Proceedings of Workshops on
Inverse Problems, Data,
Mathematical Statistics and Ecology
LiTH-MAT-R–2011/11–SE
Editors: Vladimir Kozlov, Martin Ohlson and Dietrich
von Rosen
Department of Mathematics
Linköping University
SE–581 83 Linköping, Sweden
Proceedings of Workshops on Inverse Problems, Data, Mathematical Statistics and Ecology
LiTH-MAT-R–2011/11–SE www.mai.liu.se Department of Mathematics Linköping University SE–581 83 Linköping Sweden
Contents
I
Introduction
1
II
Contributions
5
A Nest Spectrum of Symmetric Semi-Bounded Operator and Reconstruction of Manifolds - M. I. Belishev 7 1 Basic Objects . . . 10 1.1 Operator L0. . . 10 1.2 Dynamical System . . . 10 1.3 Extension E . . . 10 1.4 Collection U . . . 11 2 Nest Spectrum . . . 11 2.1 Nests . . . 11 2.2 Lattice L . . . 11 2.3 Space (ΩL0, τ ) . . . 12 3 Reconstruction of Manifolds . . . 12 3.1 Simple manifolds . . . 12 3.2 Inverse Problems . . . 13 3.3 Reconstruction . . . 14 References . . . 14
B Solving Ill-Posed Cauchy Problems in Three Space Dimensions Using Krylov Methods - L. Eldén 15 1 A Cauchy Problem on a Cylindrical Domain . . . 18
2 Numerical Implementation . . . 19
vi Contents
References . . . 19
C Dispersion Exponential Models - A. Hassairi 21 1 Natural Exponential Families . . . 24
1.1 Preliminaries . . . 24
1.2 Influence of an Affine Transformation on a NEF. . . 25
1.3 Influence of Taking Powers of Convolution in NEFs . . . 25
1.4 Action of G = GL(IRd+1) on NEFs on IRd . . . 25
2 Classifications of NEFs . . . 26
2.1 Morris Class of Quadratic Real NEFs . . . 26
2.2 Letac Mora Class of Cubic Real NEFs . . . 26
2.3 Simple Quadratic NEFs. (Casalis) . . . 26
2.4 Simple Cubic NEFs. (Hassairi) . . . 27
References . . . 27
D An Inverse Problem in Glaciology - V. Kozlov 29 References . . . 33
E Key Issues of Modelling Extreme Floods - V. Kuzmin et al. 35 References . . . 40
F The Inverse Problem for the Wave Equation and Two Acquisition Geometries - M. Lassas and L. Oksanen 43 1 Introduction . . . 46
2 The Inverse Problem for the Wave Equation . . . 46
3 Two Acquisition Geometries with Disjoint Sources and Receivers . . . . 47
References . . . 48
G The Multiplicity of Positive Solutions for Boundary Value Problems to Quasi-linear Equations: A Small Survey - A. I. Nazarov 51 References . . . 56
H Testing of Exponentiality with Application to Historic Data - Y. Nikitin and I. Piskun 59 1 A Review of Exponentiality Tests . . . 62
2 Efficiency of Tests . . . 62
3 Treatment of Historical Data . . . 63
References . . . 65
I More on the Kronecker Structured Covariance Matrix - M. Ohlson et al. 67 1 Introduction . . . 70
2 Model . . . 70
3 Estimation . . . 71
vii
J The Inverse Problem, EEG Data and Multivariate Statistics - M. Ohlson and
D. von Rosen 75
1 Introduction . . . 78
2 Assessing Local Complexity of EEG Signals . . . 79
3 Multivariate Statistical Analysis . . . 80
References . . . 80
K Comparison of Two Measurement Models of Convolution Type - P. Piiroinen 83 1 Introduction . . . 86
2 LeCam Theory and Comparison of Measurements . . . 87
3 Notes on the Continuous Model and Related Technical Issues . . . 88
References . . . 88
L Inverse Scattering Problems that have Applications in 2D Non-linear Optics - V. Serov 91 1 Formulation of the Problem . . . 94
2 Born Approximation . . . 96
3 Fixed Energy Problem . . . 96
M Robust Minimax Bias Estimation of the Correlation Coefficient - G. L. Shevlyakov 99 1 Introduction . . . 102
2 Main Result . . . 102
3 Conclusion . . . 104
References . . . 104
N Partial Balayage and the Inverse Problem of Potential Theory - T. Sjödin 105 1 Introduction . . . 108
2 Partial Balayage . . . 108
3 Application to the Exterior Inverse Problem . . . 109
References . . . 111
O Methane Armageddon: Is it Possible? - I. Sudakov and S. Vakulenko 113 1 Introduction . . . 116
2 Phase Transition Approach . . . 116
3 Lake Area Growth and Methane Emission . . . 117
References . . . 118
P Bifurcations in Large Free-Scale Networks - S. Vakulenko and M. Zimin 121 1 Introduction . . . 124
2 Main Result . . . 125
References . . . 126
Q Goodness-of-Fit Tests Based on Distribution Characterizations, and their Ef-ficiencies - K. Volkova 129 1 Tests of Normality Based on Shepp Property, and their Efficiencies. . . . 132
viii Contents
2 Tests of Exponentiality Based on Rossberg’s Characterization, and their Efficiencies. . . 133 References . . . 134
ix
Contributors
Author Page Ahmad, M. R. 67 Belishev, M. I. 7 Eldén, L. 15 Hassairi, A. 21 Korotygina, U. 35 Kozlov, V. 29 Kuzmin, V. 35 Lassas, M. 43 Makin, I. 35 Nazarov, A. I. 51 Nikitin, Y. 59 Ohlson, M. 67, 75 Oksanen, L. 43 Piiroinen, P. 83 Piskun, I. 59 von Rosen, D. 67, 75 Rumyantsev, D. 35 Serov, V. 91 Shevlyakov, G. L. 99 Sjödin, T. 105 Sudakov, I. 113 Surkov, A. 35 Vakulenko, S. 113, 121 Volkova, K. 129 Zimin, M. 121Part I
Introduction
3
Preface
Processes in Nature may be considered as deterministic or/and random. We are observing global problems such as climate changes (e.g. warming and extreme weather conditions), pollutions (e.g. acidification, fertilization, the spread of many types of pollutants through air and water) and whole ecosystems that are under pressure (e.g. the Baltic sea and the Arctic region). To understand the processes in Nature and (predict) understand what might occur it is not enough with empirical studies. One needs theoretical fundaments including models and theories to perform correct actions against different threats or at least to carry out appropriate simulation studies. For example, extreme value theory can explain some of the observed phenomena, classical risk analysis may be of help, different types of multivariate and high-dimensional analysis can explain data, time series analysis is essential, for forthcoming studies the theory of experimental designs is of interest, data assimilation together with inverse problem technique is useful for adjustment of data into mathematical models and the list can be made much longer. Behind all these approaches mathematics is hidden, sometimes at a very advanced level. Chemical and physical pro-cesses influence all observations but the challenge is to do appropriate approximations so that mathematical/statistical models can be applied. The main aim of this project is to present state of the art knowledge concerning the modelling of Nature with focus on mathematical modelling, in particular "inverse and ill-posed problems", as well as spatio-temporal models. Inverse and ill-posed problems are characterized by the property that the solutions are extremely sensitive to measurement and modelling errors. There are established connections between inverse problems and Bayesian inference but very little has been carried out with focus on parametric inference such as the likelihood approach. Concerning spatio-temporal models these are usually extensions of classical time series models or/and classical multivariate analysis models.
From the Nordic Council of Ministers, within the program Nordic - Russian Coopera-tion in EducaCoopera-tion and Research we asked for funding of 3 preparatory meetings where the plan was to create a series of events taking place during 2011-2013. Partner organizations were
• Institute of Problems of Mechanical Engineering, St. Petersburg • St. Petersburg State University
• Helsinki University
• Swedish Agricultural University • Stockholm University
• Linköping University
However, there were also some other participants from other universities.
