review  58.2  (2016): 331-353

you will recreate calculations using methods learned during the module. Complete the following tasks:

(a)  Give a brief introduction to the model described in (3.10), describing

solution variables, model parameters, and initial conditions.                (b) Write a program in R to solve the model equations (3.10) numerically.

Note that the initial conditions are part of the right-hand side of the ODEs requiring some addional programming thought on your end. Use k  = 200 and k−  = 2 as parameter values.

The model equations form a  sti system,  meaning  instead of calling ode() to solve the system of dierential equations, you must call lsode(). Otherwise your machine may freeze during computations!!!

(c) Fit the data in Table 2 to the solution of equations (3.10a-c) to ﬁnd the best ﬁt k  and k−  according to a Levenberg–Marquardt least-squares error algorithm.  Note a simpler direct version of the sum of square errors is appropriate here:

(xout-xdata)2  + (yout-ydata)2  + (zout-zdata)2

where you dene the above based on your program. In order to solve

the system for two parameters  (k．k ), take the time interval s  e [1/30．20] and use the provided data at s = 1/30 for initial conditions.

(d) Brieﬂy discuss your implementation and results.

(e) Derive the sensitivity equations for the model (3.10), based on param- eters k  and k, using the procedure shown in the lectures. Compare with (B.1a-1f) in the article. Use either your derived sensitivity equa- tions, and/or (B.1a-1f) quoted in the article, to discuss and justify any diﬀerence in the best-ﬁt parameters you obtain from that quoted in

the article for the Table 2 dataset.

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