Modeling and solution of stochastic inverse problems in mathematical physics

Maul. Comput. Modelling Vol. 16, No. 5, pp. 37-51, 1992 Printed in Great Britain. All rights reserved

0895-7177192 $5.66 _t 0.60 Copyright@ 1992 Pergamon Press Ltd

MODELING AND SOLUTION OF STOCHASTIC INVERSE PROBLEMS

IN MATHEMATICAL PHYSICS

L. PREZIOSI, G. TEPPATI AND N. BELLOMO

Department of Mathematics

Politecnico, Torino, Italy

(Received July 1991 and in revised form October 1991)

Abstract-This paper deals with modelling aspects and solution techniques of inverse type problems for nonlinear mathematical models of continuum physics (and in general of applied sciences) defined by nonlinear partial differential equations with random coefficients. The paper deals with initial-

boundary value problems with random initial and/or boundary conditions with special attention to the analysis of nonlinearities and representation of the random fields generated by the solution of the

mathematical problem.

1. WHY STOCHASTIC INVERSE PROBLEMS IN MATHEMATICAL PHYSICS

A large class of models in mathematical physics and in applied sciences can be defined in terms of systems of partial differential equations. This generally occurs whenever the state variable, charged to describe the physical state of the system which is modelled by a suitable evolution equation, is a function of time and space. As already mentioned, this is the case of the equations of mathematical physics and of their developments to the modelling of real systems in natural sciences and technology.

The related mathematical problem is an initial-boundary value (direct) problem. In other words one needs the space distribution of the dependent variable in its domain of definition at a suitable initial time, the time evolution of the said variable at suitable boundary points of the space domain and the knowledge of all the physical parameters present in the modelling equations. Starting from this type of information, one can approach the problem of the qualitative and quantitative analysis of the initial-boundary value problem.

To be more specific, consider the nonlinear scalar initial boundary value problem in one space dimension for an abstract mathematical model structured in terms of partial (nonlinear) differential equations.

Let

u = u(t, 2) : [O, T] x [O, l] -+ R (1-I)

be the state variable defined in the time interval Dt = [0, T] and in the space domain D, = [0, 11. Moreover, let

2 = f(&x,u, 2, g; r(t, xl> (1.2)

be the evolution Equation for U, where r(t, x) is a suitable set of “known” functions of time and space. The formal structure of Equation (1.2), even if not the most general one, can include a very large class of nonlinear models including the one developed in the application.

Equation (1.2) is joined, in the formulation of the initial-boundary value problem, to the initial conditions

UlJ(2) = u(t = 0, x), vx E [O, 11 (1.3)

Partially supported by the Minister for Scientific Research and University, MURST and by the National Council for the Research, CNR.

Typeset by .4,&-w

37

38 L. hlEZlOSl et a/.

and to the boundary conditions

o(t) = u(t, I = O), p(t) = u(t, 2 = l), vt E [O,T]. (1.4

Such a problem is a direct, “well-formulated” one. The literature on this matter is classical and vast [1,2,3]. 0 ne has the proper starting point to deal with the related qualitative analysis, hopefully proving that the problem is also “well-posed”. Afterwards, a quantitative analysis can be developed in order to obtain solutions to Problem (1.2)-(1.4) on the basis of the methods of nonlinear analysis.

It is reasonable to assume, in a large variety of cases, that Equation (1.2) is nonlinear (we regard linearity as a simplification of physical reality) and that Problem (1.2)-(1.4) is of the deterministic type. In several cases linearity and nonlinearity in the model can be separated, the model is then of the semilinear type

g =a(& x) + qt, z) 21 + c(t, z) 2 + 4t, x) g + g(t, 2, u, 2; r(t, z)).

(1.2’)

This specialization of Equation (1.2) will sometimes be dealt with in what follows and, in particular, in the application,

One has to keep in mind that inverse-type problems take an important role in applied sciences. In fact, the problem itself can be such that important features of the problem, say initial and/or boundary conditions or the parameters characterizing the model are not completely known. On the other hand suitable information can be available on the solution of the problem. Therefore, one has to use “incomplete” information on the statement of the problem or on the mathematical model and the “incomplete” information on the solution in order to solve in a “complete” fashion the initial-boundary value problem.

The analysis of inverse problems can often provide useful information on the validity of the model or on important features of the model itself, e.g., the identification of parameters characterizing the model.

An important aspect to be taken into account in the modelling and analysis of inverse problems, is the fact that the information on the solution of the problem are often recovered in stochastic form, namely in terms of a superposition of deterministic functions (of time and/or space) with stochastic fluctuations defined over time and/or space.

The stochasticity is often implied by the fact that measurements refer to different physical situations whose behaviour is not directly described by the model (the model may hide some of the variables of the real system), therefore one has several trajectories corresponding to measurements of the same physical quantity. This feature has to be modelled in stochastic terms.

