5. Prohorov Metric Framework
Theory
1. Since Ω = Ω 𝑫×Ω 𝝆 is a compact set, 𝑃 Ω is a
compact metric space
2. The minimizer is continuous in 𝑃
⟹ There exists a (not necessarily unique) minimizer
Theorem
There exists a (not necessarily unique) minimizer 𝑃J (Banks, Hu, Thompson, 2015)
𝑃J = argmin
M∈MN(I)
O data 𝑡P, 𝑥Q − R 𝑢 𝑡P, 𝑥Q, 𝑫, 𝝆 𝑑𝑃(𝑫, 𝝆)
I
S
Q,P
14. Performing the Inverse Problem:
Spline Functions
Assume 𝑀 nodes equispaced over Ω 𝝆 such that
𝝆i
= 𝑠](𝝆), 𝑘 = 1, … , 𝑀 , where 𝑠] are hat
functions
where 𝑝] = 𝑎] 𝑠] 𝝆 ≥ 0 represent a probability
density function. Thus, we require
𝑃J = argmin
MN(I)
O data 𝑡P, 𝑥Q − O 𝑎] R 𝑢 𝑡P, 𝑥Q, 𝐷, 𝝆 𝑠] 𝝆 𝑑𝝆
I 𝝆
i
]ab
S
Q,P
O 𝑎] R 𝑠] 𝝆 𝑑𝝆
I 𝝆
i
]ab
= 1
20. How to choose M: the optimal number
of nodes?
10 20 30 40 50
−2.5
−2
−1.5
−1
−0.5x 10
5
Number of Nodes
AICScore
Spline
twopoint
point
normal
lognormal
bigaussian
10 20 30 40 50
−2.5
−2
−1.5
−1
−0.5x 10
5
Number of Nodes
AICScore
Discrete
22. Performing the Inverse Problem:
Delta Functions
Assume there are 𝑀k nodes equispaced over Ω 𝝆 such
that 𝝆i
= Δkl
, 𝑘 = 1, … , 𝑀k and that there are 𝑀z nodes
equispaced over Ω 𝑫 such that 𝑫i
= Δz{
, 𝑙 = 1, … , 𝑀z
where 𝜔], 𝜔} ≥ 0 represent a discrete probability
density function. Thus, we require
𝑃J = argmin
MN(I)
O data 𝑡P, 𝑥Q − O O 𝑢 𝑡P, 𝑥Q, 𝐷} , 𝜌] 𝜔] 𝜔}
i~
]ab
i•
}ab
S
Q,P
O 𝜔]
i~
]ab
= 1 O 𝜔}
i•
}ab
= 1
23. Performing the Inverse Problem:
Spline Functions
Assume 𝑀knodes equispaced over Ω 𝝆 such that 𝝆i~ =
𝑠](𝝆), 𝑘 = 1, … , 𝑀k , and 𝑀znodes equispaced over
Ω 𝑫 such that 𝑫i• = 𝑠}(𝑫), 𝑙 = 1, … , 𝑀z , where
𝑠], 𝑠} are hat functions
where 𝑝] = 𝑏] 𝑠] 𝝆 ≥ 0 , 𝑝} = 𝑎} 𝑠} 𝑫 ≥ 0
represent probability density functions.
𝑃J = argmin
MN(I)
O data 𝑡P, 𝑥Q − O 𝑎} R O 𝑏] R 𝑢 𝑡P, 𝑥Q, 𝑫, 𝝆 𝑠} 𝑫 𝑑𝑫
I 𝝆
i~
]ab
𝑠] 𝝆 𝑑𝝆
I 𝑫
i•
}ab
S
Q,P
O 𝑏] R 𝑠] 𝝆 𝑑𝝆
I 𝝆
i~
]ab
= 1 O 𝑎} R 𝑠} 𝑫 𝑑𝑫
I 𝑫
i•
}ab
= 1
30. References
1. Banks, H. T., et al. "Experimental design and estimation of growth
rate distributions in size-structured shrimp populations." Inverse
Problems 25.9 (2009): 095003.
2. Banks, H. T., and W. Clayton Thompson. "Existence and consistency of
a nonparametric estimator of probability measures in the prohorov
metric framework." International Journal of Pure and Applied
Mathematics 103.4 (2015).
3. Erica M. Rutter, H. T. Banks, and Kevin B. Flores. Estimating
Intratumoral Heterogeneity from Spatiotemporal Data. Journal of
Mathematical Biology, 77(6-7):1999–2022, 2018 .
4. Banks, H. T., et al. "The Prohorov metric framework and aggregate
data inverse problems for random PDEs." Commun. Appl. Anal 22
(2018): 415-46.
5. Sirlanci, Melike, et al. "Estimating the distribution of random
parameters in a diffusion equation forward model for a transdermal
alcohol biosensor." Automatica 106 (2019): 101-109.