UTMS 2006–32

November 8, 2006

Solution formula for an inverse problem
with underdetermining databyYu. E. Anikonov and M. YamamotoÌUNIVERSITY OF TOKYOGRADUATE SCHOOL OF MATHEMATICAL SCIENCES
KOMABA, TOKYO, JAPAN1Yu. E. Anikonov and 2 M.Yamamoto
SOLUTION FORMULA FOR AN INVERSE PROBLEM
WITH UNDERDETERMINING DATA
1Sobolev Institute of Mathematics, Acad. Koptyug prospekt 4630090 Novosibirsk Russia
anikon@math.nsc.ru
2
Department of Mathematical Sciences, The University of Tokyo
3-8-1 Komaba, Meguro, Tokyo 153 Japan
myama@ms.u-tokyo.ac.jp
tel: +81-3-5465-8328
fax: +81-3-5465-7011§1. Introduction and Key Lemma.
In an inverse problem, we are required to determine coefficients in a partial differential equation in order that the solution to the differential equation
realizes prescribed data. As the mathematical topics for an inverse problem,
we mention the uniqueness and the stability, and additionally the existenceof a solution to the inverse problem is important. Usually the solution to
an inverse problem is given not by formulae which are involved by algebraic
operations and calculi, but is found through limit processes such as iterations(for example, as a solution to an operator equation of the second kind). In
this paper, we will show a formula for solutions to an inverse problem which
is attached with underdetermining data.
Our formulation for the inverse problem is underdeterming and so cannot
guarantee the uniqueness for solutions to the inverse problem. Hence ourformula gives ”one” solution to the inverse problem under consideration,and does not describe all possible solutions but includes sufficiently manysolutions in the sense that it admits a family of coefficients parameterized byfree functions in the spatial variable.
Among various inverse problems, for our approach, we will mainly discuss
an inverse problem with data at final time. For example, in Bouchouev andIsakov [3], Isakov [5], such an inverse problem for the Black - Scholes equationis considered: Determine a, b, c by w(x, T ), x ∈ I, inα

∂w
∂ 2w
∂w
= a(x) 2 + b(x)+ c(x)w.
∂t∂x∂xHere T > 0 is fixed, I is an interval and α = const > 0. As for inverse problems with data at final time, we can further refer to Choulli and Yamamoto[4], Prilepko, Orlovsky and Vasin [6] and the references therein.
1In the present paper, we construct a family of solutions of the inverseproblems, which is based on representation of solutions and coefficients.
Let D ∈ Rn be a domain and let us consider an evolution equation of thefollowing type:mX

αk (v(x))

k=1nn
X
X
∂kw
∂2w
∂w
=a(x)
+bj (x)+ c(x)w,ij
k∂t∂xi ∂xj j=1∂xj
i,j=1(1.1)

where x = (x1 , ..., xn ) ∈ D ⊂ Rn , 0 ≤ t ≤ T , αk : Rn −→ R are smoothfunctions for 1 ≤ k ≤ m, v(x) = (v1 (x), . . . vn (x)) is a differentiable vectorvalued function such that∂(v1 , ..., vn )6= 0, x ∈ D∂(x1 , ..., xn )and aij = aji , 1 ≤ i, j ≤ n.
First we formulate a general approach for obtaining a representation formula of solution w(x, t) and coefficients aij (x), bj (x), c(x).
Note that in the case where
α1 = α = const > 0,andnX

αk = 0,aij ηj ηj > 0,i,j=1

k = 2, 3, . . . , m,η ∈ Rn , η 6= 0,equation (1.1) becomes parabolic:
αnnXX∂2w∂w∂waij (x)+bj (x)+ c(x)w.
=∂t∂xi ∂xj j=1
∂xj
i,j=1

In the case of
α2 = α = const,

αk = 0, k = 1, 3, . . . , mequation (1.1) becomes
hyperbolic when α > 0, and elliptic when α < 0.
√Moreover if α1 = − −1 and α2 = · · · = αm = 0, then (1.1) is the Schrödinger
equation.
Lemma 1. Let D1 ⊂ Rn be a domain and βkℓ , βk , β ∈ C 1 (D1 ), 1 ≤k, ℓ ≤ n such that βkℓ = βℓk , be given, and u ∈ C 2 (D) be a given function
such that u(x) 6= 0 for any x ∈ D. LetF (y, t),y ∈ D1 ⊂ Rn , 0 ≤ t ≤ T2

satisfy the equationmX

k=1

αk (y)

nn
XX
∂kF
∂2F
∂F
=
β(y)+βk (y)+ β(y)F.
kℓk∂t∂yℓ∂yk k=1∂ykk,ℓ=1(1.2)

