36
D. Parra-Guevara and Y.N. Skiba
2.4 Peculiarities of Dual Estimates and Sensitivity Formulas
We now discuss the main features of the dual estimate (2.25), or its simplification
(2.26), and show the usefulness of the adjoint estimates in the study of sensitivity
of mean concentration J i (φ) to variations in the discharge rates and positions of the
sources as well as in the initial distribution φ 0 (r ) of nutrient.
In environmental monitoring, the adjoint estimate (2.25) is a good complement
to the direct mean concentration estimate J i (φ). One can use either direct or adjoint
estimates depending on the specific situation. Assume, for example, that the mean
concentration J i (φ) of a nutrient is monitored in N ecologically important zones Ω i
of domain D (i = 1, . . . , N ). If the number of zones N is large enough then it is
better to solve problem (2.4)–(2.11) and use direct estimate of J i (φ) in each zone. On
the other hand, if number N is rather small then it is more effective and economical
to solve adjoint problem (2.18)–(2.24) and use adjoint estimate (2.25). Unlike the
direct mean concentration estimate of nutrient, the adjoint estimate (2.25) permits to
explicitly evaluate the contribution of each source to value J i (φ).
In the case of invariable emission rates (Q j (t) = Q j ), evaluation (2.26) becomes
even simpler:
J i (φ) =
N
j=1
Q j w i j ,
(2.33)
where
w i j =
T
0
g i (r j , t) dt.
(2.34)
Each weight w i j depends only on the adjoint solution and characterizes the contribution of the source with emission rate Q j to the mean concentration J i (φ) in
Ω i .
What is then the basic difference between the direct and adjoint estimates of the
mean concentration of nutrient J i (φ)? The direct estimate, relating to the solution
φ(r, t) of problem (2.4)–(2.11), is independent of a concrete zone Ω, but depends on
the discharge rates Q j and position r j of sources, and also on the initial distribution of
nutrient φ 0 (r ) in D. For this reason such a estimate is preferable if one needs to know
the concentration of a substance in many zones of D, or in each point of D × (0, T ).
However, in the model sensitivity study, this approach requires much computing time,
because the solution φ(r, t) of problem (2.4)–(2.11) must be recalculated whenever
new values of the parameters Q j , r j or φ 0 (r ) are used. Unlike it, the solutions of
adjoint problem g i (r j , t) depend on Ω i zone, but are independent of Q j , r j or φ 0 (r ).
In the adjoint evaluation (2.25), g i (r j , t) serves as the weight function characterizing
the model response to these three parameters. Since the problem is linear, Eq. (2.25)
leads to the main sensitivity formula
δ J i (φ) =
N
j=1
T
0
g i (r j , t)δ Q j (t) dt +
D
g i (r, 0)δφ
0
(r ) dr
(2.35)
D. Parra-Guevara and Y.N. Skiba
2.4 Peculiarities of Dual Estimates and Sensitivity Formulas
We now discuss the main features of the dual estimate (2.25), or its simplification
(2.26), and show the usefulness of the adjoint estimates in the study of sensitivity
of mean concentration J i (φ) to variations in the discharge rates and positions of the
sources as well as in the initial distribution φ 0 (r ) of nutrient.
In environmental monitoring, the adjoint estimate (2.25) is a good complement
to the direct mean concentration estimate J i (φ). One can use either direct or adjoint
estimates depending on the specific situation. Assume, for example, that the mean
concentration J i (φ) of a nutrient is monitored in N ecologically important zones Ω i
of domain D (i = 1, . . . , N ). If the number of zones N is large enough then it is
better to solve problem (2.4)–(2.11) and use direct estimate of J i (φ) in each zone. On
the other hand, if number N is rather small then it is more effective and economical
to solve adjoint problem (2.18)–(2.24) and use adjoint estimate (2.25). Unlike the
direct mean concentration estimate of nutrient, the adjoint estimate (2.25) permits to
explicitly evaluate the contribution of each source to value J i (φ).
In the case of invariable emission rates (Q j (t) = Q j ), evaluation (2.26) becomes
even simpler:
J i (φ) =
N
j=1
Q j w i j ,
(2.33)
where
w i j =
T
0
g i (r j , t) dt.
(2.34)
Each weight w i j depends only on the adjoint solution and characterizes the contribution of the source with emission rate Q j to the mean concentration J i (φ) in
Ω i .
What is then the basic difference between the direct and adjoint estimates of the
mean concentration of nutrient J i (φ)? The direct estimate, relating to the solution
φ(r, t) of problem (2.4)–(2.11), is independent of a concrete zone Ω, but depends on
the discharge rates Q j and position r j of sources, and also on the initial distribution of
nutrient φ 0 (r ) in D. For this reason such a estimate is preferable if one needs to know
the concentration of a substance in many zones of D, or in each point of D × (0, T ).
However, in the model sensitivity study, this approach requires much computing time,
because the solution φ(r, t) of problem (2.4)–(2.11) must be recalculated whenever
new values of the parameters Q j , r j or φ 0 (r ) are used. Unlike it, the solutions of
adjoint problem g i (r j , t) depend on Ω i zone, but are independent of Q j , r j or φ 0 (r ).
In the adjoint evaluation (2.25), g i (r j , t) serves as the weight function characterizing
the model response to these three parameters. Since the problem is linear, Eq. (2.25)
leads to the main sensitivity formula
δ J i (φ) =
N
j=1
T
0
g i (r j , t)δ Q j (t) dt +
D
g i (r, 0)δφ
0
(r ) dr
(2.35)