20.18 Optimization Problems In Air Pollution Modeling

20.18 Optimization Problems in Air Pollution Modeling
Ivan Dimov, and Zahari Zlatev
ABSTRACT Environmental problems are becoming more and more important for modern
society, and their importance will certainly be increased in the next century. High pollution
levels (high concentrations and/or depositions of certain chemical species) may cause damage
to plants, animals, and humans. Moreover, some ecosystems can also be damaged (or even
destroyed) when pollution levels are very high. This is why pollution levels must be carefully
studied in order to be able to predict the appearance of high pollution levels and/or to decide
what can be done to prevent exceeding prescribed critical levels. Mathematical models can suc-
cessfully be used to resolve these problems. Very often optimization problems have to be
applied. The appearance of optimization problems in the field of air pollution modeling and
their importance will be discussed in this paper. Some applications of adjoint equations in the
treatment of optimization problems arising in air pollution modeling will be considered. We
shall present a review of some approaches that are based on the adjoint equations formu-
lation.
1. Need for Optimal Environmental Solutions
The control of the pollution levels in different highly developed and densely populated regions
of Europe and North America is an important task for modern society. Its importance has been
steadily increasing during the last two decades. The need to establish reliable control strategies
for air pollution levels will become even more important in the next century. Large scale air
pollution models can successfully be used to design reliable control strategies.
The nature of the environmental problems is such that it is highly desirable (and some-
times necessary) to find optimal solutions. This fact can be illustrated by the following simple
example. It is very easy to suggest a reduction of all emissions by some large amount (say, by
50 percent everywhere in Europe). Most of the environmental problems would certainly be
resolved if such a decision was carried out in practice. However, there is a serious danger of
creating some new and even greater problems, because the reduction of the emissions is an
expensive process and could cause a great economic crisis in the society if unjustified large
reductions were carelessly enforced. This is why optimal solutions are needed in this situation.
One has (i) to justify the need for reductions and (ii) to find out where to reduce the emissions
and by how much to reduce them. These tasks can be solved successfully by developing and
using reliable mathematical models for studying different pollution phenomena (Zlatev, 1995;
McRae et al., 1984). These models must satisfy several important requirements (Zlatev et al.,
1996):
1. The mathematical models must be defined on large space domains, because the long
range transport of air pollution is an important environmental phenomenon and high
pollution levels are not limited to the areas where the high emission sources are located.
2. All relevant physical and chemical processes must be adequately described in the models
used.
3. Enormous files of input data (both meteorological data and emission data) are needed.
4. The output files are also very big, and fast visualization tools must be used in order to
represent the trends and tendencies, hidden behind many megabytes (or even many giga-
bytes) of digital information, so that even nonspecialists can easily understand them.
2. Mathematical Formulation of an Air Pollution Model
All important physical and chemical processes must be taken into account when an air pol-
lution model is to be developed. Systems of partial differential equations (PDEs) are often used
to mathematically describe an air pollution model. Consider a three-dimensional space domain
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1

O and assume that cðr; tÞ ðc1ðr; tÞ; c2ðr; tÞ; ... cqðr; tÞÞT
; where r ðx; y; zÞ 2 O: Then the PDE sys-
tems are of the following type:
dcðr; tÞ
dt
Aðr; tÞcðr; tÞ ¼ fðr; tÞ; ð1Þ
where
Aðr; tÞ ðA1ðr; tÞ; A2ðr; tÞ; ...; Aqðr; tÞÞT
; ð2Þ
Asðr; tÞcs ¼
@ðucsÞ
@x

@ðvcsÞ
@y

@ðwcsÞ
@z
þ
@
@x
Kx
@cs
@x

þ
@
@y
Ky
@cs
@y
!
þ
@
@z
Kz
@cs
@z

ðk1s þ k2sÞcs; ð3Þ
f ð f1; ...; fqÞT
; ð4Þ
and
fs ¼ Es þ Qsðc1; c2; ...; cqÞ; s ¼ 1; 2; ...; q: ð5Þ
The different quantities that are involved in the mathematical model have the following
meaning:
. the concentrations are denoted by cs;
. u, v, and w are wind velocities;
. Kx, Ky, and Kz are diffusion coefficients;
. the emission sources in the space domain are described by the functions Es;
. k1s and k2s are deposition coefficients;
. the chemical reactions are described by the nonlinear functions Qs(c1, c2, . . . , cq).
The nonlinear functions Qs, representing the chemical reactions in which the sth pollutant
is involved, are of the form:
Qsðc1; c2; ...; cqÞ ¼
X
q
i¼1
asici þ
X
q
i¼1
X
q
j¼1
bsijcicj; s ¼ 1; 2; ...; q:
This is a special kind of nonlinearity (it is seen that the chemical terms are described by
quadratic functions), but it is not clear if this property can efficiently be exploited. To the
authors’ knowledge, it is not exploited in the existing large-scale air pollution models.
The models defined by (1)–(5) are traditionally used to calculate some concentration
fields by using both meteorological and emission data as input (Zlatev et al., 1996). This gives
an answer to the question: what are the concentration levels and/or the deposition levels
caused by the existing emissions under the particular meteorological conditions that take place
in the time-period under consideration? However, it is much more important to study the
question: how can the concentrations be kept under certain critical levels? Different types of
optimization problems must be solved when this and similar questions are to be answered.
The formulation of some of the optimization problems arising in air pollution modeling will be
studied in the rest of this paper.
3. Data Assimilation in Air Pollution Models
Initial concentration fields are needed when the model defined by (1)–(5) is to be treated
numerically (and this is, of course, true for any other air pollution model). The problem of find-
ing good initial concentration fields is very difficult. Normally, one starts with some back-
ground concentrations and runs the model over a certain sufficiently large period (say, five
days) to generate initial concentration fields. It is expected that the initial fields found in this
way are good, but there is no guarantee that this is so. Moreover, this is an unnatural solution
in the case where an air pollution forecast for the next two–three days is to be produced
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20 APPLICATION AREAS
2