The planned events should be connected to the following fields: applied mathematics, biophysics and mathematical statistics. Within applied mathematics: mathematical mod-elling and partial differential equations, inverse and ill-pose problems, data assimilation, dynamical systems, linear algebra, matrix theory; within biophysics; neural networks and
4
inverse modelling of objects; within mathematical statistical; analyses of stochastic pro-cesses, spatio-temporal modelling, experimental design, where considered. There exists a wide overlap between these areas and it is challenging to systemize this overlap and transmit this knowledge to students and stakeholders. However, due to unsure funding it was decided to discuss what can be presented during a one-year program. Moreover, due to practical reasons only 2 meetings/workshops were held:
1. Workshop on Inverse Problems, Data, Mathematical Statistics and Ecology: May 20-21, 2010 at Department of Mathematics, Linköping University.
2. Workshop on Inverse Problems, Data, Mathematical Statistics and Ecology, Part II: August 25-27, 2010 at Department of Mathematics, Helsinki University.
The output from the above events can be summarized as follows:
• We have identified a number of different areas which can be taught on from different perspectives depending on students background of mathematics.
• We have learned to know many interesting researchers who are willing to share there experiences when for example creating a summer school.
• There is no doubt that we can organize cross-disciplinary summer/winter schools with focus on either the Baltic or Archtic regions.
This booklet is also part of the deliverables. It comprizes extended abstracts of the major-ity of the talks of the participants showing their great interest. It is in some way a unique cross-disciplinary document which has joined researchers from different areas from Rus-sia, Finland and Sweden.
We are extremely grateful for the support given by the Nordic Council of Ministers (NCM-RU-PA-2009/10382) and all the enthusiastic contributions by the participants, in-cluding our host in Helsinki, professor Lassi Päivärinta.
Vladimir Kozlov Martin Ohlson Dietrich von Rosen Linköping University Linköping University Linköping University
Part II
Contributions
Paper A
Nest Spectrum of Symmetric
Semi-Bounded Operator and
Reconstruction of Manifolds
Author: M. I. Belishev
Supported by RFBR grants 08-01-00511 and NSh-4210.2010.1.
Nest Spectrum of Symmetric Semi-Bounded
Operator and Reconstruction of Manifolds
Mikhail I. Belishev
St. Petersburg Department of the Steklov Mathematical Institute,
St. Petersburg, Russia, E-mail: belishev@pdmi.ras.ru
10 Paper A Nest Spectrum of Symmetric Semi-Bounded Operator
1
Basic Objects
1.1
Operator L
0Let L0be a densely defined positive definite symmetric operator in a separable Hilbert
space H with nonzero defect indexes:
(L0y, y) ≥ γkyk2 (γ ≥ 0), 1 ≤ dim KerL∗0≤ ∞ .
Let L be its extension by Friedrichs, so that L0⊂ L ⊂ L∗0and one has
L = L∗ =
∞
Z
0
λ dQλ, (Ly, y) ≥ γ kyk2, (1)
where dQλis the spectral measure of L.
1.2
Dynamical System
With L0one associates an evolutionary dynamical system
vtt+ Lv = h , t > 0 (2)
v|t=0 = vt|t=0 = 0 , (3)
where h ∈ Lloc2 ((0, ∞); H) is a H-valued function of time (control). Its finite energy class solution v = vh(t) is represented by the Duhamel formula
vh(t) = t Z 0 L−12sin h (t − s)L12 i h(s) ds = h see (1) i = t Z 0 ds ∞ Z 0 sin√λ(t − s) √ λ dQλh(s) , t ≥ 0 (4) (see, e.g., [3]). Fix a subspace A ⊂ H; the set
VAt := {vh(t) | h ∈ Lloc2 ((0, ∞); A)} , t > 0
of all states produced by A-valued controls is called reachable (at the moment t, from the subspace A). Reachable sets increase as A increases and/or t grows.
1.3
Extension E
Let LatH be the lattice of the (closed) subspaces in H1. Define a family E = {Et} t≥0
of the maps Et: LatH → LatH by
E0A := A, EtA := clos Vt
A, t > 0 .
As one can show (see [2]), t ≥ t0 and A ⊆ A0 imply EtA ⊆ Et0A0, i.e., E extends
subspaces. By this we call it a space extension.
M. I. Belishev 11
1.4
Collection U
Define a class
M := {h ∈ C∞([0, ∞); KerL∗0) | supp h ⊂ (0, ∞)}
of smooth KerL∗0-valued controls vanishing near t = 0. The sets
Ut:= h(t) − vh00(t) h ∈ M = h see (4) i = h(t) − t Z 0 L−12sin h (t − s)L12 i h00(s) ds h ∈ M , t ≥ 0 (5) (here ( · )0 := d
dt) are said to be reachable from boundary
2. The sets Utincrease as t
grows. We put U := {clos Ut}t≥0⊂ LatH.
2
Nest Spectrum
2.1
Nests
A family of subspaces n ⊂ Lat H is a nest if for any A, B ∈ n one has A ⊆ B or A ⊇ B (see [4]). Here we deal with the parametrized nests n = {Nt}t≥0: Nt⊆ Nt0 as t ≥ t0.
These nests are partially subordinated by
{n n0} ⇔ {Nt⊆ Nt0, t ≥ 0} .
For a linear set A ⊂ H, by PAwe denote the (orthogonal) projection in H onto closA.
A nest n = {Nt}t≥0is said to be the limit of a nest sequence n1 = {Nt1}t≥0, n2 =
{N2
t}t≥0, . . . if PNt = s−lim
j→∞PN
j
t; in such a case we write nj→ n.
2.2
Lattice L
For each subspace A ∈ Lat H, the space extension E determines a parametrized nest nA:= {EtA}t≥0. The operator L0determines a (sub)lattice L ⊂ Lat H such that
1. U ⊂ L
2. EtL⊂ L for all t ≥ 0
3. L is closed w.r.t. the nest convergence introduced above 4. L is the minimal lattice obeying 1 − 3.
The lattice L is well defined and determined by the operator L0.
12 Paper A Nest Spectrum of Symmetric Semi-Bounded Operator
2.3
Space (Ω
L0, τ )
A nest m ∈ L is said to be minimal if the relations n ∈ L and n m imply n = m. By ΩL0 we denote the set of all minimal nests of the lattice L and call it a nest spectrum of
the operator L0.
The collection U is also a parametrized nest. The set ∂ΩL0 := {m ∈ ΩL0| m U }
is said to be the boundary of ΩL0.
The nest spectrum can be endowed with an intrinsic topology. At first, define a func-tion (quasi-distance) ρ : ΩL0 × ΩL0 → [0, ∞) ∪ {∞} by the following rule. For
m, m0∈ Ω L0 : m = {Mt}t≥0, m 0 = {M0 t}t≥0we put ρ(m, m0) := ( ∞ if PMtPM0t = O for all t 2 inf{t ≥ 0 | PMtPM0t 6= O} otherwise.
Also, introduce the quasi-balls
Br[m] := {m0∈ ΩL0| ρ(m, m
0) < r} , r > 0 .
Let τ be the (weakest) topology on ΩL0generated by the system of quasi-balls {B
r[n] |
n ∈ ΩL0, r > 0}.
So, (ΩL0, τ ) is a topological space associated with the operator L0in a canonical way.
A fact, which easily follows from the definitions, is that the unitarily equivalent operators have the homeomorphic nest spectra, i.e., if eH = U H and eL0 = U L0U∗with a unitary
U : H → eH then (Ω
e L0,eτ )
hom
= (ΩL0, τ ). In other words, a nest spectrum is a unitary
invariant of the operator class under consideration.