Then the function
w(x, t) = u(x)F (v(x), t)
and the coefficients aij (x), bj (x), c(x) consecutively defined by linear systemsof algebraic equations (1.3), (1.4), (1.5), satisfy equation (1.1):nXi,j=1n
Xaij (x)∂vk ∂vℓ= βkℓ (v(x))∂xi ∂xj

nX∂vk1
bj (x)=aijuβk (v) −∂xju
j=1i,j=1(1.3)!∂vk ∂u∂vk ∂u∂ 2 vk ++u,∂xi ∂xj ∂xj ∂xi∂xi ∂xj
(1.4)


nn
2XX
∂ u
∂u 1aij+bj (x).
(1.5)
c(x) = uβ(v) − u∂xi ∂xj j=1
∂xj
i,j=1

The proof of the lemma is done by direct substitution of the solution
w(x, t) = u(x)F (v(x), t) into equation (1.1) in terms of (1.2) - (1.5).
We note that if we choose v(x) = x, u(x) = 1, βkℓ = akℓ , βk = ak and
β = c for 1 ≤ k, ℓ ≤ n, then (1.3) - (1.5) are true.
After choosing v = v(x), we set A = {u, {βkℓ , βk }1≤k,ℓ≤n , β}. Then, byaij (A), bj (A), c(A), 1 ≤ i, j ≤ n, we denote aij , bj and c defined by (1.3)
2- (1.5). We note that A is composed of n +3n+4functions in x. Then,
22n2 +3n+2by Lemma 1 we can represent
functions aij , bj , c in x by n +3n+4
22functions in x. In other words, our lemma
gives representation
formulae
22free functionof coefficients aij , bj , c which contains 1 = n +3n+4− n +3n+222in x. Thus by our representation formula, we can give a pair of solution(aij , bj , c) which realizes one extra data w(x, T ) = w1 (x), x ∈ D. We noticethat our formula (1.3) - (1.5) are not involved with limit processes.
In particular, if coefficients αk , βkl , βk , β are constant in (1.2), then insome cases it is possible to represent F (y, t) which is a solution to an initialvalue problem. Moreover, if we a priori know F (y, t), then problem of searchfor coefficients and solution obviously turns to the determination of only
functions u(x), v(x) = (v1 (x), . . . , vn (x)).
We will consider a one-dimensional variant of Lemma 1 more precisely.
Let w(x, t) satisfy the following equation:mX

k=1

αk (v(x))∂kw∂ 2w∂w+ c(x)w,=a(x)+b(x)∂tk∂x2∂x3(1.6)

x0 ≤ x ≤ x1 ,0 < t < T,

where αk (y), y ∈ R1 , v(x), a(x), b(x), c(x) are some differentiable functionsand αk may be constant.
Lemma 2. Letu(x), v(x), αk (y), β1 (y), β2 (y), β3 (y), k = 1, 2, . . . , m
be some twice differentiable functions and let
x0 ≤ x ≤ x1 ,y ∈ R1 ,u(x) 6= 0,dv≡= v′ (x) 6= 0.
dx

We assume that F (y, t), y ∈ R1 , 0 < t < T , satisfies the equationmX

αk (y)

k=1

∂kF
∂ 2F
∂F+ β3 (y)F,=β(y)+ β2 (y)1k2∂t∂y∂yy ∈ R, t > 0.

(1.7)

Then the functions w(x, t), a(x), b(x), c(x) defined by the following formulaew(x, t) = u(x)F (v(x), t),a(x) =β1 (v(x)),v′2v′2 uβ2 (v) − β1 (v)(2u′ v′ + uv′′ )uv′3
u2 v′3 β3 (v) − uu′′ v′ β1 (v) − u′ v′2 uβ2 (v) + 2u′2 v′ β1 + u′ uv′′ β1c(x) =u2 v′3satisfy the equationb(x) =mX

k=1

αk (u(x))

∂kw
∂ 2w
∂w
+ c(x)w.
=a(x)+ b(x)
k2∂t∂x∂x

As for other approaches to inverse problems by means of formulae, we
refer to Anikonov [1], [2].
§2. One-dimensional parabolic inverse problem with data at
final time.
As an example of using this way we consider an inverse problem for a onedimensional parabolic equation or a modifed Black-Scholes equation (e.g.,[3]): Find functions
w(x, t), a(x), b(x), c(x), 0 ≤ t ≤ T, x ∈ R,
such that
α∂w
∂w
∂2w
= a(x) 2 + b(x)+ c(x)w,∂t∂x∂x4x ∈ R, t > 0

(2.1)

andw

t=0

= w0 (x), w

t=T= w1 (x),x ∈ R.