(because the time to start-up the model, five days, is much larger than the period of two–three
days that is needed for the forecast).
An alternative approach is based on the use of time series measurements to analyze the
initial data that is to be used in the air pollution model under consideration. The experiments
indicate the remarkable fact that time series of only a few key chemical species are sufficient to
produce initial fields (also for species that are not measured; by exploiting their coupling with
the measured species).
It must be emphasized here that data assimilation is a very powerful approach. Not only
can it be used to analyze the initial data (as sketched above), but also to obtain optimal esti-
mations of (i) emission fields, (ii) deposition rates, and (iii) other model parameters. The use of
data assimilation can be considered as applying a sequence of corrections performed at succes-
sive times in the treatment of the model. Each correction combines the appropriate background
field of results produced by the model with new observations.
Data assimilation has been used successfully in meteorology. The basic ideas can be
explained by sketching the application of data assimilation in this field. Before the development
of data assimilation techniques in meteorology, the most successful device used was optimal
analysis or optimal interpolation. Such a technique is presented in the works of McPherson
(1979), Lorenc (1981), and Hollingsworth et al. (1985). The advantage of optimal analysis is that
it provides a simple and internally consistent procedure for treating a large number of obser-
vations with different distributions, nature, and accuracy. However, optimal analysis, as it has
been used, only takes into account the dynamics of meteorological processes very indirectly. It
is because the temporal evolution of the statistical covariances of the predicted error is rep-
resented by a very simple low. This low is taken independently or almost independently of the
current state of the wind field which determines the real evolution of the predicted error. In
particular, it ignores all effects due to advection. In the paper by Talagrand and Courtier (1987)
a variational data assimilation technique is considered. This technique uses the adjoint
equations of the model. This approach is based on the Lions’ theory of optimal control (see
Lions, 1971). Generally speaking, the theory of optimal control deals with the problem of how
the output parameters of a given numerical model can be controlled by acting on the input
parameters of the model. Among different approaches to control theory the approach of the
adjoint equations is one of the most efficient. For meteorological problems this idea was first
suggested by Marchuk (1974) to compute the local gradient of a multi-variable function. The
computed gradient is then used for performing a descent step in the space of initial conditions
and the process is continued until some satisfactory approximation of the initial conditions
minimizing the distance function in the corresponding functional space has been fulfilled. This
specific application of adjoint equations was implemented in a model by Penenko and Obrazt-
sov (1976), but the feasibility and usefulness of this approach was demonstrated in the works
of Derber (1985) and Lewis and Derber (1985). In the works of Le Dimet and Talagrand (1986)
and Agoshkov (1994) one can find a rather general presentation of the use of the adjoint
equation technique in data assimilation analysis.
There are several different ways of implementing the data assimilation approach in the
field of air pollution modeling. Since the mid-eighties an approach based on the principle of
optimal control theory has become more and more popular. In this approach the initial values
are viewed as control parameters. Then a distance function is defined. This function provides
weighted and accumulated distances between the available measurements and the values of
the corresponding state variables calculated by the model during a predefined assimilation
window. An optimization procedure is applied to minimize the distance function. This requires
some knowledge of local gradients with respect to the initial state. To obtain these gradients
one has to use the adjoint equation of (1). The procedure is normally called a four-dimensional
variational data assimilation.
3.1. Introduction of the Distance Function
Consider a state variable x which is an element of a Hilbert space H. The evolution in time of
the state variable x can be expressed by the differential equation:
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20.18 OPTIMIZATION PROBLEMS IN AIR POLLUTION MODELING 3

dx
dt
¼ MðxÞ; ð6Þ
where M is a nonlinear operator. Denote the measurements by x̂ and let
OðxðtÞÞ ¼
def 1
2x̂
xðtÞ xðtÞÞO 1
ðx̂
xðtÞ xðtÞÞ;
where O is the covariance matrix of the observation and representativeness errors. By using
this notation and Lagrangian multipliers l(t), the distance function can be defined by
JðxðtÞÞ ¼
def
ZtN
t0
OðxðtÞÞ þ lðtÞ;
dxðtÞ
dt
MxðtÞ