3
Reconstruction of Manifolds
3.1
Simple manifolds
Let Ω be a C∞-smooth (possibly noncompact) Riemannian manifold with the boundary ∂Ω, ∆ the (scalar) Laplace operator. The minimal Laplacian on Ω
L0= −∆|C∞ 0 (Ω\∂Ω)
is a densely defined symmetric positive definite operator in H := L2(Ω); its Friedrichs
extension is the Dirichlet Laplacian
L = −∆|H2(Ω)∩H1 0(Ω)
(Hk are the Sobolev classes), whereas
L∗0= −∆|{y∈H | ∆y∈H}
M. I. Belishev 13
A class S of simple manifolds is introduced in [2]. Roughly speaking, a simplicity means that the symmetry group of Ω ∈ S is trivial. The class S is generic: any Ω can be made into a simple manifold by arbitrarily small smooth variations of its boundary3. As
is shown in [2],
• for any Ω, the quasi-distance ρ on the nest spectrum of its minimal Laplacian is a metric (distance), so that (ΩL0, ρ) is a metric space
• for Ω ∈ S, there is an isometry (of metric spaces) i : (ΩL0, ρ) → Ω such that
i(∂ΩL0) = ∂Ω.
Therefore, a simple Ω is determined by any unitary copy eL0of its minimal Laplacian up
to isometry.
3.2
Inverse Problems
With the manifold Ω one associates an evolutionary dynamical system
utt− ∆u = 0 in Ω × (0, ∞) (6)
u|t=0= ut|t=0= 0 in Ω (7)
u|∂Ω= f (t) for 0 ≤ t < ∞ (8)
with a boundary control f ∈ F := Lloc2 ((0, ∞); L2(∂Ω)); the solution u = uf(x, t)
describes a wave, which is initiated by boundary sources and propagates from the bound-ary into the manifold. The collection U defined in sec 1.4 consists of the reachable sets Ut := {uf( · , t) | f ∈ F } (see (5)). The input/output correspondence is realized by a
response operator
R : f 7→ ∂νuf|∂Ω×[0,∞)
defined on smooth controls vanishing near t = 0 (here ν is the outward normal to ∂Ω, ∂ν
the normal derivative). The time-domain (dynamical) inverse problem is
IP 1: given the response operator R of the system (6)–(8), to recover the manifold Ω. With the manifold Ω one associates a stationary dynamical system
(−∆ + z) w = 0 in Ω (9)
w = ϕ on ∂Ω , (10)
where z ∈ C\spec L is a complex parameter (frequency), w = wϕ
z(x) is the solution.
The Weyl-Titchmarsh function of the system is
M (z)ϕ = ∂νwϕz|∂Ω,
which is a L2(∂Ω)→L2(∂Ω)-operator valued function. The frequency-domain inverse
problem is
IP 2: given the W-T function M of the system (9)–(10), to recover the manifold Ω. 3There are easily checkable sufficient conditions of geometric character, which provide Ω ∈ S: see [2] for
14 Paper A Nest Spectrum of Symmetric Semi-Bounded Operator
Assume that the manifold Ω is compact. Let {λk}∞k=1: 0 < λ1< λ2≤ λ3≤ · · · →
∞ be the spectrum of the Dirichlet Laplacian L, {φk}∞k=1: Lφk = λkφkits eigen basis
in H normalized by (φk, φl) = δkl. The set of pairs
Σ := {λk; ∂νφk|∂Ω}∞k=1
is called the (Dirichlet) spectral data of the manifold Ω. The spectral inverse problem is IP 3: given the spectral data Σ, to recover the manifold Ω.
3.3
Reconstruction
Setting the goal to determine an unknown manifold from its boundary inverse data, one has to keep in mind the evident nonuiqueness of such a determination: all isometric mani-folds with the mutual boundary have the same data. Therefore, the only reasonable under-standing of "to recover" is to construct a manifold, which possesses the prescribed data [1].
An affirmative common feature of the problems IP 1–3 is that each kind of their data (i.e., R, M , and Σ) determines the minimal Laplacian L0up to unitary equivalence: see
[1], [2]. By this, each kind of data determines the wave spectrum ΩL0 up to isometry.
Assume that Ω is a simple manifold. Given its inverse data, one can • determine a relevant unitary copy eL0of the minimal Laplacian
• find its nest spectrum Ω
e
L0 and endow it with the distanceρe and hence get an isometric copy (Ω
e
L0,ρ) of the original manifold Ω. By isometry, thee copy has the same boundary data as the original. Thus, the inverse problems are solved.
The above described scheme of reconstruction elucidates the operator background of the boundary control method, which is an approach to inverse problems based upon their relations to system and control theory [1].
References
[1] M.I.Belishev. Recent progress in the boundary control method. Inverse Problems, 23 (2007), no 5, R1–R67.
[2] M.I.Belishev. A unitary invarian of semi-bounded operator and its application in inverse problems. http://www.arXiv:1004.1646v1[math.FA]9Apr2010
[3] M.S.Birman, M.Z.Solomyak. Spectral Theory of Self-Adjoint Operators in Hilbert Space. D.Reidel Publishing Comp., 1987.
[4] K.R.Davidson. Nest Algebras. Pitman Res. Notes Mthh. Ser., v. 191, Longman, London and New-York, 1988.
Paper B
Solving Ill-Posed Cauchy Problems in
Three Space Dimensions Using
Krylov Methods
Author: L. Eldén
Solving Ill-Posed Cauchy Problems in Three
Space Dimensions Using Krylov Methods
Lars Eldén Department of Mathematics, Linköping University, Linköping, Sweden. E-mail: laeld@mai.liu.se 17
18 Paper B Solving Ill-Posed Cauchy Problems Using Krylov Methods
1
A Cauchy Problem on a Cylindrical Domain
Let Ω be a connected domain in R2 with smooth boundary ∂Ω, and assume that L is a linear, self-adjoint, and positive definite elliptic operator defined in Ω. We consider the ill-posed Cauchy problem,
uzz− Lu = 0, (x, y, z) ∈ Ω × [0, z1],
u(x, y, z) = 0, (x, y, z) ∈ ∂Ω × [0, z1],
u(x, y, 0) = g(x, y), (x, y) ∈ Ω,
uz(x, y, 0) = 0, (x, y) ∈ Ω. (1)
The problem is to determine the values of u on the upper boundary, f (x, y) = u(x, y, z1),
(x, y) ∈ Ω.
In this talk we describe the main ideas behind the paper [6].
The problem (1) is ill-posed in the sense that the solution (if it exists), does not depend continuously on the data. It is a variant of a classical problem considered originally by Hadamard, and it is straightforward to analyze it using an eigenfunction expansion.
Since the domain is cylindrical with respect to z, we can use a separation of variables approach, and write the solution of (1) formally as
u(x, y, z) = cosh(z √
L)g. (2)
The operator cosh(z√L) can be expressed in terms of the eigenvalue expansion of L. Due to the fact that L is unbounded, the computation of cosh(z√L) is unstable and any data errors or rounding errors would be blown up, leading to a meaningless approximation of the solution.
The problem can be stabilized (regularized) if the operator L is replaced by a bounded approximation, see [3, 4, 5, 9, 10, 2], where similar problems were treated. Since it is the large eigenvalues of L (those that tend to infinity) that are associated with the ill-posedness, it would be natural to devise the following regularization method:
• Compute approximations of the smallest eigenvalues of L and the corresponding eigenfunctions, and discard the components of the solution (2) that correspond to large eigenvalues.
It is straightforward to prove that such a method is a regularization method in the sense that the solution depends continuously on the data. However, in the direct implementa-tion of such a method one would use unnecessarily much work to compute eigenvalue-eigenfunction approximations that are not needed for the particular data function g. Thus the main contribution of the paper [6] is a numerical method for approximating the regu-larized solution that has the following characteristics:
• The solution (2) is approximated by a projection onto a subspace computed by means of a Krylov sequence generated using the operator L−1. The hyperbolic cosine of the restriction of the operator L−1to that subspace is computed.
