THE FLUX RECOVERING AT THE ECOSYSTEM-ATMOSPHERE BOUNDARY BY INVERSE MODELLING Текст научной статьи по специальности «Математика»

MSC 35R30, 35J25, 35K57

DOI: 10.14529/ mmp240303

THE FLUX RECOVERING AT THE ECOSYSTEM-ATMOSPHERE BOUNDARY BY INVERSE MODELLING

E.I. Safonov1, S.G. Pyatkov1

1Yugra State University, Khanty-Mansiysk, Russian Federation E-mail: [email protected], [email protected]

We consider the heat and mass transfer models in the quasistationary case, i. e., all coefficients and the data of the problem depends on time while the time derivative in the equation is absent. Under consideration is the inverse problem of recovering the surface flux through the values of a solution at some collection of points lying inside the domain. The flux is sought in the form of a finite segment of the Fourier series with unknown Fourier coefficients depending on time. The problem of determining the Fourier coefficient is reduced to a system of algebraic equations with the use of special solutions to the adjoint problem. The equation is considered in a cylidrical space domain. We prove the existence and uniqueness theorems for solutions of the corresponding direct problem. The results are employed in the proof of the corresponding results for the inverse problem. The corresponding numerical algorithm in the three-dimensional case is constructed and the results of the numerical experiments are exhibited. It is demonstrated that the algorithm is stable under random perturbations of the data. The finite element method is used. The results can be used in the problem of the determination of the fluxes of green house gases from soils from the concentration measurements.

Keywords: inverse problem; flux; parabolic equation heat and mass transfer.

Introduction

In general, the problem of calculating the dynamics of an admixture in the atmosphere can be mathematically defined as a solution under given initial and boundary conditions of the differential equation [1-4]

Mu = du/dt + (a, Vu) = div[KVu] + f,K = diag(c1, c2,..., cn). (1)

Here u is the pollutant concentration minus the background value; a is the direction of the wind; the axis Xn is directed vertically upward; ci = Ki + D (i = 1, 2,...,n), with Ki, D the coefficients of turbulent and molecular diffusion (see [5]) and t is time. In view of applications, the equation (1) is often considered in some domain G of the form G = Q x (0,H) (Q is a bounded domain of the class C2). Assume that S0, S1 are the lower and upper bases of the cylinder G, r = dG, S = dQ x (0, H). The following initial-boundary conditions are examined: (see [1,6,7])

u|s = 0, u|Sl = 0, cndu/dxn + ^u|so = g, u|i=o = uo(x). (2)

Sometimes, it is reasonable to assume that the flux is given on the lateral surface or on the upper cover of the cylinder G rather than the concentration. In some practical problems, the problem (1), (2) can be simplified. Studying the surface emission of gases, it is possible to observe that the nonstationary summand dC/dt is essential in some special cases, in particular, in conditions of very weak wind or a low intensity turbulent exchange. The concentration changes are often of quasistationary character and thereby we can exclude

the summand dC/dt equating it to zero and assume only that the coefficients of the equation (1) are known functions of time and space variables [8, p. 19]. The statement of the inverse problem in the general case is as follows. Given the values of concentrations measured at some points yi = (yn, y2i,..., yni) (i = 1, 2,... ,r), find the function g and a solution C to the problem (1) - (3) such that the given values (t) are close to C(t,yi) or (in the ideal case)

u(t,yi)= fa(t), i = 1, 2,... ,r. (3)

We look for the function g in the form g = ^"=1 qi(t)^(x) + g0, where $i is a collection of basis functions, the function g0 is a given function and the functions qi are unknowns.

There are two different cases. In the former, the points yi lie on the boundary of the space domain. In this case the problem is well-posed in the Hadamard sense. In the case of n = 1, the uniqueness theorem in this case is established in [9] and the uniqueness and existence theorem of a classical solution in [10] (here the heat flux and the higher-order coefficient depending on time are determined). However, the case of n =1 is rather simple. Probably, the first article devoted to the problem (1) - (3) in the multidimensional case is the article [11] (see also [12]), where, in the case of Mu = ut — Au and r =1, the existence and uniqueness theorem of classical solutions to the problem was established. More general results were obtained in the article [13]. In the latter case, the points yi lie in the interior of the domain G. In this case the problem is ill-posed (see some existence and uniqueness results in [14]). At present, there are a series of articles devoted to numerical solving the problems (1) - (3) in different statements and the points {yi} in (3) can be interior [1,2,5,10] or boundary [15,16]. The main approach is reducing the problem to a control problem and minimization of the corresponding cost functional (see, for instance, [15]). The articles [17-20] are devoted to numerical solving the problem on describing green house gases emission from soils.

Here we examine the quasistationary case, i. e., the equation (1) is replaced with the equation

Mu = —div(c(t,x)Vu) + b(t,x)Vu + a(t,x)u = f, c = diag(c1,c2,...,cn), (4) and the boundary conditions are of the form

r

cnuxn |so = g, Ruls = h, u|Sl = gi, g = ^2 qi(t)$i(x) + go, (5)

i= 1

where Ru = u or Ru = du/dN + au. The quasistationary case is considered in [17,19, 20] and many other articles. The most popular idea of constructing a solution to the inverse problem belongs to Marchuk G.I. [21]. It is also described in [17] and it is based on constructing some particular solutions to the adjoint problem. In the article [20] the question on dependence of a solution g = g(x,y) on the parametrization of the coefficients of the equation is treated, and the function g = const is numerically determined in [19]. In contrast to the other articles, we look for the flux g in the form of a finite segment of the Fourier series. We expose sharp results on the existence and uniqueness of the inverse problem. The corresponding numerical algorithm and the results of the numerical experiments are exhibited in the case of the problem (3) - (5) and n = 3. The algorithm is based on the finite element method. It is demonstrated that the problem is stable under random perturbations of the data. The results can be used in the problem of the determination of the fluxes of green house gases from soils (see the statement in [1]).