dt; ð7Þ
where the inner product
R
O fðxÞgðxÞdx of two functions f and g in the Hilbert space H is
denoted by h f; gi: It should be mentioned here that the background concentrations can also be
used in the definition of the distance function by adding an extra term
1
2ðxb xðt0ÞÞT
B 1
ðxb xðt0ÞÞ
to the right-hand-side of (7).
3.2. Calculating the Gradient of the Distance Function
A perturbation equation (or a variational equation), corresponding to (6), can be defined by
dðdxÞ
dt
¼ M 0
ðdxÞ; ð8Þ
where dx is some small deviation of x and M 0
is the tangential linear operator of M. Denote by
g1 and g2 the inner products:
g1 ¼ dlðtÞ;
dxðtÞ
dt
MxðtÞ

;
g2 ¼
dlðtÞ
dt
;
dðdxðtÞÞ
dt
M 0
dxðtÞ

:
In this notation, the gradient (the variation) of the distance function introduced in the
previous paragraph can be defined as follows:
dJ ¼
def
ZtN
t0
½hdO; dxðtÞÞi þ g1 þ g2 dt:
After some modifications of this equation (the most important of which is the tradition-
ally used integration by parts), the application of the extremal principle dJ ¼ 0 results in the
adjoint equation:

dlðtÞ
dt
M 0
*lðtÞ ¼ O 1
½x̂
xðtÞ xðtÞ ;
where the adjoint operator M 0
* is defined by hy; M 0
zi ¼ hM 0
*y; zi:
In order to estimate the evolution in time of the given initial perturbation of the data at
time t0 we have to integrate equation (8):
dxðtnÞ ¼ Rðtn; t0Þdxðt0Þ; n ¼ 1; ...; N; ð9Þ
where R(tn, t0) is the resolvent operator of M 0
. In general, we have to use some numerical inte-
gration scheme in order to calculate the resolvent operator R(tn, t0). If this is done, then the
resolvent operator is expressed as
Rðtn; t0Þ ¼
Y
n 1
i¼0
Rðtiþ1; tiÞ; n ¼ 1; ...; N;
where Rðtiþ1; tiÞ is a sufficiently accurate numerical operator for stepwise integration.
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4

Let us now consider the operator S as the resolvent of M 0
*. From the duality principle it
follows that
hlðttN
Þ; RðtN; t0Þdxðt0Þi ¼ hSðt0; tNÞlðtNÞ; dxðt0Þi
and
rxðt0ÞJ ¼
X
N
m¼1
Y
m 1
i¼0
Sðti; tiþ1ÞrxOðtÞ:
It remains to be shown that rxðt0ÞJ ¼ lð0Þ: After defining the backward initial condition
lðtNÞ ¼ 0 and applying the duality principle, one can get