• At each step of the Krylov recursion, dealing with L−1 corresponds to solving
a well-posed two-dimensional elliptic problem involving L. Any standard (black box) elliptic solver, derived from the discretization of L, can be used.
L. Eldén 19
• The method takes advantage of the fact that the regularized solution operator is applied to the particular data function gm.
It is demonstrated in [6] that the proposed method requires considerably fewer so-lutions of two-dimensional elliptic problems, than the approach based on the eigenvalue expansion.
There are many engineering applications of ill-posed Cauchy problems, see [7, 8, 11] and the references therein. A standard approach for solving Cauchy problems of this type is to apply an iterative procedure, where a certain energy functional is minimized; a recent example is given in [1]. Very general (non-cylindrical) problems can be handled, but if the procedure from [1] were to be applied to our problem, then at each iteration four well-posed elliptic equations would have to be solved over the whole three-dimensional domain. In contrast, our approach for the cylindrical case requires the solution of only one two-dimensionalproblem at each iteration1. A few details of the method are described below.
2
Numerical Implementation
Assume that the problem has been discretized with respect to (x, y) so that now the el-liptic, self-adjoint operator L is represented by a symmetric matrix, for simplicity also denoted L. Solving the 2D elliptic problem Lv = w is equivalent to applying the the inverse v = L−1w. Using the Lanczos procedure we compute an orthonormal basis (q1, q2, . . . , qk) for the Krylov subspace
Kk(L−1, gm) = span(gm, L−1gm, L−2gm, . . . , L−(k−1)gm),
where gmis a discrete version of the data function in (1). This gives a low-rank
approxi-mation of L−1,
L−1Qk≈ QkTk,
where Tk is a tridiagonal matrix. It is very likely that some of the large eigenvalues of L
are rather well approximated by Tk. Therefore we compute the eigenvalue decomposition
of Tk, and replace those eigenvalues by zero, giving a “regularized” matrix bTk. We then
compute an approximation of (2) by projection to the k-dimensional Krylov subspace, uk(z) = Qkcosh( bTk†)Q
T kgm,
where bTk†is the Moore-Penrose pseudoinverse of bTk.
The procedure described can be generalized in a straightforward way to problems in more than three space dimensions.
References
[1] Andrieux, S., Baranger, T. N., and Abda, A. B. (2006). Solving Cauchy problems by minimizing an energy-like functional. Inverse Problems, 22:115–133.
1In one of our examples the two-dimensional problem had 8065 degrees of freedom and the
20 Paper B Solving Ill-Posed Cauchy Problems Using Krylov Methods
[2] Berntsson, F. and Eldén, L. (2001). Numerical solution of a Cauchy problem for the Laplace equation. Inverse Problems, 17:839–854.
[3] Eldén, L. (1995). Numerical solution of the sideways heat equation by difference approximation in time. Inverse Problems, 11:913–923.
[4] Eldén, L. and Berntsson, F. (1998). Spectral and wavelet methods for solving an inverse heat conduction problem. In International Symposium on Inverse Problems in Engineering Mechanics, Nagano, Japan.
[5] Eldén, L., Berntsson, F., and Regi´nska, T. (2000). Wavelet and Fourier methods for solving the sideways heat equation. SIAM J. Sci. Comput., 21(6):2187–2205. [6] Eldén, L. and Simoncini, V. (2009). Numerical solution of a Cauchy problem for an
elliptic equation by Krylov subspaces. Inverse Probl., 25(6):065002.
[7] Natterer, F. and Wübbeling, F. (1995). A propagation-backpropagation method for ultrasound tomography. Inverse Problems, 11:1225–1232.
[8] Natterer, F. and Wübbeling, F. (2005). Marching schemes for inverse acoustic scat-tering problems. Numerische Mathematik, 100:697–710.
[9] Regi´nska, T. and Eldén, L. (1997). Solving the sideways heat equation by a wavelet-Galerkin method. Inverse Problems, 13:1093–1106.
[10] Regi´nska, T. and Eldén, L. (2000). Stability and convergence of a wavelet-Galerkin method for the sideways heat equation. J. Inverse Ill-Posed Problems, 8:31–49. [11] Woodfield, P. L., Monde, M., and Mitsutake, Y. (2006). Implementation of an
ana-lytical two-dimensional inverse heat conduction technique to practical problems. Int. J. Heat Mass Transfer, 49:187–197.
Paper C
Dispersion Exponential Models
Author: A. Hassairi
Dispersion Exponential Models
Abdelhamid Hassairi
Laboratory of Probability and Statistics, Sfax Faculty of Sciences,
Sfax, Tunisia.
E-mail: abdelhamid.hassairi@fss.rnu.tn
24 Paper C Dispersion Exponential Models
1
Natural Exponential Families
1.1
Preliminaries
Let E be a linear space dimension d, let E∗be its dual E∗× E → IR : (θ, x) 7→ hθ, xi be the duality bracket. We denote by Ls(E∗, E) the space of the symmetric linear maps
from E∗to E . If µ is a positive Radon measure on E, we denote
Lµ : E∗→ ]0, +∞[ : θ →
Z
exphθ, xiµ(dx),
Θ(µ) = interior {θ ∈ E∗; Lµ(θ) < +∞} ,
kµ= logLµ
M(E) is the set of measures µ such that Θ(µ) is not empty and µ is not concentrated on an affine hyperplane of E. To each µ in M(E) and θ in Θ(µ), we associate the probability distribution on E
P (θ, µ)(dx) = exp{hθ, xi − kµ(θ)}µ(dx).
The set
F = F (µ) = {P (θ, µ); θ ∈ Θ(µ)}
is called the natural exponential family generated by µ. If µ and µ0 are in M(E), then F (µ) = F (µ0) if and only if there exists (a, b) in E × IR such that µ0(dx) = exp{ha, xi +
b}µ(dx). Therefore, if µ is in M(E) and F = F (µ),
BF = {µ
0
∈ M(E); F (µ0) = F }
is the set of basis of F.
The function kµ is strictly convex and real analytic. k0µ is a diffeomorphism between
Θ(µ) and its image MFcalled the domain of the means of F .
The inverse function of k0µ is denoted by ψµ and setting P (m, F ) = P (ψ(m), µ) the
probability of F with mean m, we have
F = {P (m, F ); m ∈ MF}.
The density of P (m, F ) with respect to µ is
fµ(x, m) = exp{hψµ(m), xi − kµ(ψµ(m))},
Now the covariance operator of P (m, F ) is denoted by VF(m). Clearly
VF(m) = k 00 µ(ψµ(m)) = (ψ 0 µ(m)) −1 ∈ L s(E∗, E).
The variance function m 7→ VF(m) characterizes the family F in the following sense:
If F and F0 are two NEFs such that VF(m) and VF0(m) coincide on a nonempty open
A. Hassairi 25
1.2
Influence of an Affine Transformation on a NEF.
Let ϕ(x) = δ(x) + γ where δ is in GL(E) and γ is in E, and let F = F (µ) be a NEF on E.
ϕ(F ) = F (ϕ(µ)), Mϕ(F )= ϕ(MF),
Vϕ(F )(m) = δ VF(ϕ−1(m))tδ.
1.3
Influence of Taking Powers of Convolution in NEFs
Let α be a positive number, not necessarily an integer. If µ is in M(E), let us introduce the Jorgensen set
Λ = {α > 0; there exists µαin M(E) | Θ(µα) = Θ(µ) and kµα(θ) = αkµ(θ)} .
Denoting Fα= F (µα) where α is in Λ, one has
MFα = αMF and VFα(m) = α VF
m α
.
1.4
Action of G = GL(IR
d+1) on NEFs on IR
dAn element g =hαγ βδiof G = GL(IRd+1) is defined by its blocs (α, β, γ, δ) in IR × IRd× IRd× L(IRd). We denote
dg(m) = α+ < β, m > et hg(m) = (dg(m))−1(γ + δ(m)).