This paper deals with the mathematical modelling and with the related solution methods of inverse-type problems for nonlinear stochastic equations of mathematical physics and, more in general, of the applied sciences.

The stochastic modelling refers both to the modelling of stochastic processes (function of time) and to stochastic fields (function of space). The output, say U(W, t,z), will be, in general, a random time-dependent field defined in a complete probability space (Q, 3, P), where Q is the space of the elementary events w E R, 3 is a a-algebra and P the induced probability density.

The modelling can, in principle, undergo either real noises (either white or coloured) or rela- tively easier noises such that a sample continuous treatment, in the terms defined by Kampee de Feriet [4,5] can be used. This latter is the case of separable processes of the type

(1.5)

or

b(W, x) = c ~ZY(W)i7”(~). ”

(1.6)

Stochastic inverse problems 39

In this case the randomness, rrv and rzv , is separated by the time or space behaviour fy and gv , respectively.

The analysis developed throughout the paper refers to models involving stochasticities of the type (1.5) and (1.6). Certainly this type of modelling excludes several interesting cases, say models where real noises play an important role. On the other hand, the solution techniques for these problems are not yet sufficiently developed to be taken into account for the analysis of inverse, ill-posed, problems. Moreover, one has to accept the idea that in the mathematical modelling of systems, smooth “quasi-deterministic” noises can still be taken into account for a large variety of models.

2. INDEX OF THE PAPER

After the general speculation of the first section we can now examine the content of this review paper in more detail.

Besides the statement of the index, this second section deals with an overlook of the pertinent literature, the third section with a classification of some classes of inverse-type problems, the fourth section with the solution technique for problems in one-space dimension, the fifth section with the moment approximation of the probability densities linked to the dependent variable, the sixth section with the analysis of problems in twospace dimensions and finally the seventh section with an application.

The paper will provide a unified presentation and reorganization of some results already known in the literature and some new contributions to the analysis of the problems which have been listed above. The literature on inverse problem governed by partial differential equations is almost entirely restricted to the linear deterministic case [5-71.

Deterministic nonlinear inverse problems governed by partial differential equations have been dealt with in [8-91 by a collocation-interpolation technique introduced by Bellman and cowork- ers [lo] with the name of “differential quadrature method”. This technique has been developed

to the analysis of stochastic direct initial-boundary value problems in [ll-131. In particular, paper [ll] deals with the development and application of the method, [12] with the analysis of the convergence aspects and [13] with the analysis of the time evolution of the probability density and entropy functions joined to the dependent variable. The stochastic development of Bellman’s method has been called the SAI method.

The first steps towards the application of the SAI method to inverse problems are contained in papers [14,15]. Such a method is the natural development of the stochastic case of Bellman’s differential quadrature method. These papers will be reviewed in what follows.

3. CLASSIFICATION OF INVERSE-TYPE PROBLEMS

This section provides a conceivable classification of inverse-type stochastic problems related to the initial-boundary value problem (1.2)-( 1.4). This classification does not claim to be complete since this topic cannot be regarded yet as a completely well-structured one.

It will be assumed, in what follows, that all “known” inputs of the problem, either parameters and/or boundary and initial conditions are given in a suitable deterministic fashion. On the other hand, the information on the solution of the initial-boundary value problem will be assumed to be provided in terms of random fluctuations of time and/or space. These assumptions are, at least in part, consistent with the general framework proposed in the first section.

It is well understood that the solution of the problem will be a time-dependent random field.

For simplicity of notations we will refer, in this section, to scalar problems in one-space dimension. The generalization to systems of equations and to problems in several-space dimensions is, at least as far as the classification is concerned, a technical problem which is left to the reader. The classification which is proposed here consists in the mathematical formulation of four problems:

PROBLEM 3.1. (Unspecified initial conditions)

The boundary conditions o(t), P(t) are given, whereas the initial condition is unknown. The solution t~(~,t*, Z) is given Vx E [0, l] at a fixed time t* of the interval [0, T]. Then the problem

40 L. PREZIOSI et al.

consists in finding the initial conditions

u&,z) = t+,t = 0,x), t E [O, 11 (3.1)

and the solution of the initial-boundary value problem.

PROBLEM 3.2. (Underspecified boundary conditions) The initial conditions us(~) and one of the boundary conditions, say o(t), are given, whereas

,!?(t) is not specified. On the other hand the solution of the problem u(~,t, 2’) is given for t E [0, T] in a given point X* E (0,l). The problem consists in finding the unspecified boundary condition

o(w, t), t E P,Tl (3.2)

and the solution of the initial boundary value problem.