@
@t
Sðt; t 0
Þrxðt 0ÞOðt 0
Þ ¼
dlðtÞ
dt
¼ M 0
*Sðt; t 0
Þrxðt 0ÞOðt 0
Þ: ð10Þ
After the integration of (10) one can get
lðtÞ ¼
ZtN
t
Sðt; t 0
Þrxðt 0ÞOðt 0
Þdt 0
:
The computations concerning the calculation of the values of the distance function
J(x(t0)) and of its gradient dJ(x(t0)) can be carried out in two steps.
. Forward step. Some approximations of the state variable x and dx are calculated, from (6)
and (8), respectively, for t ¼ t1; t2; ...; tN: The values of x are used to calculate the values
of the distance function J(x(t0)) in the same points.
. Backward step. The adjoint equation (or some transformed form of this equation) is inte-
grated backward for t ¼ tN 1; tN 2; ...; t0 to obtain values of the perturbed state variable
l(t) starting with lðtNÞ ¼ 0: The values of l(t) and dx(t) are used to calculate values of
dJ.
The values obtained for J and dJ are needed in the minimization procedure which will
be outlined in the next section.
3.3. Minimizing the Distance Function and Beneficial Impact from the Assimilation of the
Selected Species
Different standard optimization algorithms can be applied to minimize the distance function.
The algorithm proposed by Broyden–Fletcher–Goldfarb–Shanno (BFGS) is often used in this
field in its limited memory version. It should be stressed, however, that nonnegative solutions
are required. Therefore, it is necessary to insert some constraints which ensure nonnegative sol-
utions. A procedure of this kind is proposed by Bertsekas (1982).
Let us describe briefly a practical minimization procedure of the distance function in
order to demonstrate to what extent measurements of a limited set of species can reduce the
manifold of possible initial perturbations to fit to the observations. Let the observations of
species 1; ...; p out of p 4 q be available at N time-steps. Denote by dpx̂
xðtnÞ the following vector:
dpx̂
xðtnÞ ¼ ðdx̂
x1ðtnÞ; ...; dx̂
xpðtnÞÞT
; n ¼ 1; ...; N; ð11Þ
where p is the number of the components of the vector and dx̂
xiðtnÞ ¼ x̂
xiðtnÞ xiðtnÞ; i ¼
1; 2; ...; p: The system (9) can be rewritten as
dpx̂
xðtnÞ ¼ ½I; 0 Rðtn; t0Þ; n ¼ 1; ...; N; ð12Þ
where ½I; 0 2 Rp q
is the truncation operator composed of the identity matrix I 2 Rp p
and the
zero matrix 0 2 Rp ðq pÞ
(for the case q pÞ: In system (12) the rows of unobserved species are
truncated. It is convenient to comprise the matrices of the resolvent operator and the truncation
operator in a matrix R(tn, t0)
dNpx̂
xðtÞ ¼ Rðtn; t0ÞdNpxðt0Þ; n ¼ 1; ...; N; ð13Þ
where Rðtn; t0Þ 2 RNp q
; dNpxðt0Þ 2 RNp
; and
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dNpx̂
xðtÞ ¼ ðfdpx̂
xðt1ÞgT
; ...; fdpx̂
xðtNÞgT
ÞT
2 RNp
(the row vectors dpx̂
xðtnÞ; n ¼ 1; ...; N are defined by (11)).
It is easy to see that for Np q the system (13) is underdetermined. In the case where
Np q the system (13) is overdetermined. In this case one can estimate the condition numbers
of the matrix R 0
ðtn; t0Þ 2 Rq q
; part of R(tn, t0), which is a square matrix of size q q; that is
cond(R 0
(tn, t0)) with any value of mhi ¼ q; where hi is any combination of measured species
and m is the number of the time-steps. The condition numbers cond(R 0
(tn, t0)) for different com-
binations give us an estimate of what beneficial impact may be expected from the assimilation
of the selected species i and observation times m. These condition numbers can provide some
evidence of reasonable weighting of the observations, which is an important task. For more
details see Elbern et al. (1997).
4. Keeping the Concentrations in a Given Sensitive Area Under a Prescribed Level
The problem of environmental protection of a given sensitive area is an optimization problem.
This problem can be formulated as follows. Let a sensitive point r1 2 O be given. Assume that
the concentrations at point r1 2 O in time t1 2 ½0; T should not exceed some critical (for point
r1) level C. This means that the requirement
cðr1; t1Þ 4 C ð14Þ
must be satisfied. Assume also that some source of a prescribed power E(t) is to be located
somewhere in O. The problem is to find a subdomain o of O where the source is located, so
that condition (14) is satisfied. This problem could also be considered as a problem of optimal
planning in an effort to achieve sustainable development.
It is convenient to simplify the problem in order to facilitate the understanding of the
main principles which can be used to find the fundamental solution of the adjoint problem.
This can be done as follows. Consider the solution of 1D–transport diffusion problem:
@
@t
A

c ¼
@c
@t

@
@x
ðaxcÞ D
@2
c
@x2
¼ EðtÞdðx x0Þ; ð15Þ
c ¼ c0ðxÞ; for t ¼ 0: ð16Þ
Assume that cðx; tÞ 2 W2
1; where W2
1 is a Sobolev space, cðx; tÞ 0; D 0; a ¼ const; 1 x
1; t 2 ½0; T ; and the conditions providing the existence of the unique solution of the problem
(15)–(16) in a weak formulation are fulfilled.
It is not possible to directly obtain an analytical solution to the problem defined by (15)–
(16), due to presence of the term EðtÞdðx x0Þ: In addition, standard numerical methods used to
solve the equation (15) are not efficient because there is a d-function in (15).
The condition cðx; tÞ 2 W2
1 implies that
RT
0 dt
R1
1cð2Þ
ðx; tÞdx 1; which is a natural con-
dition.
In fact, the operator used in (15) contains a rather artificial wind field defined by u ¼ ax:
Nevertheless, we demonstrate in this way how the approach under consideration can be used
for problems with a nonconstant advection.
The problem of finding the subdomain o is not so complicated if the solution of the
adjoint formulation of the original problem (15)–(16) is used. The conditions formulated here
will be relaxed in Section on Application to Other Problems.
4.1. Adjoint formulation
Multiply equation (15) by a function c*, which will be defined later (see the paragraph after
equation (25–26)). Integrate the resulting equation in time and space:
ZT
0
dt
Z1
1
c*
@c
@t

@
@x
ðaxcÞ D
@2
c
@x2
!
dx ¼
ZT
0
EðtÞdt
Z1
1
c*dðx x0Þdx: ð17Þ
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6