1. Preliminaries

Let G be a domain in Rn. By Lp(G) and Wp(G) (1 < p < to) we mean the Lebesgue and Sobolev spaces, respectively [22]. Let E be a Banach space. Denote by Lp(G; E) the space of measurable functions defined on G with values in E and the finite norm

II II^O^II-eII r [22]- We also use the space Ck(G; E) of ¿"-valued functions continuous in

ii nLp\G)

G together with their derivatives up to the order k admitting continuous extensions on the closure G. The definitions of the Sobolev space Wp(G]E) is standard (see [23]).

Proceed with some auxiliary results. Some of them are of interest themselves. Consider the auxiliary problem

Lu = dXj(aij(t,x)uxi) + щ(t,x)uxi + ao(t,x)u + Xu = f, x E G, t E (0,T), (6) i,j=1 i=1

Ru|r = h, Riu(t,x',ri) = gi(t,x'), i = 0,1, x' = (x1 ,...,xn-1), (7)

where ani = ain = 0 for г = 1, 2,..., n - 1, r0 = 0, гг = Я, Ru = Ya,J=i x)uîÈ~ + a(t, x)u (V is the unit outward normal to S) or Ru = u, R0u = annuXn + a0u or R0u = u, R1u = annuXn + a1 u or R1 u = u, X > 0, X > 0 is a real parameter. Describe the conditions on the data. In what follows, we always assume that the operator L is elliptic, i. e., for some constant ô0 > 0, the inequality Y^j=1 aijùîj > |2 holds for all x e G,t e (0,T) and £ E Rn. Assume that

aij E C([0, T]; W^(G)), at E C([0,T]; Lq(G)) (q>n,q > p),

ao E C ([0, T ]; Lq! (G)), (q1 > n/%qx > p), i,j = 1, 2,...,n; (8)

f E C([0, T]; Lp(G)), h E C([0, T]; Wpk-1/p(S)),g E C([0,T]; Wpki-1/p(Si)), i = 0, 1, (9)

where k (or ki) is equal to 2 if Ru = u (or Riu = u) otherwise, k =1 and, respectively, ki = 1. Moreover, we suppose that if Ru = u or (and) Riu = u (i = 0,1) then, respectively,

a E C([0, T]; W^(S)),<n E C([0,T]; W^(Si)), i = 0,1. (10)

The consistency conditions are as follows: if Ru = u and Riu = u (i = 0,1) then h(t,x', 0) = gi(t,x')lon; if Ru = u and Riu = u (i = 0,1) then, for p > 2, R-ih(t,x', 0) = gi(t,x')|dn; if Ru = u and Riu = u (i = 0,1) then, for p > 2, R(t,x',r.i)gi(x')|dn = h(t,x',rj,). It is possible that the statement of the following theorem is already known. But we did not find direct references and thus the proof is exposed below.

Theorem 1. Let the conditions (8) - (10) and the consistency conditions for every t E (0,T) hold. Assume also that p =2. Then if a solution to the problem (6), (7) is unique in the class W^(G) for every t E [0,T] then a solution u exists for every t, u E C([0,T]; Wp(G)) and satisfies the estimate

1

\\u\\c([0,T];WyG)) <c(\\f\\C([0,T];Lp(G)) + \\h\\ M, + E H&H - , ^, J' (П)

C([0,T];WP P(S)) C([0,T];WP p (Si))

If h = 0,g1 = 0,g2 = 0, and p G (1, to) then there exists a parameter A0 > 0 such that for all A > A0 there exists a unique solution u G C([0,T]; W%(G)) to the problem (6) - (7) satisfying the estimate

\\u\\o ([0,T ];W2(G)) + AHu||c ([0,T];Lp(G)) < c\\f ||c([0,T];Lp(G)), (12)

where the constant c is independent of A.

Proof. First of all, we note that under the consistency conditions there exists a function $ G C([0,T]; Wp(G)) satisfying the boundary conditions (7) (see Theorem 7.3 in [25]). After the change of variables u = v + $, we arrive at the problem

Lv = f, Ru|s = 0, Riu(t,x',ri) = 0, i = 0,1, (13)

where the same the notation for the new right-hand side is employed. Without loss of generality, we can assume that if Riu = u then ai = 0. Since the summands aiu in the boundary condition are lower-order terms, the case of ai = 0 can be easily considered with the help of the method of continuation in a parameter, for example. Consider the case of the boundary conditions R0u = uXn, R1 u = uXn Ru = u. The remaining cases are treated by analogy. Construct a function fa0(xn) G C^ (R) even in the variable xn G (—H,H) and such that suppfa0 G (-2H/3, 2H/3), = 1 for xn G [0, H/2]. Define also a function (xn) even with respect to the point xn = H and such that fa1(xn) = 1 — (xn) for xn G (0,H) and fa1(xn) = 0 for xn < 0. Construct also functions ai(xn) G C^(R) with the same properties such that suppa0 C (-2H/3, 2H/3), suppa1 C (H/3, 5H/3), ai = 1 on suppfai (i = 0,1) and a0 =1 on suppfa1. Construct domains G0 and G1 such that G0 D Q x [0, 2H/3], G0 C Q x [0, 3H/4), G1 D Q x [H/3, H], G1 c Q x [H/4, H], and the parts of the boundaries dG0 and dG1 lying in the domains xn > 0 and xn < H, respectively, belong to the class C2. Next, we construct the extensions of G0 and G1 symmetric with respect to the planes xn = 0 and xn = H. Denote these extensions by the same symbols. By construction, dG0 G C2 and dG1 G C2. Given a function p G Lp(G), extend it to QH = Q x (—H, 2H) taking (p(t,x',xn) = p(t,x',xn) for xn G (0, H), Cp = p(t,x', —xn) for xn G (-H, 0), Cp(t,x',xn) = g(x', 2H — xn) for xn G (H, 2H). We have Cp G Lp(QH). Extend the coefficients aij ,ai(i = n) as even functions and an as an odd function with respect to the points xn = 0 and xn = H into QH. Consider the problems