PROBLEM 3.3. (Problem with an unspecified source term) The initial and boundary conditions are given. However a source term s(w,~, II) in a given

point ~0 E (0,l) is not specified, while one knows the solution u*(~,t, z*) for ~1 # Z* and z* E (0,l). Then the problem consists in finding the source term

and the solution of the initial-boundary value problem.

PROBLEM 3.4. (Problem with unspecified parameters) The initial and boundary conditions are properly given. However a parameter (random field)

characterizing the system, say T = r(w, z), is not known and one knows the solution u(w, t = t*, Z) for fixed time t = t’ and z E [0, 11. Then the problem consists in finding the random field

P(W, z) : s-l x [O, l] + R (3.4)

and the solution of the initial-boundary value problem. As already mentioned the classification which has been provided above is not a complete one.

A broadening consists in dealing with more general boundary conditions: e.g., Newmann or mixed-type Dirichlet-Newmann boundary conditions can replace the twopoint conditions used in the classification. This generalization will not modify, as we shall see, the solution technique.

A second step is the generalization to systems in several space variables. However, such a generalization is technical only at a formal level, while, if quantitative results are desired, the analysis has to be handled with care as shown in Section 6.

Additional modifications will be indicated in the framework of the development of the paper. It is important to acknowledge, in all cases, that the proposed classification has to be regarded as originated by a conceivable mathematical modelling and, in particular, is based upon the assumption that all information on the solution of the initial-boundary value problem are given as random functions of time or space.

4. SOLUTION METHODS IN ONE SPACE DIMENSION

This section deals with the analysis of suitable mathematical methods for the solution of the

inverse-type problems, classified in Section 3, when the dependent variable is defined over time and one, only, space variable.

The general idea is to develop a suitable re-formulation of the problem in order to decompose it into a certain number of direct problems properly connected in a fashion equivalent to the overall inverse problem. In particular decomposition of domain methods will be used for the analysis of Problems 3.2 and 3.3.

A brief survey of the SAI method, already announced in the preceding sections and developed through papers [Il-131, is useful towards the understanding of the mathematical method for

inverse problems. The reader is also referred to a forthcoming book [16] for an overall presentation of such a method. The various steps of the application of the method can be summarized as follows:

41

STEP 1. The space interval [O,l] is partitioned into (n - 1) intervals by n nodes ti, i = 1, , . . , n, such

that 11 = 0 and zn = 1. The solution of the initial-boundary value problem is interpolated by

u(wt,x) = &)ui(w,t), Ui = U(W,t,X = Xi), (4.1) i=l

where Li denote Lagrange-type polynomials [17].

STEP 2.

The space derivatives $$ and $$, respectively, are directly derived from (4.1)

( ) 2 i

(WJ) = 2O’j Uj(W,t),

j=l

( > g i

(W,t) = e*ij Uj(Wpt),

j=l

with

(4.2a)

(4.2b)

(4.3)

STEP 3. Expressions (4.2)-(4.3) are substituted into into Equation (1.2) thus obtaining a system of

ordinary differential equations

U((W, t) = &O(W) + J ‘f&,s,u,a,b)ds,

0 (4.4

where u = {ui}, a = {aij }, b = {bij } and subscripts i or j denote, a~ usual, z = xi or x = xi.

The equations for i = 1 and i = n are replaced by the boundary conditions, i.e. ~1 = a(t),

ufl = P(t). Equation (4.4) is then solved by sample treatment in order to recover moments and correlations

on the variable u. Finally, interpolation (4.1) provides the space continuous behaviour of u(w, t, x), at each fixed w E R, once the ui(w, t) have been computed for i = 1,. . . , n.

REMARK 4.1. The SAI method cannot be regarded as a Galerkin-type stochastic approximation, rather it can

be classified as a collocation-interpolation method which reduces the solution of the mathematical problem (initial-boundary value problem) for a continuum system to the solution of an initial- value problem for a discrete system.

REMARK 4.2. The solution of the initial value problem (4.4) can be obtained on the basis of mathematical

methods, well-established in the pertinent literature [18-201. Such a problem may, in the nonlinear case, involve serious difficulties to be handled carefully after the pertinent qualitative analysis.

Having in mind the solution technique for direct-type problems, we can now deal with the solution methods for the various problems listed in Section 3.

4.1. Solution of Problem 9.1

Some aspects of this problem have been dealt with in [14] and its conclusions are here summarized. If u* = u(w, t*, x) is given, then Equation (4.4) can be interpreted, changing the sign of f ,(i.e. -f replaces f), as an initial value problem in the integration from t* to t = 0. In order to define u+ we need the following:

42 L. PREZlosI et a/.

DEFINITION 4.1. Consider the discretization

I, = {Xl = 0, x2,. . . ,X,-l, 2, = 1)

of the space interval [O,l], then u*(w,z) is regarded as a “known” random field if, for every discretization I,, the probability distribution of the variable

u* = {u(x1,t*), . . . ) u(xn,t*))

is known and is defined by the density

Pt*n = P(ul,...,un;t*).