Assume that c ¼ c* ¼ 0 when x 4 1 or x 4 1: Apply integration part by part to (17).
The following results will be obtained in this way:
ZT
0
dt
Z1
1
c*
@c
@t
dx ¼
Z1
1
cc*dxjt¼T
t¼0
ZT
0
dt
Z1
1
c
@c*
@t
dx
¼
Z1
1
c0ðxÞc*
0ðxÞdx
ZT
0
dt
Z1
1
c
@c*
@t
dx; ð18Þ
ZT
0
dt
Z1
1
c*
@2
c
@x2
dx ¼
ZT
0
c*
@c
@x
c
@c*
@x

dxj1
x¼ 1 þ
ZT
0
dt
Z1
1
c
@2
c*
@x2
dx; ð19Þ
and
ZT
0
dt
Z1
1
c*
@ðxcÞ
@x
dx ¼
Z1
1
xcc*dxjt¼T
t¼0
ZT
0
dt
Z1
1
xc
@c*
@x
dx
¼
Z1
1
xc0ðxÞc*
0ðxÞdx
ZT
0
dt
Z1
1
xc
@c*
@x
dx: ð20Þ
From (18), (19), and (20), one can obtain:
ZT
0
dt
Z1
1
c
@c*
@t
þ ax
@c*
@x
D
@2
c*
@x2
!
dx
Z1
1
ð1 xÞc0ðxÞc*
0ðxÞdx ¼
ZT
0
EðtÞc*ðx0; tÞdt: ð21Þ
Assume that function c* satisfies the following adjoint equation:

@c*
@t
þ ax
@c*
@x*
D
@2
c
@x2
¼ cðx; tÞ;
with an initial condition c*ðx; TÞ ¼ 0; where the function c(x, t) will be defined later.
It is possible to rewrite (21) in the following form:
ZT
0
dt
Z1
1
cðx; tÞcðx; tÞdx ¼
ZT
0
EðtÞc*ðx0; tÞdt þ
Z1
1
ð1 xÞc0ðxÞc0
* ðxÞdx ¼ J ð22Þ
In fact, functions c(x, t) and c*(x, t), as well as functional J, depend on parameters x0, x1, and t1,
so that cðx; tÞ ¼ cðx; t; x0; x1; t1Þ; c*ðx; tÞ ¼ c*ðx; t; x0; x1; t1Þ; and J ¼ Jðx0; x1; t1Þ: When we do not
use these parametric dependences, they will be omited.
It is necessary to find an efficient numerical method to compute J. One can choose the
function c(x, t) in the form of a product of two d-functions: cðx; tÞ ¼ dðx x1Þdðt t1Þ:
Let us also assume that E(t) can be represented as
EðtÞ ¼ eðtÞHðtÞ; ð23Þ
where H(t) is a Heaviside function ðHðtÞ ¼ 1 for t 0 and HðtÞ ¼ 0 for t 4 0Þ: The second term
of the right-hand-side of (22) vanishes when such an assumption is made.
Let us now consider the following linear functional of the solution of the original prob-
lem (15)–(16):
J ¼
ZT
0
dt
Z1
1
cðx; tÞcðx; tÞdx: ð24Þ
Under the above assumptions, we have J ¼
RT
0 dt
R1
1cðx; tÞdðx x1Þdðt t1Þdx ¼ cðx1; t1Þ:
Obviously, the value of the functional (24) is equal to the value of the concentration in
point x ¼ x1 at moment in time t ¼ t1: From the representation (22) it follows that instead of
solving the original problem (15)–(16) one can solve the adjoint problem for c*:

@c*
@t
þ ax
@c*
@x
D
@2
c*
@x2
¼ dðx x1Þdðt t1Þ ð25Þ
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c* ¼ 0; for t ¼ T: ð26Þ
From the solution of the problem formulated by (25)–(26) one can estimate the functional
(22) as a function depending of a parameter x0 and solve the problem under consideration. In
fact, if the function c*(x, t, x0, x1, t1) is the solution of the adjoint problem (25)–(26), then the
presentation (22) defines a function of x0 Jðx0; x1; t1Þ: Now, the set o of points x0 can be deter-
mined by solving the inequality Jðx0; x1; t1Þ 4 C: That is the solution of the problem under con-
sideration.
4.2. Evaluating the functionals
One needs an efficient method for evaluating the linear functionals
Zti
0
eðtÞc*ðx0; tÞdt ð27Þ
of the solution to the adjoint problem (25)–(26) (e(t) is defined by (23)).
It is efficient to apply a statistical numerical method in the evaluation of (27). This is
because the complexity of the statistical numerical methods for computing a linear functional
of the solution is very low; of the same order as the complexity of the solution at only one
point of the domain (see Sobol, 1973; Dimov and Tonev, 1993). The statistical methods give
statistical estimates for the functional of the solution by performing samples of a certain ran-
dom variable, the mathematical expectation of which is the desired functional. These methods
have proved to be very efficient in solving multi-dimensional problems in composite domains
(Curtiss, 1954, 1956; Hammersley and Handscomb, 1964; Sobol, 1973; Ermakov et al., 1984).
Moreover, it has been shown that for some problems (including one-dimensional ones) the stat-
istical methods have a better convergence rate in the corresponding functional spaces than the
optimal deterministic methods in the same functional spaces (Bahvalov, 1964; Dimov and
Tonev, 1989, 1993b). It is also very important that the statistical methods are very efficient
when parallel or vector processors or computers are available, because the above mentioned
methods are inherently parallel and have loose dependencies. They are also well vectorizable.
By using powerful parallel computers, it is possible to apply a Monte Carlo method or a par-
ticle-tracking method for evaluating large-scale nonregular problems which are sometimes diffi-
cult to solve by traditional numerical methods.
The difficulties with finite difference and finite element methods are caused by the nonre-
gularity of the right-hand-side of equation (25) and by artificial diffusion (or diffusion caused
by the numerical scheme). This explains why the functional (27) can be evaluated by using a
statistical numerical method when A is a general linear operator. In our case, the fundamental
solution of the adjoint equation can be used.
In fact, introduce a new variable t* ¼ T t; where t* 2 ½0; T and present the adjoint prob-
lem in the following form:
@c*
@t*
þ ax
@c*
@x
D
@2
c*
@x2
¼ dðx x1ÞdðT t* t1Þ ð28Þ
c* ¼ 0; for t* ¼ 0; 8x:
The fundamental solution of problem (28) can be found by using the techniques dis-
cussed by Bitzadze (1982) and Tikhonov and Samarski (1977):
ĉ
c*ðx; tÞ ¼
1
½paðtÞ 1=2
exp
½x bðtÞ 2
aðtÞ
( )
; ð29Þ
where aðtÞ ¼ 2D
a ð1 e 2at
Þ and bðtÞ ¼ x0e at
:
The solution of the adjoint problem can be expressed in the following form:
c*ðx; t*Þ ¼
ZT
0
dt*
Z1
1
ĉ
c*ðx x; t* tÞdðx x1ÞdðT t t1Þdxdt: ð30Þ
The result of integration (30) is the function defined by (29) with new arguments:
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8