Lui = pa, x G Gi,, i = 0,1, (14)

uildGi = 0, i = 0,1. (15)

By Theorem 8.2 [24], there exists A0 > 0 such that the problems (14), (15) are uniquely solvable for A > Ao and the estimate

KWw^Gi) + A\\ui\\Lp(Gi) < c\g\Lp(Gi), i = 0, 1 (16)

holds. Due compactness of the segment [0,T], we can assume that the constant c is independent of time. In our case G0 is symmetric with respect to the planes xn = 0 and a solution u0 possesses the property u0Xn |Xn=0 = 0. Indeed, consider the function u0(t,x) = u0(t,x', —xn) for x G G0. In this case u0|dGo = 0, u0Xn = —u0Xn(t,x', —xn), uoxi = uoXi (t,x', —xn), uoXnXn = uox„x„ (t,x', —xn), lLoXiXj = uoXiXj (t,x', —xn) for i,j < n—1.

This function u0 is also a solution to the problem (14), (15). The uniqueness theorem implies that U0 = u0. Thus, the function u0 is even in xn and thereby u0xn (t,x', 0) = 0. Similarly, u1xn (t,x', H) = 0 and ui is an even function in the variable xn with respect to the point xn = H .A solution to our problem is sought in the form v = u0ip0 + u1^1. The function v meets the boundary conditions in G, i. e. the Dirichlet condition on dQ x (0, H) and the Neumann condition on the planes xn = 0 and xn = H. Inserting v in the equation, we have Lv = 10Lu0 + [L,10]u0 + 11 Lu1 + [L,11]u1 = g + [L,10]u0 + [L,11 ]u1, where [L,1i]b = L(1ib) — 1Lb. The function g must satisfy the equation

g + V(g) = f, V(g) = [L,l0]u0 + [L,l1]u1. (17)

Estimate the norm ||V(g)||Lp(G). By definition, [L,1k]uk = L(1kuk) — IkLuk = — 2ann1kxnukxn — a,nnuk1kxnxn + an1kxnuk (k = 0,1). We have the embeddings Wp(G) C Lq0 (G) (q0 < np(n — p) for p <n and arbitrary for p = n), Wp(G) C L^ (G) for p > n [22]. Let, for example, p < n. In this case (see (8))

11

11 V (g)|Lp(G) < Y^ cllukxn ¡Lp (Gk) + c1 hn ¡L, (G) ||uk ||Lp,/(,_p) (Gk) < c2 ^ ||uk lW1(Gk )• (18) k=0 k=0

The case p > n is treated by analogy. The interpolation inequality ||u||Wji(Gk) < c|u|W/2(G)|u|L/2g) and the estimate (16) yield

||V(g)||Lp(G) < C31A| 1/21g||Lp(G), t G [0,T], (19)

where the constant c2 is independent of A > A0 and t G [0, T]. Fix q G (0,1). Increasing the parameter A0 if necessary, we can assume that c3 | A | 1/2 < q. In this case the equation (17) is uniquely solvable and a solution satisfies the estimate ||g||c([0,T];Lp(G)) < c4|f ||C([0,T];Lp(G)), c4 = 1/(1 — q). Moreover, the function v = u111 + u2ip2 is a solution to the initial problem

satisfying the estimate (12). The first part of the theorem results from the Fredholm theory.

□

Next, we discuss the generalised solvability of the problem (6), (7) in the case of g,gi = 0. Denote by W1,B(G) the space of functions in Wp(G) satisfying those boundary conditions in (7) that have a sense. Therefore, if Ru = u, R0u = u, and R1 u = u then Wp1;B(G) = Wp1(G). Denote by W-B*(G) 1/p + 1/q =1) the dual space to W^(G) with respect to the duality pairing defined by the inner product in L2(G). The adjoint problem to the problem (6), (7) with the homogeneous data is written as:

L*v = f, Rvls = 0, Rv(x', ri) = 0, i = 0,1, (20)

where Rv = Rv + a • vv if Rv = v and Rv = v if Rv = v, R0v = R0v + anv if R0v = v and R0v = v if R0v = v, R1 v = R1v + anv if R1v = v and i?1v = v if R1v = v. Here

n

a •v = }^i=1 aiVi.

Denote by W-B(G) (1/p + 1/q = 1) the dual to Wq1B* (G) with respect to the duality pairing defined by the inner product in L2(G), where Wq1B* (G) is the space of functions in Wq1(G) satisfying those boundary conditions in (20) that have a sense. The condition on the data are as follows:

aij G C([0, T]; W^(G)), a G C([0,T]; W^(S)), a(x') G C([0,T]; W^(Si)), i = 0,1, (21)

where the last inclusions are fulfilled when Ru = u or R^u = u for some i = 0,1, respectively;

ao G C ([0, T ]; Lqi (G)),a G C ([0,T]; Wlqi (G)) (qi > max(n,p,q)), i = 1,...,n, (22)

where q = p/(p — 1), p G (1, œ). Define a generalized solution to the problem (6), (7) in the case of h,gi = 0. Let, for instance, Ru = u and Ri u = u (i = 0,1). By a generalized solution to the problem, we mean a function u G C([0,T]; WpB(G)) such that

/ ^^ aijUxi<P*i + aiUxi + a0u)pdx + awpdS — a0up dx' + autpdx' =

i j — 1 i—1 J J J

G ij — 1 i—i S So Si

= J f (t,x)ip(t,x) dx, Vp G C([0, T]; (G)), t G [0,T].