The problem is then to compute the time evolution of the probability density linked to the probability distribution of the vector variable u.

Consider first the case when both are constant in time and assume that the map

uo = al*, (4.5)

defined by Equation (4.4) is a one-to-one map. Then, under suitable continuity and regularity properties, see for instance [13], one obtains (cf.[18, Ch. 6; 20, Ch. 3) the density Pon(uo;t = 0)

by the classical formula of change of variable

Po,(uo = @II’) = Jt(u0 --t u*)pon(u*), (4.6)

where 51 is the Jacobian of the inverse mapping of the initial value problem derived from (4.4) after the change of sign in f.

As known, Jt satisfies the initial value problem

dJt n-l afi x=JtCz.

i=2 ’ (4.7)

The generalization of this result to the case of time-dependent boundary conditions is not immediate. In line of principle one can recover, by the sample solutions of the initial value

problem, suitable moments of the variable u in order to obtain a polynomial simulation of the probability density function. This aspect is dealt with in the next section.


Consider the second problem listed in Section 3. It will be shown that suitable techniques of decomposition of domains allow to transform the original inverse-type problem into two direct, properly linked problems, which can be solved by the SAI method. This approach was developed in [15].

Consider then the decomposition of domains

D, = [0, l] = D1 U D2 = [0,x*] U [x*, l].

The problem in D1 is well formulated with the proper initial and boundary conditions. The solution of the problem in D1 leads to the time and space evolution of u and, in particular, to theflowofuinz=x*,sayto

c#J(u,t) = $,t;x*). (4.8)

If Equation (1.2) can be written as

a2u au au - = g(t,x,u,)t, g$> ax2 (4.9)


which is possible in several cases, and in particular in the application developed in the last section of this paper, then Equation (4.9) can be rewritten as a system of two first order equations

(4.10)

which can be regarded as an evolution equation with initial conditions

U(W,1,E = 2*) = U*(W,t), +,t,x = XC) = 4(w,t) (4.11)

and boundary conditions u(t = 0,x) = W)(x), 2 E [x*, 11. (4.12)

Therefore, the exchange of integration variables produces an exchange between the boundary and initial conditions. Of course it is necessary to exchange the space interpolation into a time interpolation.

Let then the time interval [O,T] be discretized by m nodes th such that tl = 0 and t, = T, then m .

u(u,t,x) = ~Lh(t)“h(w), l&=1

u(u,t, x) = 2 Lh(t)wl(W, x), h=l

(4.13)

where the terms Lh denote the Lagrange type polynomials in the variable t and

u/I = +,tII, x), vh = ‘+, t,g, x).

The time derivative in the nodal points th is given by

( > $ (W,t*,X) = eajh u*. j=l

Then we have all tools to obtain a system of m(m - 1) ordinary differential equations of the type

h= 1: u1 = %3(x) I

h=2,...,m: Uh(W, x) = Uf(W, th) + J

va(w, 4 & 0 (4.16)

h=l,...,m: v~(w,x)= m(w,th)+~~g(th,u,uh,a,rl)dq.

The integration of Equation (4.16) yields the space behaviour of the variables u~(w,x) and v,,(w,x). Then the time interpolation (4.13) provides the time-space behaviour of the variables u and v.

One of the crucial problems is, once more, the representation of the random fields. The analysis can be developed in a fashion analogous to the one followed for Problem 3.1. The starting point is the definition of the random processes u*(w, t) and $(w, t). Assume that they are known according to the following:

DEFINITION 4.2. Consider the discretization

It = (t1 = 0, ta,. . . , tn-1,tn = T}

of the time interval [0, T], then U* (w, t) is regarded as a “known” stochastic process if, for every discretization It, the probability distribution of the variable

u* = {u(t1,2*),..., 4tn, x*)1


known and is defined by the density

PZ, = P(u(h), . * *, +); t’).

An analogous definition holds for the process 4(w,t), which will be known if the density Pzn(+) known where 4 is the vector of the values of #J in the discretization points. Considering that Equation (4.15) defines the map

w = wrJ; w = {WV); wo = {u*&, (4.17)

the problem requires essentially the same analysis and has the same difficulties of Problem 3.1.


The solution of this problem exploits the analysis developed for Problem 3.2. If z* < 211 < 1,

one can decompose D, into the three subdomains

D, = DiUDZU D3,

where Di = [O, x*], Dz = [z’, z”] and 03 = [d, 13.

The SAI method can be applied to solve the direct problem in D1 in order to obtain (g)‘.