c*ðx x1; t1 tÞ ¼
1
½paðt1 tÞ 1=2
exp
½x x1 bðt1 tÞ 2
aðt1 tÞ
( )
: ð31Þ
Substituting (31) into the functional (27) one can get:
Zt1
0
eðtÞ
1
½paðt1 tÞ 1=2
exp
½x0 x1 bðt1 tÞ 2
aðt1 tÞ
( )
dt ¼ Jðx0; x1; t1Þ;
Under the assumptions formulated in this section the problem has been reduced to a
problem of numerical integration. We have to evaluate a 1-D integral in our particular case.
For this purpose we do not need to use a special statistical numerical method since the pro-
blem is simplified. The proposed numerical method is based on a numerical integration scheme
for evaluating the last parameterized functional. Using a Newton–Cotes quadrature one can
get
J ¼
X
k 1
i¼0
ei
1
½paðt1 tiÞ 1=2
exp
½x0 x1 bðt1 tiÞ 2
aðt1 tiÞ
( )
ðtiþ1 tiÞ þ OðDtÞ; ð32Þ
where Dt ¼ maxiðtiþ1 tiÞ is the maximal time-step and ei ¼ eðtiÞ: The numerical scheme (32) is
of the order O(Dt). It is possible to apply more complicated schemes of high order accuracy
(say, O((Dt)2
), or O((Dt)4
)). But from computational point of view, when the integrand is a suf-
ficiently smooth function, which corresponds to the case under consideration, it is better to use
an adaptive numerical scheme. In our practical computations an adaptive numerical scheme is
used. The integration starts with a regular coarse grid with a low number k of grid points.
After evaluating the first coarse approximation to the functional (32), the total and local a pos-
teriori variances are estimated. The information obtained is used for refinement of the mesh—
in the subregions in which the local variance is too large the mesh is refined, such that after
each step of refinement the number of nodes (mesh-points) increases two times. The refinement
procedure is continued until the required accuracy of the integration is reached or until the
maximum number of evaluations is executed.
4.3. Application to Other Problems
The approach studied can be easily applied to transport problems with another linear operator
A. One can for example include deposition terms kc in the operator considered in the previous
subsections:
@
@t
A