We define the action of the group G on a NEF F by,
(TgVF) (m) = (dg(m)) −1 h0g(m)−1 VF(hg(m)) h0g(m) ∗ −1 .
If α = 1 et β = 0, the image F1of F by the affinity x 7→ δ(x) + γ satisfies
VF1= TgVF.
Also if α is in the Jorgensen set of F , then for g =α 0
0 I, Tg corresponds to the
Jor-gensen transformation of parameter α. In particular, when d = 1, the action of an element of GL(IR2) on a real NEF F is given by
(TgVF) (m) = (α + βm)3 (αδ − βγ) VF γ + δm α + βm .
Let G0the subgroup of G such that β = 0 and α > 0.
All the classifications of NEFs are done up to affine transformations and Jorgensen trans-formations, that is up to G0orbits.
26 Paper C Dispersion Exponential Models
2
Classifications of NEFs
2.1
Morris Class of Quadratic Real NEFs
The class of NEFs with quadratic variance functions contains exactly six G0orbits.
Name
M
FV
F(m)
Gaussian
IR
1
Poisson
]0 ; +∞[
m
Binomial
]0 ; n[
m − m
2Negative binomiale
]0 ; +∞[
m + m
2Gamma
]0 ; +∞[
m
2Hyperbolic-cosine
IR
1 + m
22.2
Letac Mora Class of Cubic Real NEFs
The class of NEFs with variance functions of degree three contains six G0orbits.
Type µ VF Abel P∞ k=0p(p + k) k−1 δk k! m(1 + m p) 2 Takàcs P∞ k=0ap Qk−1 j=1(a(p + k) + j) δk k! (m + m2 p ) 1 +a+1 a m p Strict arcsine P∞ k=0p (∗) k (p) δk k! m(1 + m2 p2) Large arcsine P∞ k=0 p p+kp (∗) k (a(p + k)) δk k! m +2m2 p + a2+1 a2 m3 p2 Ressel p xΓ(x+p+1)x+p−1e−x1[0,+∞[ m 2 p (1 + m p)
Inverse Gaussian x−32exp
−2xp2√p 2π1[0,+∞[ m3 p2 (∗)p2n(a) =Qn−1k=0(a2+ 4k2) et p2n+1(a) =Qn−1k=0(a2+ (2k + 1)2
2.3
Simple Quadratic NEFs. (Casalis)
The multivariate version of the Morris class is the class of NEFs on IRd with variance functions of the form
VF(m) = a m ⊗ m + B(m) + C,
where m ⊗ m (θ) =< θ, m > m and B.
The class of simple quadratic NEFs on E, contains (2d + 4) G0−orbits:
(d + 1) Poisson-Gaussian G0−orbits,
(d + 1) negative multinomial-gamma G0−orbits,
a multinomial G0−orbit,
an hyperbolic G0−orbit built from particular mixtures of families of normal, Poisson,
A. Hassairi 27
2.4
Simple Cubic NEFs. (Hassairi)
The twelve G0−orbits of the Letac-Mora are divided into 4 G-orbits
Gaussian Inverse Gaussian
1 m3
Poisson Gamma Abel Ressel m m2 m(1 + m)2 m2+ m3 Binomial Negative-Binomial Takàcs
m(1 − m) m(1 + m) m(1 + m) 1 + 1+aa m a > 0
Hyperbolic Large arcsine Strict arcsine 1 + m2 m1 + 2m +1+a2
a2 m
2 m + m3
a > 0
Let F be simple quadratic NEF and g =hαγ βδiin the group G. Then TgVF is a
polynomial in m of degree ≤ 3 VF1 = ( a m ⊗ m + [I + m ⊗ β][B(m) + (< β, m > +1)C] ) [I + β ⊗ m].
The obtained NEF F1= TgF is said simple cubic.
The class of simple cubic NEFs on IRdis distributed into 3d + 1 G-orbits
References
[1] Casalis, M. (1996). The 2d+4 simple quadratic natural exponential families on IRd. Ann. statist.24, 1828–1854.
[2] Feinsilver, P. (1986). Some classes of orthogonal Polynomials associated with mar-tingales. Proc. A.M.S. 98, 298–302.
[3] Hassairi, A. and Zarai, M. (2004). Characterization of the cubic exponential families by orthogonality of polynomials. Ann. Probab. 32, 2463–2476.
[4] Hassairi, A. and Zarai, M. (2006). Characterization of the simple cubic multivariate exponential families. Journal of Functional Analysis. 235, 69–89.
[5] Letac, G. and Mora, M. (1990). Natural real exponential families with cubic variance functions. Ann. statist. 18, 1–37.
[6] Morris, C.N. (1982). Natural exponential families with quadratic variance functions. Ann. statist.10, 65–82.
Paper D
An Inverse Problem in Glaciology
Authors: V. Kozlov
This talk is based on joint work with S. Avdonin, D. Maxwell, and M. Truffer [1].
An Inverse Problem in Glaciology
Vladimir Kozlov Department of Mathematics, Linköping University, Linköping, Sweden. E-mail: vladimir.kozlov@liu.se 3132 Paper D An Inverse Problem in Glaciology
Many problems in geophysics are ill-posed. For example, there is no known method to measure basal velocities, but surface velocities can be measured directly on the ground or by a variety of remote sensing methods. Basal velocities must then be found through inverse methods. We will consider an ice flow model suggested in [2]. They treated a first order model of planar ice flow along a longitudinal cross section of a glacier, and showed that the longitudinal velocity component obeys a non-linear Poisson equation.
Let Ω be the domain in the xy−plane with Lipschitz boundary, which consists of the upper boundary S = (0, l), a surface of the ice sheet, and B, a bottom, connecting end points of S and lying below S. We consider the following system of equations in Ω:
−∇ · (G(|∇u|)∇u) = f in Ω (1)
∂yu|S = 0 (2)
where f is a given function. We suppose that
G(t) =F
0(t)
t ,
where F is a convex function for t ≥ 0, F0(0) = 0 and F0 is Lipschitz continuous, i.e. |F0(t) − F0(τ )| ≤ C|t − τ |. For such F , F00(t) exists almost everywhere and we suppose
that
ν(1 + t)1n−1≤ F00(t) ≤ µ(1 + t) 1
n−1 (3)
for some positive ν, µ and n ≥ 1.
A finite viscosity version of Glen’s flow law, which is often used in glaciology (e.g. [3]), gives an example of the function G(t) as the solution of the equation
1 G(s) =T 2 0 + G 2(s) s2n−12 with T06= 0. Then F (t) = t Z 0 G(τ ) τ d τ
is strictly convex and satisfies (3), see [2].
Conditions (1)–(2) do not distinguish the unique solution of equation (1), so we add an additional condition on S which can be interpreted as a surface measurements:
u|S = ϕ (4)
Problem (1)–(2), (4) is ill-posed: its solution exists not for every (even smooth) functions ϕ and one cannot expect continuous dependence of the solution on ϕ.
The alternative method for solving ill-posed linear problem was introduced in [4] and [5]. For our nonlinear problem it runs as follows. The approximation u0is found by
solv-ing the boundary value problem (1), (2) supplied with the Dirichlet boundary condition on B: u|B = ψ , where ψ is an arbitrary function from an appropriate Sobolev space on
B. In order to find the (k + 1)th iteration one must solve the nonlinear equation (1) for u = uk+1supplied with the Dirichlet-Neumann (or Neumann-Dirichlet, depending on k
V. Kozlov 33
is odd or even) boundary conditions on S and B. The boundary condition on B is taken from the given Cauchy data and the boundary condition on S is taken from the previous iteration step. It is possible some variations in this procedure, for example instead of the nonlinear equation (1) we can solve the (linear) equation
−∇ · (G(|∇uk|)∇uk+1) = f (5)
with the same boundary conditions.