(2.2)

More precisely, represent the four functions w(x, t), a(x), b(x), c(x) by w0 (x),w1 (x) and one real-valued auxiliary function. Here we assume thatw0 (x) > 0,x ∈ R.

(2.3)

Then we note that w1 (x) > 0 for x ∈ R by the maximum principle.
Theorem. We choose δ ∈ R and a smooth function f0 > 0 for x ∈ Rsuch thatf0 (x)(2.4)
Φ(x) = R ∞ 1y2√
exp−f0 (x − y)dy−∞ 2 πT
4T
is an injective function in x ≥ δ. Furthermore we assume that"#w0 (x)0, sup⊂ Φ([δ, ∞)).
x∈R w1 (x)We set−1v(x) = Φ!

w0 (x),
w1 (x)

u(x) =Then1a(x) =b(x) = −c(x) =w1 (x),w1 (v(x))(v ′ (x))2(2.5)

x ∈ R.

(2.6),2u′ (x)v ′ (x) + u(x)v′′ (x),u(x)(v ′ (x))3−u(x)v′ (x)u′′ (x) + 2(u′ (x))2 v′ (x) + u(x)u′ (x)v ′′ (x),(u(x))2 (v ′ (x))3u(x)w(x, t) = √2 πt(x − y)2w0 (v(y))dy,
exp −4t−∞!

Z ∞x ∈ R, t > 0,satisfies (2.1) and (2.2).
Example. Setting
2f0 (x) = e−x ,x > 0,

we see that for any δ > 0, the function Φ is injective in x ∈ R. In fact,y21√exp −4T−∞ 2 πTZ ∞!

−(x−y)2ex2
1exp −,
dy = √
4T + 1
4T + 15!

and soΦ(x) =

√

4T x2
4T + 1 exp −.
4T + 1!

Thus under the assumption thatw0 (x) √< 4T + 1,
x∈R w1 (x)supwe can rewrite (2.6) as(v(x) =√4T + 1w1 (x)
4T + 1log
4T
w0 (x)

and!) 12w1 (x),
w1 (v(x))u(x) =so that the conclusion of the theorem holds.
Proof of Theorem. In Lemma 2, we set α1 = 1, α2 = · · · = αm = 0,β1 = 1, β2 = β3 = 0. Then
1F (y, t) = √2 πt

Z ∞

−∞

y2

e− 4t f0 (x − y)dy

satisfies (1.7). We havew0 (x) = u(x)f0 (v(x)),

w1 (x) = u(x)F (v(x), T ).

Eliminating u(x) in these equations, we obtainw0 (x)F (v(x), 0)== Φ(v(x)).
w1 (x)F (v(x), T )Therefore we have−1v(x) = Φandu(x) =w0 (x)FΦ−1

w0 (x)
w1 (x)

w0 (x)
w1 (x)

 =,0!

w1 (x)F

Φ−1

w0 (x)
w1 (x)

Hence, by Lemma 2, the proof of the theorem is complete.
ACKNOWLEDGEMENTS.

6

.

,0

This work has been done during the stay of the first author at Graduate
School of Mathematical Sciences of the University of Tokyo February of 2004
which was supported by the 21st Center of Excellence Program and he gratefully acknowledges the support. The second author was partly supported byGrants 15340027 from the Japan Society for the Promotion of Science andGrant 17654019 from the Ministry of Education, Cultures, Sports and Technology.
REFERENCES
1. Yu. E. Anikonov, Formulas in Inverse and Ill-posed Problems, VSP,Utrech, 1997.
2. Yu. E. Anikonov, Inverse Problems for Kinetic and Other Evolution
Equations, VSP, Utrech, 2001.
3. I. Bouchouev and V.Isakov, Uniqueness, stability, and numerical methods for the the inverse problem that arises in financial markets, Inverse Problems, 15(1999), R95-R116.
4. M. Choulli and M. Yamamoto, Generic well-posedness of an inverse
parabolic problem - the Hölder space approach, Inverse Problems, 12 (1996),
195–205.
5. V.Isakov, The inverse problem of option pricing, in Recent Development in Theories and Numerics, World Scientific, Singapore, 2003, pp.47-55.
6. A.I. Prilepko, D.G. Orlovsky and I.A. Vasin, Methods for Solving
Inverse Problems in Mathematical Physics, Maecal Dekker, New York, 2000.

7

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The Graduate School of Mathematical Sciences was established in the University ofTokyo in April, 1992. Formerly there were two departments of mathematics in the University of Tokyo: one in the Faculty of Science and the other in the College of Arts andSciences. All faculty members of these two departments have moved to the new graduate school, as well as several members of the Department of Pure and Applied Sciencesin the College of Arts and Sciences. In January, 1993, the preprint series of the formertwo departments of mathematics were uniﬁed as the Preprint Series of the Graduate
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