c ¼
@c
@t
u
@c
@x
þ kc D
@2
c
@x2
¼ EðtÞdðx x0Þ; ð33Þ
c ¼ c0ðxÞ; for t ¼ 0; c ¼ 0; for x 4 1; and x 4 1:
Using the technique of the adjoint equations described above one can reconstruct the
location and power of a steady state point source from two observations c(x1, t1) and c(x2, t2).
A possible application would be the identification and characterization of a pollutant source in
an inaccessible area. This formulation leads to a well conditioned problem and can be general-
ized to a large class of more complicated cases (see Dimov et al., 1996).
Here we consider the following
Problem. Find the place of location x0 and the power e of the source EðtÞ ¼ E ¼ eHðtÞ
ðHðtÞ is a Heaviside function) if two measurement data cðx1; t1Þ and cðx2; t2Þ are available
(it is assumed, that x1$x2Þ:
Let the power of source e, as well as its location x0 be unknown. Two measurement data
points—c(x1, t1) and c(x2, t2) are assumed to be available. One can then define two functions of
x0, F1(x0) and F2(x0), using the following formal presentations:
Fiðx0Þ ¼
Ĵ
Jðx0; xi; tiÞ
cðxi; tiÞ
; i ¼ 1; 2;
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where
Ĵ
Jðx0; xi; tiÞ ¼
Jðx0; xi; tiÞ
e
; i ¼ 1; 2:
(It is possible to do this because e is strongly positive.)
One can see, that x0 is the solution of equation
F1ðxÞ ¼ F2ðxÞ;
being the point of the location of the source of pollutants. The value of the power of the source
e can be determined using the expression:
e ¼
1
Fiðx0Þ
:
The problem consists of using an efficient numerical method for evaluating the functional
Zti
0
ec*ðx0; tÞdt:
Such a method for finding the place of location and the power of the sources for problem
(33) is formulated and experimented with in Dimov et al. (1996, 1997). Computational results
are given in Dimov et al. (1996) for different values of D/u, where D is the diffusion coefficient
and u is the wind velocity (see (33)).
5. Optimization Problems in the Theory of Measurements and Planning
the Experiments
Suppose that n functionals Jpi
; i ¼ 1; ...; n; are available. For every functional Jpi
we consider the
corresponding weight function upi
*; i ¼ 1; 2; ...; n for a nonperturbed problem. (It should be
emphasized that in this section the functionals Jpi
are measurements, u and v are weight func-
tions (not wind velocities), and pi are also functions.) We consider a model in which the oper-
ator L, as well as the domain of definition are supposed to be known. Consider the following n
problems:
L*upi
* ¼ pi; i ¼ 1; 2; ...; n:
Let us find n weight functions upi
*; i ¼ 1; 2; ...; n and solve the following problem:
Lu ¼ E: ð34Þ
The nonperturbed operator L is adjoint to the operator L*. In our consideration, we assume
that u 2 U and u* 2 U*; where U and U* are the domains of definition of the operators L and
L*, respectively.
The goal is to reconstruct the coefficients a
0
k and b
0
k in the expression, by which the per-
turbed operator L 0
is defined:
L 0
¼
X
m
k¼1
fa
0
kAk þ Bkðb
0
kCkÞg; ð35Þ
where Ak, Bk, and Ck are simple linear operators (for example, operators of differentiation, inte-
gration, and so on), a
0
kðxÞ and b
0
kðxÞ are perturbation coefficients.
Use the theory of small perturbations to construct n formulas:
ðupi
*; dLuÞ ¼ dJpi
; i ¼ 1; 2; ...; n; ð36Þ
where dL is the difference between the perturbed operator L 0
and the nonperturbed operator L.
Assume that the operator L can be presented as
L ¼
X
m
k¼1
fakAk þ BkðbkCkÞg; ð37Þ
0 HAO HAO071—28/2/2001
10

where ak(x) and bk(x) are unknown coefficients which are usually approximately known for
nonperturbed (model) problem.
Using expressions (37) and (35) one can get
dL ¼
X
m
k¼1
fdakAk þ BkðdbkCkÞg; ð38Þ
where
dak ¼ a
0
k ak and dbk ¼ b
0
k bk:
Substituting (38) into (36) one can get the following system of equations:
X
m
k¼1
½ðupi
*; dakAkuÞ þ ðB k
* upi
*; dbkCkuÞ ¼ dJpi
; ð39Þ
i ¼ 1; 2; ...; n:
Now the problem consists of parameterization of the variances dak and dbk. It has been
shown (Marchuk, 1980) that if dbk ¼ 0 and dak are constants, then (39) is reduced to the linear
algebra problem
Ly ¼ F; ð40Þ
where L is a matrix with elements lik ¼ ðupi
*; AkuÞ; F is a vector with components dJpi
and y is
a vector with components dak.
If the number of functionals (measurement data) n is equal to the number of varied coef-
ficients ak, then the system allows us, in principle, to determine dak. If n is greater than m the
solution could be found by minimizing the following quadratic functional:
kLy Fk2
2 ¼ min; ð41Þ
where k · k2
2 is the Euclidean norm.
This optimization problem can be efficiently solved using a gradient optimization pro-
cedure like the procedure described in Section 3.2.
This approach could be extended for the case when dak(x) and dbk(x) are functions. In
this case, the solution of the inverse problem could be found using the parameterization
approach. In this approach we suppose that as a result of the statistical and correlation anal-
ysis, orthogonal systems of functions ukl(x) and vkl(x) could be found. Using such functions
one can obtain good approximations to functions ak and bk for a sufficiently small n(k), (n
depends on k), so that
dakðxÞ ¼
X
nðkÞ
l¼1
ak;luk;lðxÞ; ð42Þ
dbkðxÞ ¼
X
nðkÞ
l¼1
bk;lvk;lðxÞ; ð43Þ
where ak,l and bk,l are unknown coefficients. Substituting (42) into (39) one can get
X
m
k¼1
X
nðkÞ
l¼1
½ak;l½upi
*; uk;lAkuÞ þ bk;lðB k
*upi
*; vk;lCkuÞ ¼ dJpi
ð44Þ
i ¼ 1; 2; ...; n:
It is easy to see that system (44) is equivalent to system (40). Then one could again con-
sider the minimization problem (41).
Consider the problem of planning a complicated experiment in environmental science.
Let us formulate the following
0 HAO HAO071—28/2/2001