Our numerical experiments with the nonlinear problem (1), (2), (4) indicate that this procedure appears to have similar convergence properties as for the linear case.
A Landweber-type iterative procedure for solving the above non-linear problem is also discussed.
References
[1] Avdonin, S.; Kozlov, V.; Maxwell, D.; Truffer, M. Iterative methods for solving a nonlinear boundary inverse problem in glaciology. J. Inverse Ill-Posed Probl. 17 (2009), no. 3, 239–258.
[2] J. Colinge and J. Rappaz, A strongly nonlinear problem arising in glaciol-ogy, M2AN Math. Model. Numer. Anal. 33 (1999), no. 2, 395–406.
[3] W.S.B. Paterson, The physics of glaciers. 3rd edition. Oxford, New York, Tokyo, Pergamon, 1994.
[4] Kozlov, V. A. and Maz’ya, V. G., On iterative procedures for solving ill-posed boundary value problems that preserve differential equations, Algebra Analiz, 1 (1989), 144–170. English transl.: Leningrad Math. J. 1 (1990), 1207–1228.
[5] Fomin, A. V., Kozlov, V. A., and Maz’ya, V. G., An iterative method for solving the Cauchy problem for elliptic equations, U.S.S.R. Comput. Math. and Math. Phys., 31 (1) (1991), 45–52.
Paper E
Key Issues of Modelling Extreme
Floods
Authors: V. Kuzmin, U. Korotygina, I. Makin, D. Rumyantsev and A. Surkov
Key Issues of Modelling Extreme Floods
Vadim Kuzmin∗, Ulyana Korotygina, Ivan Makin, Denis Rumyantsev and
Aleksandr Surkov
Russian State Hydrometeorological University, St. Petersburg, Russia.
∗E-mail: vknoaa@hotmail.com
38 Paper E Key Issues of Modelling Extreme Floods
In extreme floods modelling and forecasting, a large number of mathematical issues usu-ally arise. Most of them are associated with solution of the incorrect inverse problem, which should be more or less efficiently solved for identifying the applied model pa-rameters. Efficient parameters are those providing acceptable accuracy of floods forecast that allow undertaking various preventive measures to decrease or even avoid significant losses.
Let us consider in details such issues as
• high uncertainty of the model input data and data used for evaluation of the model output;
• selection of the effective and efficient forecasting model; • selection of an objective function;
• developing objective methods of smoothing a response surface;
• selection of a simple yet efficient optimization procedure, which can be applied in conjunction with methods of handling the above issues.
The first issue in our list is high uncertainty of the observational data obtained from the national hydrometeorological networks. Often, such data are quite inconsistent or have irregular spatial and temporal resolution; stochastic structure of errors associated with data collection and processing also can vary in time and space; even in ”successful” countries, available data sets are incomplete in terms of their ability to satisfy our un-derstanding of the modeled hydrometeorological processes. This issue generally means that hydrologists cannot use physically and mathematically rigor models that consist of some number of differential equations, because these equations cannot be supplied with necessary data. Alternatively, hydrologists have to use so-called conceptual model, which describe the modeled process less rigorously, however, they can be forced by using avail-able data. Thus, selection of an appropriate model, in fact, is searching for a compromise between our understanding of the modeled process and ability to obtain something useful from available observations.
That is why correct selection of the best forecasting model, which can guarantee that issued forecasts are accurate enough, becomes a serious problem. Formally, the most efficient model should be that providing the highest success rate (say, in terms of Nash-Sutcliff criteria [2]) estimated for an independent validation period. However, nobody can guarantee that such a model will perform well in case of natural disasters (even if some disasters were involved in the model calibration). Our experience shows that the most reliable and robust model should be efficient for the entire spectrum of possible scenarios of the modeled process. In this case, the model robustness means that structure of the considered model and its parameters should not be too sensitive to the forcing data.
Selection of an appropriate objective function represents another problem. In gen-eral, a hypothetically ”perfect” objective function should provide the absolute fit of the modeled and actual (observed) processes, what never happens due to model and data un-certainties. Thus, we do not pretend to get the perfect fit of two or more plots and agree to get some compromise. It’s quite interesting, that matrices of losses caused by incor-rect forecasts are, perhaps, the worst objective function for the model calibration and, certainly, the best criterion to evaluate the model performance.
V. Kuzmin et al. 39
Uncertainty in the model and data and various disadvantages of the selected objective function make the response function’s bottom extremely irregular. The forecasting model calibration becomes a quite rough attempt to find out a unique solution of an essentially incorrect inverse problem. Often, hydrologists try to smooth the response surface by applying various ways of regularization, penalty functions etc. After all, the modified surface becomes more or less smooth, yet quite useless, because any physical sense of the response surface is lost.
Finally, regardless of our success in making the response surface more smooth and regular, we still need to find out a physically correct optimum parameter set, and now we encounter another problem: most of hydrological models have approximately 8-20 parameters, hence, all the said above was about multidimensional parameter space. If so, any, even quite simple procedure becomes more complicated, because even simple algorithms now require tremendous processor resources. In addition, local minima are not stable in time (to imagine the response surface behaviour, we often use a visual model of a ”floating udder”).
In our opinion, all the described problems cannot be resolved separately. Let us de-scribe several innovations which allow us to reach certain success in handling them.
First, we select a model or an optimum set of N parameters by using indices of F -robustness, which reflect relative stability of the local optimum’s neighbourhood. In gen-eral, F -index is defined as a (N + 1)-dimensional integral of the response surface in the interval [Pi− r; Pi+ r], (where Pidenotes an examined parameter set). In the simplest
case, N -dimensional F -index FN is the average value of an objective function in radius
r around Pi. Fncan be found by using the following equation:
Fn,r =
P +r
Z
P −r
J dP, (1a)
where J denotes the objective function and r denotes radius of the objective function averaging (F-radius). In practical tasks, when function J (P ) is discrete, index Fn can be interpreted as the average value of J (P ) within a given distance from the inspected parameter set (for example, 1, 2, 3 or more steps in each direction):
Fn= 1 n i=n X i=1 j=s+1 X j=1 Ji,j/(2s + 1) , (1b)
where s denotes a number of steps made in each direction, and Ji,jis the objective
func-tion for n parameters and s-step radius around each of them. In this case, any small shifts of ”pits” do not cause significant worsening of corresponding objective function values. As for the model selection, the best model should have the smallest F-index for the effec-tive region of used parameters. If an alternaeffec-tive or potentially suitable model or parameter sets have a ”deeper” optimum, but greater F-index, they should be declined (Figure 1).
Second, we perform ”natural” smoothing of the response surface by using multi-scale objective functions (MSOF). The particular objective function used in this work has the
40 Paper E Key Issues of Modelling Extreme Floods following form: J = v u u t n X k=1 σ1 σk 2 mk X i=1 (qo,k,i− qs,k,i(X)) 2 , (2)
where qo,k,iand qs,k,idenote the observed and simulated flows averaged over time
inter-val k (i.e. the k-th aggregation scale), σk denotes the standard deviation of discharge at
that scale, n denotes the total number of scales used, and mkis the number of ordinates at
the scale k. In this work, we used hourly, daily, weekly and monthly scales corresponding to k = 1, 2, 3 and 4, respectively. A value of k may not be restricted and can be assigned any number of scales (in this case, we get so-called all-scales objective function (ASOF). Note in Eq. (2) that the weight associated with each term is given by the inverse of the standard deviation of the flow at the respective scales. This weighting scheme assumes that the uncertainty in modeled streamflow at each scale is proportional to the variability of the observed flow at that scale. Another important motivation for using the multi-scale objective function (MSOF) is that it smoothes the objective function surface, and hence reduces the likelihood of the search getting stuck in tiny ’pits’ [1].