G

Similar definitions are introduced in the remaining cases.

Theorem 2. Let the conditions (21), (22) hold and let f G C([0,T]; W~B(G)) and h,gi = 0. Then if a solution to the problem (6), (7) is unique in the class Wp, B(G) for every

t G [0,T] then a solution u to this problem exists for every t and u G C([0,T]; Wp,B (G)).

There exists a parameter A0 > 0 such that, for all X > A0, there exists a unique solution u G C([0,T]; WpB(G)) to the problem (6), (7) satisfying the estimate ||u||C([0)T];Wpi(G)} + A||u||C([0;T];W-i (G)) < c||f ||W-i (G), where the constant c is independent of A.

Proof. The proof is carried out along the same lines as those in the previous theorem.

Instead of the results in [24], we refer to Theorems 12.2, 14.2 [26].

r □

Corollary 1. Let f = Y1 ¿—i Qi(t)S(x — yi) (yi G G, and 6 is the Dirac delta-function) and qi(t) G C([0,T]). Assume that the condition (22) holds and

a.

j G C([0, T]; Wl(G)), a+a^v G C([0,T]; W^(S)), ^(x')+an G C([0,T]; W^(Si)), (23)

where the last inclusions are fulfilled when Ru = u or Riu = u for some i = 0,1, respectively. Then f G C([0,T]; W~B* (G)) with q G [1,n/(n — 1)), where q = p/(p — 1), p G (1, to), and if a solution to the problem (20) is unique in the class Wq1B» (G) for every t G [0,T] then there exists a unique solution v to the problem (20) such that v G C([0,T]; WIb.(G)).

Remark 1. It is sometimes possible to take Q with a Lipschitz boundary. For example, if n = 3 and Q = (a,b) x (c, d) then all statement of Theorems 1-2 are valid whenever the operator L agrees with the operator M in (4). We only must take into account that additional consistence conditions can appear at the points x1 = a,b,x2 = c,d. The remaining statement are of the same form.

2. Existence and Uniqueness Theorems

Under consideration is the inverse problem

n n

Lu = — dxj (aij(t,x)uxi) + ai(t,x)uxi + ao(t,x)u + Au = f, (24)

i,j—1 i—1

n

n

Ru|r = h, Rlu(t,x',H)= gl(t,x'), R0u(t,x', 0) = g(t,x'), u(t,yi) = 1(t), (25)

where i = 1,.. .r, R0u = annuXn + a0u, and g = [=1 q.i(t)$i(x') + go ($i is a collection of linearly independent functions on Q and the functions qi(t) are unknowns). The adjoint problem to the problem (6), (7) is written as

L*v = f, Rvls = 0, Riv(t, x', ri) = 0, i = 0,1. (26)

Fix p > n/2, p = 2. Next, let the conditions (9), (10), (22), (23) hold. Assume also that $j G Wp l/p(Q) for all j. If Ru = u and p > 2 then we suppose that $i(x')|dQ = 0, i =1, 2,... ,r. The consistency conditions are written as follows: for every t G [0,T], if Ru = u and Rlu = u then h(t,x',H) = gl(t,x')|dQ; if Ru = u, Riu = u (i = 0,1) and p > 2 then Rih(t,x', 0) = gi(t,x')|dQ (i = 0,1); if Ru = u, Rlu = u, and p > 2 then R(t,x',r.i)gl(t,x')|dQ = h(t,x',rl). Under the above conditions, there exists a function $ G C([0, T]; W2(G)) such that R$|s = h, Rl$(x',n) = gb Ro$ = go [25, Theorem 7.3]. Making the change of variables u = v + $, we arrive at the problem

n

Lv = - dxj (aij (t,x)vXi ) + a (t,x)vxi + ao(t,x)v + Xv = f, x G G, (27)

i,j=l i=l

Rv|r = 0, Rlv(t,x',ri) = 0, R0v(t, x', 0) = qi(t)$i(x'), (28)

i=l

v(t, y3) = 1 (t) - $(t, y3) = i^j, j = 1, 2,..., r. (29)

Let v G C([0,T]; Wp(G)) be a solution to this problem. The condition p > n/2 ensures the embedding Wp(G) C Wql0(G), with q0 > n. In this case q0 G (1,n/(n — 1)), with q0 = q0/(q0 — 1). If the conditions of Corollary 1 are fulfilled then there exist solutions vj G C([0, T]; Wql^ (G)) to the problems (26), with f = 5(x — y3) G C([0,T]; W-B (G)). It is not difficult to justify multiplying (27) by vj and integrating by parts that

J R0v(t,x', 0)vj(t,x', 0) dx' + ij (t) = J fvj dx,

Q G

where the right-hand side is the value of the functional f on the function vj. The boundary

condition (28) leads to the system

|>/$i(x>j(t'x'0)dx' = /fvjdx — jthj = 1'2--'r' (30)

i=l Q G

which can be written in the form

Aa = F, Fi = J fvi dx — (t), aij = J Фjvi(t,x', 0) dx'. (31)

g n

Assume that

det A = 0 Vt G [0,T], ti(t) G C ([0,T]) (г = 1, 2,...,r). (32)

n

The matrix A has the entries aj = f $jVi(t, x', 0) dx'. Thus, to solve the inverse problem,

n

we need to determine the functions Vj and to solve the system (31).