Problem. Consider a set of all possible (practically realizable) experiments and choose the
subset of it which is the best, from the point of view of solving the given inverse problem
of reconstruction of the parameters of the environment (that is coefficients of the
equations).
Such a problem is very difficult and does not have a solution in the general case. But
there are some special approaches to this problem. Suppose that one considers the model of
nonperturbed problem (34) before doing the experiment and that one describes linear func-
tionals of the solution. One can find the necessary accuracy for measuring the functionals by
taking into account the information about measurement accuracy. Assume that some pre-
scribed requirements to the accuracy of the measurements are fulfilled, i.e. kdJpi
k e; where
e 0 is a given small value. Then one could consider different sets of experiments and choose
the set which leads to the matrix L defined by (40) with the best condition number (i.e. the L
with the smallest kLk2kL{
k2; where kLk2 is any Euclidean norm of the matrices L and L{
is
the pseudo-inverse of the matrix L defined by L{
¼ ðLT
LÞ 1
LT
Þ: In fact, by all this we are
assuming that the system of linear algebraic equations can be solved and the solution found is
sufficiently accurate. Such a plan for the experiments (the subset of the experiments leading to
matrix L with the smallest condition number) can be considered as optimal among the set of
subsets considered.
More details about this approach, as well as some applications can be found in the
works of Marchuk (1994) and Marchuk and Agoshkov (1993).
6. Optimization of Emissions from Enterprises by Using Some Economical Criteria
Here we consider the problem of determining the quantity of enterprise emissions, such that
the concentrations of any pollutant of a given region will be less than or equal to the pollution
level which ensures that the ecosystem will not be destroyed. Another problem which is very
close to this one is the problem of optimization of the location of new enterprises so that the
pollution of a given region will not exceed prescribed critical levels which are damaging for
plants, animals, and humans. We are trying to determine emission levels which do not damage
the ecosystem, but are not too low, in order to achieve the maximum economical efficiency for
the given restrictions. It is important to consider such kinds of optimization problems
especially for Central and East European countries, which must quickly achieve sustainable
development of their industry and agriculture.
Let us consider the region O and its boundary @O. Let n enterprises (or sources of pollu-
tants) Ai be located in the points ri, i ¼ 1; 2; ...; n: Assume that all sources can be considered as
point sources Ei ði ¼ 1; 2; ...; nÞ: Introduce m ecological domains ok, k ¼ 1; 2; ...; m: Suppose that
the limits of the pollution levels for the integrated interval of time [0, T] are defined for every
ecological domain.
In this case the model of air pollution transport for a fixed sth pollutant (1) can be pre-
sented in the following form
dcðr; tÞ
dt
Aðr; tÞcðr; tÞ ¼
X
n
i¼1
Eidðr riÞ: ð45Þ
Consider the functional induced by the domain ok.
Ysk ¼
ZT
0
dt
Z
ok
pccsdo; k ¼ 1; ...; m: ð46Þ
Thus, functional (46) describes the admissible pollution levels in the ecodomain ok. The
problem is to find the set of planned emissions Ei ensuring
Ysk 4 ðCsÞk; k ¼ 1; 2; ...; m; ð47Þ
for minimal economical losses for technical reconstruction of the enterprises, where (Cs)k is the
critical level of the considered sth pollutant for the domain ok.
0 HAO HAO071—28/2/2001
12

It is clear that in this case we have to consider the minimizing functional
I ¼
Zn
i¼1
xiðÊ
Ei EiÞ; ð48Þ
together with constraints (47), where Êi and Ei are the initial and planning power of the emis-
sions, xi describes the economical investments in technologies needed to achieve the same level
of production with the reduced emissions (calculated per unit of emission power). Then the
functional (48) describes the losses needed to improve the technologies of all Ai enterprises in
order to switch from emission powers Êi to the planned emission powers Ei. This means that
we can formulate the following minimization problem: find the power of the sources Ei in
model (45) such that
I ¼
Zn
i¼1
xiðÊ
Ei EiÞ ¼ min; ð49Þ
Ysk 4 ðCsÞk; k ¼ 1; 2; ...; m: ð50Þ
The problem (45), (49), (50) can be considered as a linear programming problem (Gill et al.,
1982). There are two approaches to solving this problem: the first one uses the original
equations, the second one is based on the theory of the adjoint equations. The approach which
is based on the theory of the adjoint equations can be derived by using a technique which is
similar to that presented in Section on Keeping the Concentrations in a Given Sensitive Area
under a Prescribed Level.
ACKNOWLEDGMENTS
This study was partially supported by grants from NATO (OUTR. CGR.960312), the European
Union (WEPTEL Esprit 22727 and EUROAIR Esprit 24618), and the Bulgarian Ministry of Edu-
cation, Science and Technology (I-811/98). A grant from the Danish Natural Sciences Research
Council gave us access to all Danish supercomputers.
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