on the holomorphic recovery and some applications to ... · the idea of the algorithm for the...
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On the holomorphic recovery and some
applications to parabolic problems
C. Palencia (University of Valladolid)
BCAM, December 10, 2012
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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.
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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.
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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.
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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.
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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 .
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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.
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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.
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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.
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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.
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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.
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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:
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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.
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Domain Ω and Γ1 in the theorem. The angle θ′ is chosen
in (0, θ).
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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.
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Domain Σ and future nodes tn
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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
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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 .
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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.
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Sideways heat equation. Problem: to obtain the temperature
in the unaccessible part of a conducting beam
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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.
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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.
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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.
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1.2
1.4
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b(t) a(t)
WELL POSED
ILL POSED
WELL POSED
L0 L
space
time
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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b
L0 L
x1 x2 xN-1 xN
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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 .
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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).
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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 .
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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.
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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].
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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.
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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.
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Hidden initial datum u0(x) and available approximate datum
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uT (x) + δuT (x).
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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
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Approximations to u(tn, x) at future nodes tn, 1 ≤ n ≤ N = 9.
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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
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Numerical solution (dotted line) and exact solution (solid
line) are hard to distinguish
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Zoom of the numerical solution (dotted line) and exact
solution (solid line) at T/16.
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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.
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The plot of the approximation at x = 0.25 versus different time
levels is:
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If we plot the approximation versus x, for a fixed t ≥ 0, we obtain
the plot of the continuous output:
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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.
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Plot of u(t,0.1 ∗ L0) versus different time levels.
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