Theorem 3. Assume that the conditions (8) - (10), (22), (23), (32) and the consistency conditions hold, p > n/2, and a solution to the problem (6),(7) from the class W£(G) is unique for every t G [0,T]. Then there exists a unique solution u,qi,q2,... ,qr to the problem (24),(25) such that u G C([0,T]; Wp2(G)), q G C([0,T]) (i = 1,...,r). If the conditions of the theorem hold and there exists a segment [ti ,t2] (ti < t2) such that det A = 0 Vt G [ti,t2] then a solution to the problem (24),(25) is not unique in the class C([0,T]; W2(G)).

Proof. First of all, we note that the duality arguments allow to show that if a solution to the problem (6),(7) from the class Wp(G) is unique for every t G [0,T] then a solution to the problem (26) from the class Wq1B* (G) is unique for every t as well. In this case we can construct the functions Vj (j = 1, 2,..., r) to the problem (26) with f = 8(x — yj) and the former part of the theorem results from the fact that the system (31) is uniquely solvable. Next, we recover the function v as a solution to the problem (27), (28), and determine the function u = v + $. Demonstrate that a solution v to the problem (27), (28) meets (29). The definition of a generalized solution to the problem (27), (28) and the properties of the functions Vj imply the equalities

J(annvXn (t,x', 0) + a0v(t,x', 0))vj(t,x', 0) dx' + v(t,yj) = J fvj dx,

n G

subtracting which from the equalities (30), we justify (29).

Now, assume that det A = 0 Vt G [ti,t2]. Take t0 G (ti,t2) Let r(A(t0)) = 3 < r. In this case, either there exists a neighborhood about t0 in which r(A(t)) = /3 < r (the rank of A(t)) or in any neighborhood there exists a point ti such that 3 < r(A(ti)) < r. In the latter case, choose ti G (ti,t2) such that 3 < r(A(ti)) < r. Repeating the arguments finitely many times, we can find a point tk G (ti,t2) with a neighborhood U C (ti,t2) such that r(A(t)) = 30 < r in U. Without loss of generality, we can assume that the matrix lying at the first 3o columns and rows has the rank /0 and its determinant does not vanish in U. Instead of the system (31) with the matrix A = {aj}, we take the system

ßo r

qj = - Y^ aij qj ■

j=l j=ßo+l

Take arbitrary functions qj (j = ß0 + 1,..., r) of the class ). The remaining functions

{qj} are solution to this system. These functions belongs to the class C(U) and are compactly supported in U. Extending them by zero on the segment [0,T] and solving the direct problem (27), (28) with these functions we determine a nontrivial solution to the homogeneous problem.

□

3. Numerical Algorithm

We take n = 3 and G = Q x (0,Z). We examine the problem (3) - (5), where the boundary conditions are of the form

r

C3UX3|so = g(t,x), uls-1 = 0, uls = 0, g = qi(t)^i(x'). (33)

i=i

Let ym = (yim, y2m, y3m). The finite element method is employed. Divide the domain G into tetrahedra and construct the corresponding piecewise linear basis {pi(x)}N=i. The nodes of the grid are denoted by {pi}1j=l. For convenience, we assume that the points ym are the nodes pim (m=1,2,... ,r) of the grid. An approximate solution has the form v = 1=l Ci(t)wi(x) and the functions Ck(t) are solutions to the system

13

"Y^iik Ck = F, aik = (t,x)ipkxj Wixj + bj (t,x)ipkxj Wi) + Wi dx, (34)

k=i G j=i

where F = F0 - qj (t)Fj and Fj = ^ ^ (X, 0) dx', ...J ^ tpN (x', 0) dx' )T, F0 =

n n

((f, p1),..., (f, pN))T. Thus, the system can be written in the matrix from

AC = F. (35)

Let ej be a basis vector whose jth coordinate is equal to 1 and the remaining coordinates vanish. Define the vector Vj = (A*)-1ei. (j = 1, 2,...,r), where A* is the transposed matrix. Find the quantities qi. To this end, multiply the equation (35) scalarly by Vj and use the conditions (3) which in our case have the from Cij = vpj. We derive that

r

< Fo,Vj > -^j = Y qi < Fi ,Vj >, j = 1, 2,... ,r,

i=1

where the brackets < •, • > stand for the inner product in RN. The vectors q can be determined from this system. The vector CF is a solution to the system (35).

The matrix with entries < Fi,Vj > is a discrete analog of the matrix A in (32). In the case of a regular family of finite elements, it is possible to prove the convergence of the entries < Fj,vi > to the corresponding entries of the matrix A (see Theorems 3.1.5, 3.1.6 [27]). This means that if the condition (32) is fulfilled then the determinant of A in (34) also does not vanish for sufficiently small partition of the domain.

4. Numerical Realization

In this section we present the results of numerical experiments for some collections of the data. To determine the accuracy of calculations, we take given functions c, b, a, f and the function g depending on the known functions $i and qi (see (5)). Solving the direct problem (4), (5) we can find a solution u to this problem and thereby the quantities u(t, yi) = (i = 1, 2,... ,r). Next, using this data we can solve the inverse problem (3) -(5), and find a solution u and the functions qi. Comparing the initial functions qi, u and the results of calculations, we can estimate the convergence of the algorithm.

The characteristics of the computer are as follows: Processor: Intel(R) Xeon(R) CPU E5-2678 v3 @ 2.50GHz (2 two processors); RAM: 64.0 GB; OC: Windows 10 Pro x64.

For brevity, we only display graphics and tables with the results of calculations qi. Assume that all coefficients in the equation (4) are known. Every experiment consists of the following steps: definition of the points {yi}r=i and the functions qi, $i (i =1, 2,..., r); defining the parameters of a grid; converting all used functions into arrays in accordance with grid nodes and the time step; solving the direct problem (4), (5), constructing the functions t(t) (i = 1, 2,... ,r) and adding random noise to the values of these functions; solving the inverse problem (3) - (5); comparing the initial data and the results of calculations of the functions Qi and u.

