On the holomorphic recovery and some

applications to parabolic problems

C. Palencia (University of Valladolid)

BCAM, December 10, 2012

In the first part I briefly present a numerical algorithm for the

recovery of a holomorphic mapping from knowledge of approximate

values of it at given nodes. In the second part, the algorithm

is used to numerically solve a couple of ill-posed problems: the

backwards and the sideways heat equation. I would rather to

comment on the main ideas, thus skipping many details and

comments.

Most of the contributions were in collaboration with J. M. Marban.

1

A remark on the maximum principle. Let Ω ⊂ C an open and

bounded domain with Lipschitz boundary Γ = Γ1 ∪ Γ2, with Γjclosed and Γ1 ∩ Γ2 = ∅. On the other hand, let f : Ω ∪ Γ→ C a

continuous mapping, holomorphic in Ω, and set Mj = maxz∈Γj |f(z)|.The classical maximum principle says that

|f(z)| ≤ maxM1,M2, z ∈ Ω.

2

In fact, the Two Constants Theorem guarantees that

|f(z)| ≤M1−ω(z)1 M

ω(z)2 , z ∈ Ω,

where ω : Ω ∪ Γ is the so called harmonic measure of Γ2 w.r.t Ω:

the harmonic mapping in Ω with boundary values

ω(z) = 0, z ∈ Γ1 and ω(z) = 1, z ∈ Γ2.

This result can be extended in different ways (unbounded domains

under growth conditions, overlapping Γj)and is in the heart of

the main estimates in the topic we are considering today.

3

The Analytic Recovery Problem. I proceed to briefly describe

the numerical algorithm in J. M. Marban and C. P. (Numer.

Math. 2002) for the recovery of holomorphic mappings, specialized

to the choice of Chebyshev nodes.

Fix 0 < r < 1 and set I = [−r, r]. For N ≥ 1, let sn, 1 ≤ n ≤ N ,

be the Chebyshev nodes of first kind over I:

(1) sn = −r cos

((2n− 1)π

2N

), 1 ≤ n ≤ N.

4

Let D ⊂ C be the open unit disc and denote by H2(D) the Hardy

space, formed by all the holomorphic mappings f : D → C such

that

‖f‖22 = sup0<R<1

1

2π

∫ 2π

0|f(Reiθ)|2dθ < +∞.

Given f ∈ H2(D), it is well known that the radial limits

f∗(eiθ) = limR→1−

f(Reiθ)

exist for almost every θ ∈ [0,2π], that f∗ ∈ L2([0,2π]), and that

‖f‖22 =1

2π

∫ 2π

0|f∗(Reiθ)|2dθ.

For M ≥ 0, BM stands for the closed ball in H2(D), centered at

the origin, of radius M .

5

Let f ∈ H2(D) and set w = wnNn=1 := f(sn)Nn=1 ∈ CN . Given

perturbed nodal values w + δw = wn + δwnNn=1 ∈ CN , the goal

is to recover f from knowledge of the approximate values w + δw.

We also assume that |δwn| ≤ ρ, 1 ≤ n ≤ N , and that f ∈ BM ,

where ρ and M are a priori known.

6

Let SN ⊂ H2(D) be the linear space generated by the Cauchy

kernels

Kn(s) =1

1− sns, s ∈ D, 1 ≤ n ≤ N.

The recovery of f proposed in is given by the least squares

method (LSM) as F ∈ SN satisfying

(2)

F =N∑n=1

λnKn = arg minG∈BM⋂SN

N∑n=1

|G(sn)− (wn + δwn)|2.

Working on a suitable orthonormal basis, problem (2) can be

efficiently be solved by means of the SVD.

7

Set

τ =r

1 +√

1− r2, M∗ =

4M(1 + r)

1− r2, ξ = 1 +

ln(ρ/M∗)

ln τ.