The knowledge of (g)* j oined to the datum u* allows us to solve the problem in 02 with the

same procedure used in Problem 3.2 in order to obtain ufl(w,t). The knowledge of ur allows us to solve the problem in Ds which is now well-formulated as both

boundary conditions are given. The source term s(w,-&) is defined by the change of slope of the profile of u in ~fl

s(w,t) = k(u) g (w,t,z -_, z:‘-) 1 - $w,t,z + .I+)] . (4.18)

The solution method for Problem 3.3 is then completed by following the indications given for the solution of Problem 3.2.


The solution of this problem requires a somewhat different approach with respect to the methods developed for the preceding problems. The solution which is proposed in this present paper holds for short time intervals. In particular, we shall first develop an analytic solution of the initial-boundary value problem, then this solution (which as we shall see can be developed only for short time intervals) will be used for the identification of the parameters characterizing the model.

Keeping this in mind consider Problem 3.4 assuming that P is known. This allows a “formal” solution of the initial-boundary value problem, which can be obtained by the decomposition method [19]. According to such a method, the solution is sought for in terms of power expansion of an artificial parameter X

u= 2 xi&), $ = I.

j=l j=l

The vector function f is analogously expanded in terms of the same parameter

(4.19)

(4.20)

If (4.19) and (4.20) are substituted into Equation (4.4), then one obtains a sequence of quadratures which provides in a simple way the various terms of the solution (4.19).


This method has been described in several papers and books. We essentially refer to the books [19, 20, Ch. 31 as well as to the review paper [21] and to the papers [22-241 which define the limits of applicability of the method itself.

In particular, it is proved [22] (the result is also reported in [ZO, Ch. 31 that the solution of (4.4), delivered by the decomposition method is a stochastic time expansion of the type

u(w, t) = 5 cj (r(w)) tj , (4.21) j=l

which may converge to the solution of the true problem within a suitable time-interval, defined by the convergence radius of the series (4.21), to be determined by fixed point techniques as discussed in [23,24].

Considering that for t = t*, u* is a known random vector variable, then one can obtain, using (4.21) and providing that the convergence radius is less than t*, suitable moments of r which can be used to identify r. This program has to be regarded, however, as a suggestion for future research activity. In fact, it has not been yet solved at a practical level even if simple analogous cases have been dealt with in the deterministic case.

Transforming the program suggested in this section into an operative result would certainly contribute to relevant aspects of the stochastic modelling of physical systems.

5. ON THE REPRESENTATION OF THE RANDOM FIELDS

The analysis of the preceding section has shown how one can obtain “sample” solutions, i.e., for w E R , of inverse-type problems listed in Section 3. It is necessary, in order to make practical use of this result, to transfer these sample solutions into an efficient representation of the random field u = u(w, t, 2) related to the solution of the problem.

Such a problem was dealt with exhaustively (for direct problems) in [13] and we refer to such a paper to support the discussion developed in this present paper. If our attention is limited to the first probability density, then the computing is referred to the first probability density function

P(u;t, x), (5.1)

namely to the probability density over the variable u with (t,x) as parameters. If the moments of the variable u are known by sample solution at fixed (t, I), say

E(G) = J

zP(w, t, z)P(cfw), (5.2)

n

then following [25] or [26] one can obtain an efficient representation of P by suitable expansions of orthogonal polynomials

P(u;t,z) = w(u) 2 @d@ 2 4&k E(uk)(t, I>, (5.3)

where W(U) is a Gaussian distribution the coefficient of the polynomials (f.i.

h=O q=O kc0

with given mean value and variance and the terms dhq are Hermite) Fh defined as

Fh = & dhq uq, (5.4 q=o

orthogonal with weight w. Use of different types of polynomials and convergence problems are discussed in [26].

Second order correlations can be obtained by interpreting one of the two independent variables BS fixed parameters. Then one defines the densities

P(u(tl,C),U(t2,2);tl,t2,2), vt,,tz E [O,T] (5.5)


and for every E E [0, 11, or

P(2L(t,Il),U(t,22);t,L1,22), b, 22 E p&l] (5.6)

and for every t E [O,T]. More in general,

P(U(tl,21),U(tZ,Zl)rU(t1,22),~(~2,22)),

vt1,t2,3a,22 E p,q x [O,ll.

(5.7)

Representations analogous to the ones indicated for (5.3) can also be obtained for the functions which have been defined above. One needs for (5.5) and (5.6) twodimensional expansions and for (5.7) a four-dimensional expansion.

Classically the density (5.1) g ives rise to first order statistics, the densities (5.5)-(5.6) to second order statistics and (5.7) to higher order ones. The reader is referred to [13] for a deeper insight into the general problem of the representation of random fields and to [25, 261 for the problem of the orthogonal representation and convergence criteria.