Consider the results of calculations for the first group of data. Our domain is a cube with the unit edge, whose diametrically opposite corners have the coordinates (0; 0; 0) and (1; 1; 1).

The first group of the data. Let r = 3 and let the points yi have the coordinates: (0, 25; 0, 75; 0, 25), (0, 75; 0, 5; 0, 75), and (0, 5; 0, 25; 0, 5). The functions Qi are the functions qi = 2t + 1, q2 = (t — 1)2, and q3 = t3 + 3. The number of the time steps is equal to M = 10.

1) Since the problem is three-dimensional, it is necessary to partition the domain into tetrahedra. Let us denote the steps in spatial variables by Ax, Ay, and Az. Let us construct a grid on part of the boundary x3 = 0 consisting of right triangles with legs equal to Ax and Ay. Next, we duplicate this layer by raising it to Az and connecting the points, thus obtaining rectangular tetrahedra of the height Az with a right triangle at the base. We employ three grids Z0, Zi, and Z2 with the number of nodes N equal 729, 2197, and 9261, respectively.

2) Next, we define the arrays of nodes on the faces of the cube. Note that we use the homogeneous Dirichlet condition on all faces except for the lower face of the cube.

3) The time step is equal to t = T/M. Introduce the coefficients as follows: a = (t + 1) * (x + y + z +1), bi = (t +1) * (x +1)2, b2 = (t + 1) * (y +1)2, b3 = (t +1) * (z +1)2, cii = (x + 1)/(t + 1), cii = (t +1) * (x + 1)2, C22 = (y + 1)/(t + 1), C33 = (z + 1)/(t + 1). Next, we define the right-hand side f = 1, the functions g and qi.

4) The functions $i in our calculations are actually a partition of unity on the lower face of the cube, we divide the boundary x3 = 0 into r parts according to the following rule: the nodes of the ith subdomain are closer to the ith point yi than to all other points. So $i = 1 on some collection of nodes and vanishes at the remaining nodes.

5) Next, we solve a direct problem (4), (5), as it was described in the previous section. The next step is constructing the functions t = u(t,yi) (i = 1, 2,..., r). To add a random perturbation, we employ a uniformly distributed random variable noise G (—5; 5) (5 G [0,1]) with zero mean, so 1005 is a deviation in percents. The resulting functions are of the form ti(jT) = ti(jT)(1 + noise(jT)). For the first group of the data we take 5 = 0.

6) Introduce the calculation errors as follows: the equality £q = maxi(maxj |qj (iT) — qj |) defines the calculation error for the functions qi (the numbers qj are the result of calculations, qj ~ qj (ti), j = 1,..., r); the error of calculations of a solution u is defined as £u = maxi;j luiyj — u(pi,Tj )|, where i = 1, 2,...,N and j = 1, 2,...,M. Let ts be execution time of the algorithm, including the time for solving the direct problem, in seconds.

The results of calculations for the three grids (the case of 5 = 0) show that the graphics of the initial and the constructed functions actually coincide, so we do not

display the results. The quantities ea, eu, and ts for the above three grids are as follows: (2, 4e-14, 4, 9e-15,1.67), (4, 9e-14,1, 4e-14,11, 9), (5, 6e-14,1, 3e-14, 373, 3).

For the second group of experiments, we take only one point y1 and add 1, 5 and 10 percent noise. The number of nodes of the grid is equal to 1331. Changing the coordinates of the point y1 with the step 0.1 from (0,1; 0,1; 0,1) to (0, 9; 0, 9; 0, 9), we obtain practically identical result and the average parameters are as follows: = 5,55e-16, eaUUr = 3,82e-16, T^vr = 3, 22. The largest error achieves at the point y1 = (0, 5;0, 5;0, 5). In the next table we take y1 = (0, 5; 0, 5; 0.5). The Table 1 shows the dependence of the errors on the functions qi and a random noise. Next, we take M = 100

Fig. 1. The results of calculations of q1 with 25% noise

and 5 = 0, 25. The results are displayed on Fig. 1 for the function q1 = sin(n(t + 1)). In this case eq = 0, 24, eu = 0, 091, ts = 32, 8. The calculation shows that the algorithm is stable with respect to the noise.

Table 1

T

îe results of experiments for the second group o

No 5 Qi Ts

1 0,01 log(i + 1) 0,0061 0,0033 3,41

2 0,05 log(i + 1) 0,0185 0,007 3,46

3 0,1 log(i + 1) 0,0581 0,031 3,44

4 0,01 et+l 0,044 0,24 3,65

5 0,05 et+1 0,217 0,09 3,3

6 0,1 et+1 0,398 0,19 3,33

7 0,01 Sin(7r(i + 1)) 0,0091 0,0037 3,46

8 0,05 Sin(7r(i + 1)) 0,037 0,015 3,25

9 0,1 Sin(7r(i + 1)) 0,089 0,032 3,34

the data

For the third group of experiments, we use an array of 8 points {yi} and the corresponding functions qi below. We also slightly change the mesh construction area by stretching and compressing it by 2 times relative to the X and Z axes, respectively. Let's set the random noise to 5 percent. The results are exhibited in Table 2 and on Fig. 2. According to the results of computational experiments, it is clear that the calculation error increases as the coordinates of the overdetermination point move away from the lower face. The results shows good convergence of the algorithm as a whole.