A result in MP shows that if ρ < M∗ and N = [ξ], then for s ∈ Dwe have

(3) |f(s)− F (s)| ≤ 3(Mγ(s))1−ω(s)(Nρ)ω(s), s ∈ D.

where

γ(s) = (1 + |s|)(1− |s|)−1(1− ω(s))−1,

and ω : cl (D)→ [0,+∞) is the harmonic measure of I with respect

to D \ I, i.e., the continuous mapping in cl (D) that is harmonic in

D \ I and such that ω(s) = 1, for s ∈ I, and ω(s) = 0 for |s| = 1.

8

Backwards parabolic problems (J.M. Marban & C. P. SINUM,

2002). Let A : D(A) ⊂ X → X be the infinitesimal generator of

a C0, holomorphic semigroup S(t), t ≥ 0, of linear and bounded

operators in a Banach space X. Without loss of generality,

we are assuming that the semigroup is also bounded. This all

means that, for some angle θ ∈ (0, π/2), the semigroup admits a

holomorphic extension to the sector

Σθ = z ∈ C : |arg(z)| < θ

and that for some Cθ > 0 the extension (denoted again by S)

satisfies

‖S(z)‖ ≤ Cθ, θ ∈ Σθ.

9

It is well-known that the backward parabolic problemu′(t) = Au(t), 0 ≤ t ≤ T,u(T ) = uT given in X,

is, in general, an ill-posed problem.

We assume that uT ∈ R(S(T )) (here R stands for the range of)

but, on the other hand, we are given an observed approximate

datum uT + δuT , with δT ∈ X and the only information we have

is that ‖δuT‖ ≤ ε, for a known ε > 0.

10

Notice that:

a) The value uT + δuT likely does not belong to R(S(T )).

b) The uncertainty δuT likely propagates uncontrolled for 0 < t < T .

To make some progress we need a sort of regularizing hypothesis:

we will assume that the ideal initial datum u0 = u(0) is a priori

bounded by some known quantity, i.e.

‖u(0)‖ ≤M.

This a priori bound regularized the problem, since there holds

the next:

11

Theorem. Assume that uj, j = 1, 2 are two solutions of the IVP

corresponding to A such that ‖uj(0)‖ ≤M . Then

‖u2(t)− u1(t)‖ ≤ Cθ(2M)1−ω(z)‖u2(T )− u1(T )‖ω(z), 0 ≤ t ≤ T,

where ω is the harmonic measure of Γ1 w.r.t. Ω defined in in

the next figure.

12

Domain Ω and Γ1 in the theorem. The angle θ′ is chosen

in (0, θ).

13

Algorithm for the backward parabolic problem. Select R > T

and set

Σ = Σθ,r = z ∈ Σθ : |z| < R.

Given 0 < r < 1, let Ψ : Σ→ D be the conformal transformation

such that Ψ(0) = −1, Ψ(R) = 1 and Ψ(T ) = −r.

For N ≥ 1, let −r < s1 < s2 < · · · sN < r be the Chebyshev nodes

on [−r, r] and, finally set tn = Ψ−1(sn), 1 ≤ n ≤ N . Note that

T < t1 < t2 < · · · tN < T ′ = Ψ−1(r) < R.

14

Domain Σ and future nodes tn

15

The idea of the algorithm for the backward problem with final

datum u(T ) = uT + δuT is as follows:

a) By means of a standard time stepping method, with good

stability properties, we integrate the problem forward, so as

to obtain good approximations Un ≈ at the future nodes tn,

1 ≤ n ≤ N .

b) By means of Ψ, we translate the holomorphic recovery algorithm

on D so as to obtain a holomorphic approximation U(z) to u(z),

for z ∈ Σ, by using the translation to the discrete level of the a

priori bound ‖u(z)‖ ≤ CthetaM .

In particular, we will get U(t) ≈ u(t), for 0 < t ≤ T

16

To precise the obtained estimates, we must introduce much

more details and hypothesis. Let me just comment that the

idea can be implemented combined with finite differences or

finite elements, in the framework of the maximum-norm. With

ρ = ‖δuT‖+ error in the forward integration, the final result is

that, for N = O(| ln ρ|) tuned according to the recovery algorithm,

there holds

‖U(t)− u(t)‖ ≤ CMγ(t)1−ω(t)(Nρ)ωt

where ω is the harmonic measure of [T, T ′] w.r.t. Σ.