Finally, after all the described procedures, we apply a quasi-local optimization in a physically predetermined (or predefined) region of the parameter space by using a step-wise line search algorithm (SLS [1]). If the model input data are very uncertain, an enhanced configuration of the SLS algorithm is used: we apply a random generator to produce a number of ensembles of the input time series and perform model calibration for all the scenarios. The best (unbiased) parameters are those corresponding the smallest MSOF value.
The described measures allow us significant reducing of the mentioned issues im-pact and successful implementing the described approach for extreme floods forecasting in data scarce regions. The performed research was supported by the Federal Purpose-Oriented Program of the Russian Federation (Project P1103 ”Developing technologies of the catastrophic river floods forecasting in NW Russia and making decisions”).
References
[1] Nash, J.E. and Sutcliffe J.V., River flow forecasting through conceptual models part I - A discussion of principles, Journal of Hydrology, 1970, No. 10 (3), P.282-290. [2] Kuzmin, V. A., Seo, D.-J., and Koren, V. I., Fast and Efficient Optimization of
Hy-drologic Model Parameters Using a Priori Estimates and Stepwise Line Search (SLS), Journal of Hydrology, 2008, No. 353, P.109-128.
V. Kuzmin et al. 41
Figure 1: Illustration of robust and non-robust phases: if the added parameter pro-vides a ”deeper” optimum, but greater F -index, it should be declined, because the model becomes more sensitive to the model input data
Paper F
The Inverse Problem for the Wave
Equation and Two Acquisition
Geometries
Authors: M. Lassas and L. Oksanen
The Inverse Problem for the Wave Equation
and Two Acquisition Geometries
Matti Lassas and Lauri Oksanen
Department of Mathematics and Statistics, University of Helsinki,
Helsinki, Finland.
E-mail: {matti.lassas, lauri.oksanen}@helsinki.fi
46 Paper F The Inverse Problem for the Wave Equation and Two Acquisition Geometries
1
Introduction
We consider the inverse problem of determining the spatially varying wave speed inside a domain from measurements of acoustic pressure wave fields on the boundary of the do-main. The measurements are modelled as an operator on the wave fields between various source-receiver pairs. The arrangement of the source-receiver pairs is called the acquisi-tion geometry.
In seismic imaging, the wave fields are often stimulated by explosions, [25, 23]. In such a case receivers need to be far away from the sources. For acquisition geometries commonly employed in seismic imaging see [22].
We study two acquisition geometries with disjoint sources and receivers, and show that the inverse problem for the wave equation with these acquisition geometries has unique solution.
Inverse problems where data is given on a part of the boundary are widely studied, the Calderón’s inverse problem for the conductivity equation and the closely related inverse problem for the Schrödinger equation being the paradigm problems, [6, 24, 21, 1], [4, 9]. The inverse problem for Schrödinger equation on a bounded domain of Rn, n ≥ 3,
with receivers on an open subset Γ of the boundary is solved in [17]. The inverse problem for Schrödinger equation on a bounded domain of Rn, n = 2, with sources and receivers
on the same open subset Γ of the boundary is solved in [10]. The corresponding problem on a compact Riemannian surface is solved in [8]. For related results with measurements on a part of the boundary, see [5, 7, 11].
The inverse problems for the wave equation and for more general hyperbolic equations are studied in [2, 3, 20, 12, 15]. The inverse problem for the wave equation with sources and receivers on the same open subset Γ of the boundary is solved in [13], see also [18, 16].
2
The Inverse Problem for the Wave Equation
Let M be a bounded, connected and open set in Rn, n ≥ 2, with C∞-smooth boundary
∂M , and let c be a strictly positive C∞-smooth function on M . Let us consider an inverse problem for the wave equation
∂t2v − c(x)2∆v = 0 in M × (0, ∞), (1) v|∂M ×(0,∞)= f,
v|t=0= ∂tv|t=0= 0.
Denote by vf(x, t) = v(x, t) the solution of (1) for f ∈ C∞
0 (∂M × (0, ∞)), and
define the hyperbolic Dirichlet-to-Neumann operator
Λ : C0∞(∂M × (0, ∞)) → C∞(∂M × (0, ∞)), Λf := c−1∂νvf|∂M ×R+,
where ∂ν is the Euclidean normal derivative on ∂M . Denote by ΛTΓ1,Γ2the restriction of
the Dirichlet-to-Neumann operator ΛTΓ
1,Γ2: C
∞
M. Lassas and L. Oksanen 47
where Γ1, Γ2⊂ ∂M are open. Furthermore, denote ΛΓ1,Γ2 := Λ
∞ Γ1,Γ2.
Here ΛTΓ1,Γ2 for a certain collection G of pairs of nonempty open sets (Γ1, Γ2) can
be interpreted as a continuum idealiazation of measurements of acoustic waves. The collection G defines the acquisition geometry. The inverse problem for the wave equation (1) with the acquisition geometry G is to determine the wave speed c(x), x ∈ M , given the domain M and the data
ΛTΓ1,Γ2, (Γ1, Γ2) ∈ G. (2)
The operator Λ determines the wave speed c, [2], that is, if the acquisition geometry puts no restrictions on the inverse problem, the problem has a unique solution. Even if the wave speed is anisotropic, the Dirichlet-to-Neumann operator determines it up to a change of coordinates, [3]. Moreover, if there are sources and receivers operating simultaneously in a same place, that is if there is Γ ⊂ ∂M such that (Γ, Γ) ∈ G, then the wave speed c is uniquely determined, [13].
In [19] we considered the inverse problem for a related geometric problem with two acquisition geometries where no sources and receivers operate simultaneously in the same places. There, the inverse problem is studied on a Riemannian manifold (M, g) and the operator c2∆ in equation (1) is replaced by the operator ∆
g+ q, where ∆gis the
Laplace-Beltrami operator on (M, g) and the potential q is a C∞-function on M. It was shown that the boundary data related to ∆g+ q determine (M, g) up to a change of coordinates.
Using the above result, we will here prove the uniqueness for the inverse problem for the equation (1). This is done by showing that the data (2) determines the related data for a gauge equivalent operator of the type ∆g+ q and by observing that the obtained
Riemannian manifold (M, g) can be embedded in a unique way in the domain M ⊂ Rn.
3
Two Acquisition Geometries with Disjoint Sources
and Receivers
We consider here only the case with T = ∞. The finite time case studied in [19] can be adapted for the equation (1) in a similar manner. In the two theorems below the set M is fixed and known.
Theorem F.1
If(Γ1, Γ2) ∈ G and Γ1∩ Γ26= ∅, then the data (2) and c|∂M determine the wave speed
c(x), x ∈ M , uniquely. Theorem F.2
Let (Γ1, Γ2), (Γ1, Γ3), (Γ2, Γ3) ∈ G. Then the data (2) and c|∂M determine the wave
speedc(x), x ∈ M , uniquely.
Let us describe the proof Theorem F.1. We may write the operator c2∆ in equation (1) as a weighted Laplace-Beltrami operator
c2∆v = µ−1|g|−1/2 n X j,k=1 ∂ ∂xj µ|g|1/2gjk ∂ ∂xkv ,
48 Paper F The Inverse Problem for the Wave Equation and Two Acquisition Geometries
where the Riemannian metric g := (gjk)nj,k=1= (c −2δ
jk)nj,k=1in the coordinates of R n,
the weight µ = cn−2, |g| denotes the determinant of g and (gjk)nj,k=1is the inverse of g. Let us denote κ := c(n−2)/2. The gauge transformation of c2∆,
Av := κc2∆(κ−1v),
is of form ∆g+ q by [14] [KKL, pp. 257-258]. Let us denote the Dirichlet-to-Neumann
operator corresponding to A by eΛ. By [KKL, pp. 202-203],
e
Λf = κΛ(κ−1f ) + (∂νκ)κ−1f.
As c|∂M and the restriction ΛΓ1,Γ2are known, so are κ|∂Mand the corresponding
restric-tion of eΛ.