The resu

Table 2

ts of experiments for the third group of the data

No Vi Qi %

1 (0,2;0,1;0,45) sin(7r(10t+l))-16 2,44

2 (0,6;0,3;0,35) {t - 2)2 + 16 1,39

3 (1;0,5;0,25) (t- l)3 - 12 1,32

4 (1,4;0,7;0,35) log2(0, It + 1) - 8 4,53

5 (0,2;0,9;0,05) 2t + 12 0,72

6 (0,6;0,7;0,15) -lOt- 1 0,31

7 (1,8;0,1;0,05) - cos(71-lot) + 8 0,93

8 (1,4;0,3;0,15) _e2i+0,5 + 4 0,97

Fig. 2. The results of calculations of qi with 5% noise

Conclusions

Using theoretical results on well-posedness of the problem, we construct a numerical algorithm for recovering the surface flow on the lower face with the use of point observations of the concentration. It is based on the conventional methods (in our case FEM and difference schemes). The results of numerical experiments are presented. The obtained results reveal the accuracy, efficiency, and robustness of the proposed algorithm. It is stable under random perturbations of the data.

Acknowledgments. This work was supported by the Russian Science Foundation and the Government of the Khanty-Mansiysk Autonomous Okrug-YUGRA (Grant no. 22-1120031).

References

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14. Pyatkov S., Shilenkov D. Existence and Uniqueness Theorems in the Inverse Problem of Recovering Surface Fluxes from Pointwise Measurements. Mathematics, 2022, vol. 10, no. 9, article ID: 1549, 23 p. DOI: 10.3390/math10091549

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19. Desyatkov B.M., Borodulin A.I., Kotlyarova S.S. Determination of Aerosol Particle Flow Emitting from an Underlying Surface by Solving the Inverse Problem of their Propagation in the Atmosphere. Optics of Atmosphere and Ocean, 1997, vol. 10, no. 6, pp. 639-643. (in Russian)

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22. Triebel H. Interpolation Theory. Function Spaces. Differential Operators. Berlin, VEB Deutscher Verlag der Wissenschaften, 1978.

23. Amann H. Compact Embeddings of Vector-Valued Sobolev and Besov Spaces. Glasnik Matematicki, 2000, vol. 35 (55), pp. 161-177.

24. Denk R., Hieber M., Prüss J. R-Boundedness, Fourier Multipliers, and Problems of Elliptic and Parabolic Type. Memoirs of the American Mathematical Society, 2003, vol. 166, no. 788, pp. 1-114.

25. Grisvard P. Equations Differentielles Abstraites. Annales Scientifiques de l'Ecole Normale Superieure, 1969, vol. 2, no. 3, pp. 311-395. (in French)

26. Amann H. Nonautonomous Parabolic Equations Involving Measures. Journal of Mathematical Sciences, 2005, vol. 130, no. 4, pp. 4780-4802. DOI: 10.1007/s10958-005-0376-8

27. Ciarlet P.G. The Finite Element Method for Elliptic Problems. Amsterdam, Noth-Holland Publishing Company, 1978.

Received April 30, 2024

УДК 517.956 БЭТ: 10.14529/mmp240303

ВОССТАНОВЛЕНИЕ ПОТОКА НА ГРАНИЦЕ ЭКОСИСТЕМА-АТМОСФЕРА

Е.И. Сафонов1, С.Г. Пятков1

1 Югорский государственный университет, г. Ханты-Мансийск, Российская Федерация

Мы рассматриваем модели тепломассопереноса в квазистационарном случае, т.е. все коэффициенты и данные зависят от времени, но производная по времени в уравнении отсутствует. Исследуется обратная задача восстановления потока на границе

области по заданным значениям решения в наборе точек, лежащим внутри области. Поток ищется в виде конечного отрезка ряда Фурье, с неизвестными коэффициентами. Задача определения коэффициентов сводится с помощью специальных решений сопряженной задачи к системе алгебраических уравнений. Исходное уравнение рассматривается в цилиндрической пространственной области. Это выбор сделан в силу того, что этот случай, как правило рассматривается в приложениях. Доказана теоремы существования и единственности решений прямой задачи. Полученные результаты используются в доказательстве соответствующих результатов для обратной задачи. В трехмерном случае строится численный алгоритм и приводятся результаты численных экспериментов. Показывается, что алгоритм устойчив к случайным возмущениям данных. Используется метод конечных элементов. Результаты могут быть использованы, например, в задачах определения потоков парниковых газов из почвы по данным замерам концентраций.

Ключевые слова: обратная задача; параболическое уравнение; поток; тепломас-соперенос.

Литература

1. Glagolev, M.V. Determination of Gas Exchange on the Border between Ecosystem and Atmosphere: Inverse Modeling / M.V. Glagolev, A.F. Sabrekov // Mathematical Biology and Bioinformatics. - 2012. - V. 7, № 1. - P. 81-101.

2. Сабреков, А.Ф. Определение удельного потока метана из почвы с помощью обратного моделирования на основе сопряженных уравнений / А.Ф. Сабреков, М.В. Глаголев, И.Е. Терентьева // Доклады Международной конференции <Математическая биология и биоинформатика >. - Пущино, 2018. - T. 7. - Article ID: e94.

3. Белолипецкий, В.М. Оценка потока углерода между атмосферой и наземной экосистемой по измеренным на вышке вертикальным распределениям концентраций СО2 / В.М. Белолипецкий, П.В. Белолипецкий // Вестник НГУ. Серия: Информационные технологии. - 2011. - Т. 9, № 1. - C. 75-81.

4. Яговкина, С.В. Оценки потоков метана в атмосферу с территории газовых месторождений севера Западной Сибири с использованием трехмерной региональной модели переноса / С.В. Яговкина, И.Л. Кароль, В.А. Зубов, В.Е. Лагун, А.И. Решетников, Е.В. Розанов // Метеорология и гидрология. - 2003. - № 4. - C. 49-62.