The bound certainly deteriorates as t→ 0+, according to some

function of the ratio t/T .

17

The algorithm solves least square problems indeed but, as differentfrom standard methods, based on Tikhonov regularization, thematrix involved is of size N ×N , very moderate and independentof the problem (since it is only Ψ-dependent).

In practice, after space discretization, the recovery algorithm isused for recovering the coefficients expressing U(t) in a suitablebasis. This means solving many LS problems, let us say one pernode, but they all with the same matrix of size N ×N and andthe task can be carried out in parallel.

The algorithm produces a continuous output (rational mappings)which are used once and for all along the whole interval [0, T ].

Linearity is not required. The same ideas can be used for nonlinearproblems, as far as the propagator is holomorphic in time.

18

Sideways heat equation. Problem: to obtain the temperature

in the unaccessible part of a conducting beam

19

This problem arises in several applications

J. V. Beck, B. Blackwell & C. R. St Clair Jr, Inverse Heat

Conduction, Wiley-Interscience, New York, 1985.

L. B. Drenchev & J. Sobczak, Inverse heat conduction problems

and application to estimate of heat paremeters in 2-D experiments,

in Proc. Int. Conf. High Temperature Capillarity, 1997.

and it is challenging from a mathematical point of view.

20

Thus, we start by considering boundary problems of the kindut(t, x) = uxx(t, x), x ≥ 0, t > 0,

u(0, x) = 0, x ≥ 0,

u(t, L0) = b(t), t > 0,

suitable boundary cond. at L = 0, t > 0,

where b : [0,+∞)→ R is the solution history at L0.

Note the difference with the classical, well posed Cauchy problemut(t, x) = uxx(t, x), x ≥ 0, t ≥ 0,

u(0, x) = 0, x ≥ 0,

u(t,0) = a(t), t ≥ 0,

suitable boundary cond. at L = 0, t > 0,

where now the datum a : [0,+∞)→ R corresponds to the boundarycondition at the left end x = 0.

21

Fourier Analysis shows that our problemut(t, x) = uxx(t, x), x ≥ 0, t > 0,

u(0, x) = 0, x ≥ 0,

u(t, L0) = b(t), t > 0,

suitable boundary cond. at L = 0, t > 0,

is severely ill-posed indeed. It is called the sideways heat equation.

22

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

b(t) a(t)

WELL POSED

ILL POSED

WELL POSED

L0 L

space

time

23

The ill-posedness of the sideways heat equation means that:

i) Given two solutions uj with data bj = uj(·, L0), j = 1,2, it is

not possible to estimate the difference u2 − u1 over the unaccessible

interval 0 < x < L0, in any reasonable norm, in terms of b2 − b1.

ii) Solution may fail to exist for a given datum b, even for a very

smooth one.

24

Nevertheless, like in the case of the backward heat equation,our problem can somehow be stabilized by incorporating certainmathematical restrictions on u. Thus, in most of the referenceson the subject, it is assumed that an a priori bound

‖u(·,0)‖2 =

(∫ +∞

0|u(t,0)|2 dt

)1/2

≤M,

for some M > 0, is available.

In fact, under the above a priori bound, our problem becomeswell posed in the following sense ([Miller, 64], [Cannon andMiller 65]): In case L = +∞, given two solution uj with databj = uj(·, L0), j = 1,2, both satisfying the above bound, thereholds

‖u2(·, x)− u1(·, x)‖2 ≤ (2M)1−x/L0‖b2 − b1‖x/L02 , 0 ≤ x ≤ L0.

25

Note that the estimate

‖u2(·, x)− u1(·, x)‖2 ≤ (2M)1−x/L0‖b2 − b1‖x/L02 , 0 ≤ x ≤ L0,

holds only for two true solutions, both satisfying the a priori

bound. Therefore, it cannot be applied directly when a numerical

method is used. However, certainly it gives the flavor of the sort

of estimates we can expect after discretizing and which are really

obtained in the literature.

26

Tikhonov’s regularization has been considered in:

• A. Carasso, Determining surface temperatures from interiorobservations, SIAM J. Appl. Math., 42 (1982), pp. 558–574.