For some problems the unknown process may be functions of time or space only. For instance, in Problem 3.1 the unknown us(w, z) is a function of space only and in Problem 3.2, the unknown boundary condition is a function of time only. In the first case we only need the densities (5.1) and (5.5), whereas in the second case we need the densities (5.1) and (5.6).

6. ANALYSIS OF PROBLEMS IN TWO SPACE DIMENSIONS

The analysis developed throughout the preceding Sections was referred to problems in one- space dimension. The “formal” generalization to the case of two-space dimensions does not introduce, at least in principle, additional difficulties. In fact, if the state variable is defined over two space dimensions

‘11=U(W,t,X):~xx+xD,~R, (6.1)

where, for instance, D, = [0, l] x [0, 11, then the same methodology developed in the preceding sections can be applied in a similar fashion.

The variable u can be interpolated in two dimensions

u(w,t,Y)x~Li(r)ui(w,l,Y)~~~Li(.)Lj(Y)Uij(Wlt)r i=l i=l j=l

(6.2)

where uij = +oJi,Yj). The application of the SAI method leads to a system of n x m ordinary differential equations.

The organization of the solution of such a system, in the fashion indicated in Section 4, can lead to the solution of the inverse-type problems listed in Section 3.

However this approach is, at a practical level, computationally heavy. In fact, the system of n x m ordinary differential equations is large and alternative approaches are useful.

Keeping this in mind, it is possible to develop a stochastic formulation of Temam’s fractional steps method when the evolution equation is of semilinear type, say of the type (1.2’) with a suitable form, which allows the decomposition of the integration with respect to the two state variables z and y, respectively. In particular, we consider the two-dimensional semilinear case

dU +AIU +&u = g(t,1:,~,u;r(W,t)),

where

(6.4)

Of course the class of equation defined in (6.3)-(6.4) is a special one and therefore more

general methods are certainly important. Nevertheless, restricting our attention to such a case,


we can indicate the sequential steps needed for the solutions of the “direct” initial-boundary value problem.

The initial condition are indicated as

uo(2, Y) = u(t = 0, x, Y) (6.5)

and the boundary conditions as

cr(t, y) = u@, 2 = 0, y), P(& Y> = u(t, 2 = 1, Y),

y(t,x)=u(t,x,y=O), 6(t,x)=u(t,x,y=l). (6.6)

One can also consider, according to the line of the preceding sections, random initial and/or boundary conditions. However, considering that the problem is, for fixed w E R, deterministic, we can indicate the sequential steps of the application of the method in the deterministic case.

STEP 6.1. The time interval [0, T] is discretized by a sequence of discrete times t, E [0, T] where (t, -

t ,,_I) is the step of the time integration which will be used in the integration (numerical) of the system of ordinary differential equations.

STEP 6.2.

The space domain D, = [0, l] x [0, l] is discretized by a suitable set of discrete points (zi, yj) withi=l,..., nandj=l,..., msuchthat

Xl = yi = 0, 2, = ym = 1.

STEP 6.3.

At each time interval t, ---) tn+l one deals, for instance, with the initial-boundary value problem

~+Alu=O, (6.7)

with initial conditions ~,(t, yj) = u(tn, I, yj) and boundary conditions u(t,z = 0, yj) and u(t, 2 = 1, Yj).

Then the application of the SAI method provides the solution

u*(t,X,Yj)7 t E [tn,tn+1], j= l,...,m.

In particular, one has to solve m times a system of n ordinary differential equations.

STEP 6.4.

Still, in [t,, tn+l], one deals with the initial-boundary value problem

z+Azu=g,

(64

(6.9)

with the initial conditions u,(zi, y) = u*(&+~, xi, y), obtained by the solution of the problem in Step 6.3, and with boundary conditions u(t, zi, y = 0) and u(t, xi, y = 1). This gives u(t,+l, xi, y).

In particular, one has to solve n times a system of m ordinary differential equations. The solution is continued at each time-interval.

REMARK 6.1.

On has the practical advantage of solving systems of n or m equations rather than a (stiffer) system of n x m equations. It is plain that this simple procedure can be technically modified or improved.

REMARK 6.2. One should combine the analysis of [12] with the one of [27] in order to obtain, under reasonable

assumptions, suitable convergence criteria. This open problem is certainly a challenging topic for applied mathematicians.

MCH 16:5-D

48 L. blEZIOS1 et al.

REMARK 6.3. Once the method sketched in Steps 6.1-6.4 can be applied under suitable convergence criteria

then the solution of the inverse problem can be developed in the same fashion indicated in Section 4.

REMARK 6.4.