Hence (M, g) is determined as an abstract Riemannian manifold by Theorem 1 of [19]. This means that we know the metric g up to a change of coordinates. Using [KKL, Lemma 4.46 and the proof of Lemma 4.45] we obtain c in the original coordinates of Rn.
Here it is crucial to assume that c|∂M is known.
The proof of Theorem F.2 is similar with the above proof.
References
[1] Astala, K. and Päivärinta, L. (2006). Calderón’s inverse conductivity problem in the plane. Ann. of Math. (2), 163(1):265–299.
[2] Belishev, M. I. (1987). An approach to multidimensional inverse problems for the wave equation. Dokl. Akad. Nauk SSSR, 297(3):524–527.
[3] Belishev, M. I. and Kurylev, Y. V. (1992). To the reconstruction of a Riemannian manifold via its spectral data (BC-method). Comm. Partial Differential Equations, 17(5-6):767–804.
[4] Bukhgeim, A. L. (2008). Recovering a potential from Cauchy data in the two-dimensional case. J. Inverse Ill-Posed Probl., 16(1):19–33.
[5] Bukhgeim, A. L. and Uhlmann, G. (2002). Recovering a potential from partial Cauchy data. Comm. Partial Differential Equations, 27(3-4):653–668.
[6] Calderón, A.-P. (1980). On an inverse boundary value problem. In Seminar on Nu-merical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), pages 65–73. Soc. Brasil. Mat., Rio de Janeiro.
[7] Greenleaf, A. and Uhlmann, G. (2001). Local uniqueness for the Dirichlet-to-Neumann map via the two-plane transform. Duke Math. J., 108(3):599–617.
[8] Guillarmou, C. and Tzou, L. (2009). Calderon inverse problem with partial data on riemann surfaces.
M. Lassas and L. Oksanen 49
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Paper G
The Multiplicity of Positive Solutions
for Boundary Value Problems to
Quasilinear Equations: A Small
Survey
Author: A. I. Nazarov
The Multiplicity of Positive Solutions for
Boundary Value Problems to Quasilinear
Equations: A Small Survey
Alexander I. Nazarov
Department of Mathematics and Mechanics, St. Petersburg State University,
St. Petersburg, Russia. E-mail: an@AN4751.spb.edu
54 Paper G The Multiplicity of Positive Solutions for Boundary Value Problems
We begin with the Dirichlet problem to a simple quasilinear equation in an annulus Ω = BR2\BR1 ⊂ R n: −∆pu = uq−1 in Ω, u = 0 on ∂Ω u > 0 in Ω, (1)
(here ∆pu ≡ div |∇u|p−2∇u is p-Laplacian; we always assume that n ≥ 2 and 1 <
p < ∞). This equation describes, e.g., reaction-diffusion processes in chemistry and biology.
The classical case p = 2 corresponds to conventional linear diffusion.
The non-uniqueness of solutions to semilinear BVPs was investigated by many au-thors beginning from the classical paper [1]. The phenomenon of multiplicity was dis-covered by Coffman [2]. Namely, it turned out that, for n = 2, p = 2 and q > 2, sufficiently thin annulus admits arbitrary many different (rotationally nonequivalent) so-lutions of the problem (1). This result was generalized for n ≥ 4 in [3] and for n = 3 (the most difficult case!) in [4]. We underline that for Ω = BRthis phenomenon does not
occur, the solution to (1) is unique, as it was established in a remarkable paper [5]. For arbitrary 1 < p < ∞ the multiplicity in (1) was established in [6] for n = 2, in [7] for n ≥ 4 and in a recent paper [8] for n = 3.
Remark G.1 (Remark 1). For the slow reaction (1 < q ≤ p) there exists a unique solution to (1) for any bounded domain Ω!
We set ε := R2
R1 − 1 and denote by p
∗
nthe Sobolev critical exponent in Rn:
1 p∗ n = 1 p− 1 n.
We introduce the so-called (m, k)-decomposition of the space Rn
= (Rm)`⊕ Rk, where
m ≥ 2 and either k = 0 or k ≥ m. Denote by x = (y1, . . . , y`; z) points in Rn; here ys
belong to Rms while z ∈ Rk. Note that a nontrivial decomposition (m < n) is admissible
only if n ≥ 4. Theorem G.1 ([7])
Let 1 < p < q < ∞. Then there existsε depending only on p and q such that forb anyε < ε and any (m, k)-decomposition satisfying q < pb ∗n−m+1 there exists a weak solution of the problem (1) depending only on|y1|, . . . , |y`|, |z|. Moreover, for different
pairs(m, k) corresponding solutions are different (rotationally nonequivalent). Remark G.2 (Remark 2). Note that p∗n−m+1 ≥ p∗
n, and strict inequality holds if p < n.
In particular, a radial solution to (1) exists for all q < ∞. We recall that for Ω = BRthe
problem (1) has no solution if q ≥ p∗n, see [9] for p = 2.
Theorem 1 does not give multiplicity since the number of different (m, k)-decomposi-tions of Rnis limited. The next theorem provides multiplicity in the thin layer for n 6= 3. Theorem G.2 ([7])
Let1 < p < ∞, p < q < p∗
n. For anyt0∈ N there existsε, depending only on p, q and tb 0, such that for anyε <bε, for all (2, k)-decompositions and for all 2 ≤ t ≤ t0, the problem
A. I. Nazarov 55
(1) has a weak solutionub2,k,t(y1, . . . , y`; |z|) symmetrical with respect to arguments yj
and2π/t-periodic with respect to polar angle of yj ∈ R2. Moreover, for different pairs
(k, t) corresponding solutions are different. The only exception may be t = 2, t0 = 4, 2k0− k = n.
Corresponding result for n = 3 is more difficult to prove and reads as follows. Theorem G.3 ([8])
Letn = 3, 1 < p < ∞, p < q < p∗3. For anyt0 ∈ N there existsε, depending onlyb
onp, q and t0, such that for anyε < ε and for all 2 ≤ t ≤ tb 0, the problem (1) has a weak solutionbu2,k,t(x1, x2; |x3|), 2π/t-periodic with respect to polar angle of (x1, x2).
Moreover, for differentt corresponding solutions are different.
Remark G.3 (Remark 3). For even n ≥ 4 S. Kolonitskii recently proved multiplicity of solutions to (1) in thin layer under assumption p < q < p∗n−1only. These solutions have more complicated symmetries.
Now we discuss the ”complementary” effect of multiplicity for fixed layer thickness and large q.
Theorem G.4 ([7])
Let n be even, n ≤ p < ∞, t0 ∈ N. There existsbq depending only on p, ε and t0 such that for anyq < q < ∞ and t ≤ tb 0the problem (1) has a weak positive solution
b
u2,0,t(y1, . . . , yn/2) symmetrical with respect to arguments yj and 2π/t-periodic with
respect to polar angle ofyj. Ift 6= t0thenub2,0,tis non-equivalent toub2,0,t0. The results of Theorems 1, 2, 4 are generalized for the problem
− div a(∇u) = f (u) in Ω, u > 0 in Ω, u = 0 on ∂Ω,
where a(σ) ≡ ∇A(σ) and, roughly speaking, function A(σ) is ”not more convex” then |σ|p, and function F (s) =Rs
0f (u)du is ”more convex” then s p.
Let us explain briefly the basic ideas of proofs. The multiplicity in the thin layer is proved by a priori estimates. Namely, we consider the energy functional
J (u) =k∇ukLp(Ω)
kukLq(Ω)
and show that energy levels split for sufficiently small ε (for n = 3 the method is more complicated). In Theorem 4 we proceed by the method of local bifurcations. Namely, we show that the second differential of energy functional is not positive for sufficiently large q.
A. Shcheglova [10] transferred these results to nonlinear Neumann boundary value problem of the type
−∆pu + up−1 = 0 in BR |∇u|p−2h∇u; ni = uq−1 on ∂B 1 u > 0 in BR, (2)