• H. Levine, Continuous data dependence, regularization and athree lines theorem for the heat equation with data in a spacelike direction, Ann. Mat. Pura Appl., 134 (1983), pp. 267-286.

Filtering, in:

• L. Elden, Numerical solution of the sideways heat equation bydifference approximation in time, Inverse Problems, 11 (1995),pp. 913–923.

• L. Elden, Solving the sideways heat equation by a method oflines, Trans. Ams. J. Heat Transfer, 119 (1997), pp. 406–412.

27

Other approaches for similar equations in:

• R. E. Ewing & T. Lin, Proc. 27th IEEE Conf. on Decision

and Control (1988), pp. 240–244.

• L. Elden, Inverse Problems, 3 (1987), pp. 263–273.

• P. Manselli & K. Miller, Ann.Mat.Pura Appl., 123 (1980), pp.

161–183.

• K. Miller, SIAM J. Math. Anal., 1 (1970), pp. 52–74.

• T. I. Seidman, Inverse Problems, 6 (1990), pp. 681-696.

28

All these methods make use of some a priori information and

lead to estimates in the spirit of

‖u2(·, x)− u1(·, x)‖2 ≤ (2M)1−x/L0‖b2 − b1‖x/L02 , 0 ≤ x ≤ L0,

In its formulation, all the existing methods try to approximate

the whole temporal history u(t, x), t ≥ 0 from the whole register

b(·), t ≥ 0, rather than proceeding through a stepping scheme,

thus requiring to solve large linear systems.

29

The new method we propose proceeds in a completely different

way: we try to recover, at each fixed t > 0,

u(x, t), 0 < x < L0

from knowledge of approximate values

Un ≈ u(xn, t), 1 ≤ n ≤ N,

at suitable nodes xn, 1 ≤ n ≤ N, located in the accessible interval

[L0, L).

The approximations Un can be either the result of direct measurements

at nodes xn or to be obtained through a time stepping numerical

method applied to the standard evolution problem on [L0, L]× [0,+∞)

with boundary value b along the extreme x = L0.

30

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

b

L0 L

x1 x2 xN-1 xN

31

Instead of the L2-a priori bound previously introduced, we rather

assume that for some available M > 0 there holds

|u(t, x)| ≤M, 0 ≤ x < L, t > 0,,

which seems to be better suited to applications.

We first prove the following essential fact: for each fixed t > 0,

the mapping x→ u(x, t) admits a holomorphic extension to a

suitable domain Ω in the complex plane. The domain Ω can

be either a sector with vertex x = 0, in case L = +∞, or a

diamond region with diagonal [0, L] in case L < +∞. Moreover,

the extension is bounded in terms of some geometrical constants

and M .

32

To be more precise, for 0 ≤ φ ≤ π/4, set

Σφ = z ∈ C : | arg(z)| < φ

and let ΣLφ be the diamond like region defined by

ΣLφ = Σφ ∩ (L−Σφ).

33

THEOREM. Let u be a solution ofut(t, x) = uxx(t, x), 0 ≤ x ≤ L, t ≥ 0,

u(0, x) = 0, 0 ≤ x ≤ L,

such that |u(t, x)| ≤M, 0 ≤ x ≤ L, t ≥ 0.

Then, for each fixed t > 0, u(t, ·) admits an analytic extension,

again denoted by u(t, ·), to the open square ΣLπ/4. Moreover, for

each angle 0 < φ < π/4, there holds

|u(z, t)| ≤4M√

cos(2φ), t ≥ 0, z ∈ ΣL

φ .

34

Then, we use the algorithm for the holomorphic recovery.

To this end, we first conformally transform the domain

Ψ : ΣLφ → D

into the unit disk D.

35

In our context, we work with a real transformation Ψ such that−r = Ψ(L0) ∈ (−1,0) and such that for some L0 < Lr < L thereholds r = Ψ(Lr).

The nodes xn ∈ [L0, Lr] are selected in such a way that

Ψ(xn) ∈ [−r, r], 1 ≤ n ≤ N,are the Chebyshev nodes of first kind on [−r, r].