The whole method can be applied only if the splittings (6.3) and (6.4) hold., i.e., only for some particular cases of equations (however several equations of the mathematical physics are included). When this splitting does not hold, an alternative approach consists in representing the time varying surface ~(t; Z, y) by B-splines in two dimension. This will reduce the stiffness of the system of n x m ordinary differential equations.

7. DISCUSSION AND APPLICATION

This paper provides a unified presentation of several solution techniques of inverse type problems in mathematical physics. The analysis of the problem was developed having in mind the various aspects of the modelling of physical systems in continuum mechanics. Some speculations on this topic is presented in the first two sections of this paper.

Certainly the weak aspect of the analysis developed in this paper is the limitation to rather simple assumptions on the stochastic variables characterizing the physical system, see equations (1.5)-(1.6). On the other hand the analysis provides constructive methods which hopefully can be used in several still interesting physical situations. Moreover, the paper provides constructive methods which can provide quantitative solutions.

Certainly the generalization of the analysis of this paper to more general choices of the stochastic processes would be of great interest and deserves future research speculations. However, it does appears, at present, neither simple nor immediate.

In order to show the practical application of the solution techniques developed and surveyed in this paper, we will now deal with a classical inverse “source” problem for the heat equation. The analysis of inverse problems for the heat equation has consistent technological motivations as described in the introductory sections of the book by Beck el al., [28]. In particular, we deal with an inverse source problem for the nonlinear heat equation. Such a problem was dealt with, in the linear case, by Malishev [6].

The governing equation is

C(u) 2 = ; [h(U) 21 + r(w, t) a(z - Z”), .t = 0.75.

In particular, we deal with a problem characterized by the following coefficients

C(u) = 1, h(u) = 0.2 + y.

Moreover, the initial and boundary conditions are given by

t=o : u(0, Z) = 1 + 0.4Z(l - x)

x=0 : a(t) = 1

x = 0.5 0.1

: u* = ‘+ 1+2.5[1+y(w)]t

x=1 : p(t) = 1,

(7.1)

(7.2)

(7.3)

where r(w) is a random variable uniformly distributed over the interval [-.05, .05]; y(w) simulates the error in the temperature measurement in Z* = 0.5.

The solution scheme follows the procedure indicated in Section 4.3, where 11 collocation points

are used in each subdomain whereas the time integration in D1 and Ds, and the space integration in Dz are performed by a Runge-Kutta method of the fourth order.


Let R be the number of runs of the numerical experiment and if the calculations are referred to the value of the source, then mean value, variance and time-correlation functions are estimated by the following expressions

E(r)(t) = $2 r(t;i) i=l

’ dr)(t) = I i 2 [r(t; i) - E(r)(t)12 (7.4) i=l

Pbw> dt211 = 1

2 [r(h; 4 - E(r(h))l [r(t2; i) - E(r(t2)1, ‘~(~(‘1))~(r(t2)) i=l

where t(t; i) stands for the value of r(t) obtained at the ith run. In a similar fashion, one can obtain additional statistical measures.

OL 0 t

Figure 1. Mean value of P versus time.

The calculations developed in what follows have been obtained for R = 50 (increasing R does not improve accuracy). In particular, Figure 1 shows E(r) versus time. As far as the correlations between positions of the interface at the times ti and tj is concerned, we have the following correlation matrix for ti, tj = 0.2,0.4,0.6,0.8,1.

1 .1788938 - .5275495 -.6003805 -.1446293

1 -.4827653 1 -.8586575

calculations reported at level of show applicability of method at practical Further can be a similar Analogously can probability more than ones with in application.

now conclusions the developed in paper, we mention fact this deals, a survey and at proposal with

classes” of type which have mathematical and in sciences.

In words, aim is to with solve, at some problems may a better either of mathematical the problem

applications. problems dealt and “simple” processes sample trajectories, included in model.

50 L. PREZIOSI et a/.

The guiding line which is followed consists in the idea of decomposing the original “ill-posed” problem into several, properly connected, direct “well-formulated” problems. This technique is the one proposed in [15] and is generalized and furtherly developed in this paper.

Several problems are certainly left open. Some of them can be mentioned as indications for future research perspectives:

(i) Alternatives to the methods developed in this paper at least for deterministic problems. For instance by inverse scattering techniques [29,30].

(ii) Analysis of the stability of the numerical schemes. (iii) Analysis of models with stochasticities more general than the ones dealt with in this paper.

This last one is certainly the most interesting aspect of the problem. This aspect is not dealt with in this paper, despite the fact that the paper deals with the representation of random fields and that such a problem is not completely understood even in the case of simple stochasticities (say random initial conditions or constant random boundary conditions). This paper strongly refers, in Section 5, to the analysis developed by Bonzani [13].

Dealing with true stochastic processes would require, to deal with It&type partial differential equations, for instance in the line of paper [12], and with sharp estimates of the random fields [31,32].