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6. Бородулин, А.И. Статистические характеристики потока метана, выделяемого заболоченной подстилающей поверхностью / А.И. Бородулин, Г.А. Махов, Б.Д. Десятков,

C.Р. Сарманаев // Доклады академии наук. - 1996. - Т. 349, № 2. - С. 256-258.

7. Бородулин, А.И. О распределении потока метана над заболоченной местностью / А.И. Бородулин, Г.А. Махов, С.Р. Сарманаев, Б.Д. Десятков // Метеорология и гидрология. - 1995. - № 11. - С. 72-79.

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9. Onyango, T.M. Restoring Boundary Conditions in Heat Conduction / T.M. Onyango,

D.B. Ingham, D. Lesnic // Journal of Engineering Mathematics. - 2007. - V. 62. - P. 85-101.

10. Hussein, M.S. Simultaneous Determination of Time-Dependent Coefficients in the Heat Equation / M.S. Hussein, D. Lesnic, M.I. Ivanchov // Computers and Mathematics with Applications. - 2014. - V. 67, № 5. - P. 1065-1091.

11. Kostin, A.B. Some Problems of Restoring the Roundary Condition for a Parabolic Equation, II / A.B. Kostin, A.I. Prilepko // Differential Equations. - 1996. - V. 32, № 11. - P. 15151525.

12. Kostin, A.B. On Some Problems of Restoration of a Boundary Condition for a Parabolic Equation, I. / A.B. Kostin, A.I. Prilepko // Differential Equations. - 1996, - V. 32, № 1. -P. 113-122.

13. Pyatkov, S.G. Determination of the Heat Transfer Coefficient in Mathematical Models of Heat and Mass Transfer / S.G. Pyatkov, V.A. Baranchuk // Mathematical Notes. - 2023. -V. 113, № 1. - P. 93-108.

14. Pyatkov, S.G. Existence and Uniqueness Theorems in the Inverse Problem of Recovering Surface Fluxes from Pointwise Measurements / S.G. Pyatkov, D. Shilenkov // Mathematics. -2022. - V. 10, № 9. - Article ID: 1549, 23 p.

15. Wang, Shoubin. Solution to Two-Dimensional Steady Inverse Heat Transfer Problems with Interior Heat Source Based on the Conjugate Gradient Method / Shoubin Wang, Li Zhang, Xiaogang Sun, Huangchao Jia // Mathematical Problems in Engineering. - 2017. - V. 8. -Article ID: 2861342, 9 p.

16. Knupp, D.C. Explicit Boundary Heat Flux Reconstruction Employing Temperature Measurements Regularized via Truncated Eigenfunction Expansions / D.C. Knupp, L.A.S. Abreu // International Communications in Heat and Mass Transfer. - 2016. - V. 78. -P. 241-252.

17. Glagolev, M.V. Inverse Modelling Method for the Determination of the Gas Flux from the Soil / M.V. Glagolev // Environmental Dynamics and Global Climate Change. - 2010. -V. 1, № 1. - P. 17-36.

18. Glagolev, M.V. Methodologies for Measuring Microbial Methane Production and Emission from Soils. A Review / M.V. Glagolev, O.R. Kotsyurbenko, A.F. Sabrekov, Y.V. Litti, I.E. Terentieva // Microbiology. - 2021. - V. 90, № 1. - P. 1-19.

19. Десятков, Б.М. Определение потока аэрозольных частиц, выделяемых подстилающей поверхностью, путем решения обратной задачи их распространения в атмосфере / Б.М. Десятков, А.И. Бородулин, С.С. Котлярова // Оптика атмосферы и океана. - 1997. -Т. 10, № 06. - С. 639-644.

20. Глаголев, М.В. О методе «обратной задачи> измерения интенсивности газообмена на границе почва/атмосфера / М.В. Глаголев, А.Ф. Сабреков // Болота и биосфера: Материалы VII Всероссийской с международным участием научной школы. - Томск, 2010. -C. 31-35.

21. Marchuk, G.I. Mathematical Models in Environmental Problems. Studies in Mathematics and its Applications / G.I. Marchuk. - V. 16. - Amsterdam: Elsevier Science Publishers, 1986.

22. Triebel, H. Interpolation Theory. Function Spaces. Differential Operators / H. Triebel. -Berlin: VEB Deutscher Verlag der Wissenschaften, 1978.

23. Amann, H. Compact Embeddings of Vector-Valued Sobolev and Besov Spaces / H. Amann // Glasnik Matematicki. - 2000. - V. 35(55). - P. 161-177.

24. Denk, R. R-Boundedness, Fourier Multipliers, and Problems of Elliptic and Parabolic Type / R. Denk, M. Hieber, J. Prüss // Memoirs of the American Mathematical Society. - 2003. -V. 166, № 788. - P. 1-114.

25. Grisvard, P. Equations Differentielles Abstraites / P. Grisvard // Annales Scientifiques de l'Ecole Normale Superieure. - 1969. - V. 2, № 3. - P. 311-395.

26. Amann, H. Nonautonomous Parabolic Equations Involving Measures / H. Amann // Journal of Mathematical Sciences. - 2005. - V. 130, № 4. - P. 4780-4802.

27. Ciarlet, P.G. The Finite Element Method for Elliptic Problems / P.G. Ciarlet. -Amsterdam: Noth-Holland Publishing Company, 1978.

Сергей Григорьевич Пятков, доктор математических наук, профессор, Инженерная школа цифровых технологий, Югорский государственный университет (г. Ханты-Мансийск, Российская Федерация), [email protected].

Егор Иванович Сафонов, доцент, Инженерная школа цифровых технологий, Югорский государственный университет (г. Ханты-Мансийск, Российская Федерация), [email protected].

Поступила в редакцию 30 апреля 2024 г-