Then, we know that with N = O(log(M/ε)), a moderate numberof nodes, it is possible to recover f by means of a rationalmapping g : D → C, without poles on D, in such a way that thecomposition F = g(Ψ) satisfies

|f(z)− F (z)| ≤ γ(z)| log(z)|M1−ω(z)εω(z), z ∈ ΣLφ ,

where γ is a geometric factor and ω is the harmonic measure ofΣLφ \ [L0, Lr] with respect to [L0, Lr].

36

Thus, our algorithm proceeds, at each t > 0, simply by recovering

the holomorphic mapping u(t, ·) from knowledge of approximate

values

Un(t) ≈ u(xn, t), 1 ≤ n ≤ N.

The nodes xn are as explained and lie in the accessible zone.

Therefore, the approximate values can be obtained from either

direct measurements or by numerical computation of the

solution along the accessible part. Thus, time history is not

required. Moreover, since the recover on the disk is given in term

of a rational mapping, the method yields a continuous output

for approximating the solution u(t, x) on the whole unaccessible

zone 0 ≤ x ≤ L0.

37

Illustration 1. We consider the one-dimensional problem equation

ut(t, x) = uxx(t, x), t ≥ 0, 0 ≤ x ≤ 1,

along with homogeneous Dirichlet boundary conditions u(t,0) = u(t,1) = 0.

We solve numerically (finite differences plus EDO23s), with accuracy

10−7, the problem with initial datum

u0(x) = sin(25πx2), 0 ≤ x ≤ 1.

up to time T = 1/8192. Then we add a pseudorandom perturbation

δuT (x), uniformly distributed on [−δ, δ], with δ = 10−6, to the

numerically obtained value uT (x).

Starting from uT (x) + δuT (x), 0 ≤ x ≤ 1, we try now to backwards

integrate the problem.

38

Hidden initial datum u0(x) and available approximate datum

39

uT (x) + δuT (x).

We take θ = π/2.2, R = 4.1T , and r = 0.3. We adopt the theoretical

a priori bound M = sec1/2 θ. This results in N = 9 nodes. The

forward integration yields approximations at future nodes

40

Approximations to u(tn, x) at future nodes tn, 1 ≤ n ≤ N = 9.

41

In spite T that is small, notice that it is the ratio t/T which is

really relevant in the experiment. Now, for t = T/8, T/16, the

errors between the recovered mapping and the accurate forward

integration from u0 turn out to be 4.43e− 2, 4.7.76e− 2, in

agreement with the theory. The combined plot render mappings

hard to be distinguished

42

Numerical solution (dotted line) and exact solution (solid

line) are hard to distinguish

43

Zoom of the numerical solution (dotted line) and exact

solution (solid line) at T/16.

44

Illustration 2. On the half-axis [0,+∞) we consider the solution

u(x, t) of the problem with Dirichlet condition

u(0, t) = a(t) =

1 if 0.25 ≤ t ≤ 0.75,

0 otherwise,

This solution is obtained through a sinc-method, with very high

precision. Then we consider the sideways equation along L0 = 1

and datum

b(t) = u(1, t), t ≥ 0.

Now, at x = 0.25, we recover the solution after introducing

random perturbations of size 10−6. The number N of required

nodes is 9.

45

The plot of the approximation at x = 0.25 versus different time

levels is:

46

If we plot the approximation versus x, for a fixed t ≥ 0, we obtain

the plot of the continuous output:

47

Illustration 3. Now a very oscillatory behaviour at x = 0

u(t,0) = t(t− 2) sin(100t), t ≥ 0,

is considered. In the present example we work on [0,0.2]. At

x = L = 0.2 the boundary condition is u(t, L) = 0, t ≥ 0.

We adopt L0 = 0.1. On the interval [0, L] we approximate u,

with very low tolerance on [L0, L], by means of finite differences.

Then we randomly perturb b(t) = u(t, L0), with ε = 10−6 and

apply the algorithm to recover u(t,0.01). This leads to the next

plot of u(t,0.1 ∗ L0) versus different time levels.

48

Plot of u(t,0.1 ∗ L0) versus different time levels.

49