1. 2.

3.

4.

5.

6. 7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17. 18. 19. 20.

21. 22.

23. 24.

REFERENCES

R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience, New York, (1953). A.N. Tikhonov and A.A. &mar&ii, Equations of Mathematical Physics English translation: Pergamon Press, Oxford, Nauka, Moscow, (1972). J.L. Lions and E. Magenes, Problemes aus Limites non Homogenes ei Applicationa, Vol.f-9, Dunod, Paris, (1967- 70). J. Kampee de Feriet, Integrales aleatoires de l’equation de la chaleur dans une barre i&e, C. R. Acad. 5% Paris 240,7113712 (1955). J. Kampee de Feriet, Random solutions of partial differential equations, In Proc. 3rd Berkeley Symp. Mathematical Statislics and Probability, pp. 199-208, Univ. California Press, Berkeley, (1956). I. Malishev, An inverse source problem for the heat equation, J. Math. Anal. Appl. 142, 206-218 (1989). A. Markowski, Development and application of ill-posed problems in USSR, Appl. Mech. Review. 41,247-256 (1988). A. Repaci, On the solution of a class of inverse evolution problems by a Belhnan-Adomian method, Appl. Math. Letters 2 (2),151-153 (1989). A. Repaci, Bellman-Adomian solution of nonlinear inverse problems in continuum physics, J. Math. Anal. Appl. 143, 57-65 (1989). R. Bellman, B.G. Kashef and J. Caati, Differential quadratures: A technique for the rapid solution of nonlinear partial differential equations, J. Comp. Phys. 10, 40-52 (1972). N. Bellomo, L.M. de Socio and R. Monaco, The random heat equation: solution by the stochastic adaptative interpolation method, Comp. Math. Appl. 16 (9), 759-766 (1988). N. Bellomo and F. Flandoli, Stochastic partial differential equations in continuum physics, Math. Comp. Simd. 31,3-17 (1989). I. Bonzani, Time evolution of random fields in stochastic continuum mechanics, Comp. Math. Modelling (to

appear). R. Monaco, A. Repaci and N. Bellomo, Analysis of stochastic systems in continuum physics, Struck Safety 8,93-101 (1990). L. Preziosi and L.M. de Socio, Nonlinear inverse phase transition problems for the heat equation, M3AS:, Math. Models Melh. Appl. Sci. 1 (1991). N. Bellomo, 2. Blzezniak and L.M. de Socio, Nonlinear Sloehastie Evolution Problems in Applied Sciences, Kluwer, Amsterdam, (1992). B. Demidovich and I. Maron, Elements de Cal& Nvmerique &nch Translation, Mir, Moscow, (1979). T.T. Soong, Random Differential Equations in Science and Engineering, Academic Press, New York, (1973). G. Adomian, Stochastic Systems, Academic Press, New York, (1983). N. Bellomo and R. Riganti, Nonlinear Slochastic Systems in Physics and Mechanics, World Sci., London, New Jersey, Singapore, (1987). N. Bellomo, Book Review, Foundation of Physics 19, 443-448 (1989). I. Bonzani, On a class of nonlinear stochastic dynamical systems: Analysis of the transient behaviour, J. Math. Anal. Appl. 128, 39-50 (1987). Y. Che-ult, Convergence of Adomian’s method, Cybemelics 18, 31-38 (1989). A. Repaci, Nonlinear dynamical systems: On the accuracy of Adomian’s decomposition method, Appl. Math. Lelt 3 (4), 35-39 (1990).


25. G. Adomian and K. M&l&n, Stochastic analysis, M&L. Modelling 1 (3), 211-235 (1983). 26. C. Seize, Steady-state solution of Fokker-Planck equation in higher dimension, Prob. Eng. Me&. 3, 196-206

(1988). 27. R. Te-, Sur la stabilite et la convercence de la methode des pas fractionnaires, Theses Doctorat, Univ.

Paris, Paris, (1967). 28. J.V. Beck, B. Blackwell and CR. St. Clair Jr., Inverse Heat Condadion, Wiley, London, (1985).

29. A.G. Ftamm, Multi-dimensional inverse problems and completeness of products of solutions to PDE, J. Maih. Anal. Appl. 134, 211-253 (1988).

30. A.G. Ramm, Numerical methods for solving 3D inverse problems of geophysics, J. Math. Anal. Appl. 136, 352-356 (1988).

31. A.V. Ivanov and N.N. Leonenko, Slatis2ical Analysia of Random Fields, Kluver, Amsterdam, (1979). 32. A.G. Rams, Random Fields Estimation Theory, Pitmaun, London, (1989).

Modeling and solution of stochastic inverse problems in mathematical physics

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