Random operator equations and their approximationromisch/papers/Engl60.pdf · Random operator...
Transcript of Random operator equations and their approximationromisch/papers/Engl60.pdf · Random operator...
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Random operator equations and their approximation
- Heinz Engl’s nonrandom contributions -
W. Romisch
Humboldt-University BerlinInstitute of Mathematics
www.math.hu-berlin.de/~romisch
Workshop Heinz W. Engl 60,Johannes Kepler Universitat Linz, 28. Marz 2013
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Introduction
Differential and integral equations with random coefficients and
random right-hand sides are now within reach of efficient compu-
tational methods.
The latter require a combination of discretization and sampling
techniques and a specific theoretical justification.
New sampling methods based on randomized lattice rules (specific
Quasi-Monte Carlo methods) are available, which led to a break-
through in high-dimensional numerical integration and to lifting the
curse of dimension.
(recent work of Sloan/Kuo/Joe and Dick/Pillichshammer)
A general approximation theory for random operator equations is
already available since about 25 years and has been developed by
Heinz Engl and his collaborators.
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Recent work on random partial differential equations
Simultaneous stochastic Galerkin and FE methods
Elliptic partial differential equation with random coefficients
−∇(a(ξ, z(ω))∇x(ξ, ω)) = f (ξ) (ξ ∈ D), x(ξ, ω) = 0 (ξ ∈ ∂D).
where D ⊂ Rm, z = (zj)j∈N is uniformly distributed in [0, 1]N,
a(ξ, z) = a(ξ)+
∞∑j=1
zjψj(ξ) (ξ ∈ D) (Karhunen-Loeve expansion).
Variational formulation:
X = H10(D) ⊂ H = L2(D) ⊂X∗= H−1(D), ‖v‖X = ‖∇v‖H∫
D
a(ξ, z(ω))〈∇x(ξ),∇v(ξ)〉dξ =
∫D
f (ξ)v(ξ)dξ (∀v ∈ X).
s-term N -sample QMC scheme: a(ξ, zsN) = a(ξ) +∑s
j=1 zjψj(ξ)
FE method: Replace X by a finite element subspace Xh.
(Kuo/Schwab/Sloan: SIAM J. Num. Anal. 2012)
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Random operator equations: Existence
Let (X, d) be a separable complete metric space, Y and Z separable
metric spaces and 0 ∈ Y fixed. All metric spaces are endowed with
their Borel σ-fields. Consider the random operator equation
T (x, z(ω)) = 0 (ω ∈ Ω),
where T : X × Z → Y is a mapping and z is a Z-valued random
variable given on a probability space (Ω,F ,P).
Question: (Existence of a random solution (Hans 56))
Does there exist a measurable map x : Ω→ X such that
T (x(ω), z(ω)) = 0, P-almost surely
if the equation T (x, z(ω)) = 0 is solvable for all ω ∈ Ω ?
Note that measurability of T (x, ·) with respect to z (∀x ∈ X)
together with P -a.s. solvability of T (x, z(ω)) = 0 is not sufficient!
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Theorem: (Engl 78, Nowak 78)
Assume that S(ω) = x ∈ X : T (x, z(ω)) = 0 6= ∅ P -a.s.If T (·, z(·))−1(0) ∈ F × B(X), there exists a random solution.Proof:There is A ∈ F such that P(A) = 1 and S(ω) 6= ∅, ∀ω ∈ A.
GrS = (ω, x) : x ∈ S(ω) = T (·, z(·))−1(0) ∩ A ∩B : B ∈ F × B(X)
Apply measurable selection theorems for set-valued maps with measurable
graph on the completion of A ∩ B : B ∈ F (Saint-Beuve, Leese, Himmelberg
74/75) and modify a measurable selection on a set of measure 0.
The condition T (·, z(·))−1(0) ∈ F ×B(X) is implied by (a) or (b):
(a) T is Borel measurable.
(b) T is a Caratheodory mapping, i.e., T (·, z) is continuous for all
z ∈ Z and T (x, ·) is measurable for all x ∈ X .
Extensions due to Engl:(i) Stochastic domains, i.e., T (·, z(ω)) : C(ω)→ Y , where
C : Ω ⇒ X is a closed-valued measurable multifunction.
(ii) Set-valued operators T , i.e., T : X × Z ⇒ Y .
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A Borel probability measure µX ∈ P(X) is called weak solution of
the random operator equation
T (x, z(ω)) = 0 (ω ∈ Ω)
iff there exists µ ∈ P(X × Z) such that
µT−1 = δ0 , µX = µ p−1X , L(z) = µ p−1Z ,
where L(z) = P z−1 : B(Z)→ [0, 1] is the probability distribution
(or law) of z, pX and pZ are the projections from X×Z to X and
Z, respectively, and δ0 ∈ P(Y ) denotes the Dirac measure placing
unit mass at 0 ∈ Y .
Remark:If x : Ω → X is a random solution, L(x) is a weak solution (by
putting µ = L(x, z)).
A weak solution of a random operator equation is a random solution
on some probability space.
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Random operator equations: Approximations
Let X , Y and Z be complete separable metric (Polish) spaces,
T : X × Z → Y be Borel measurable, z be a Z-valued random
variable (on (Ω,F ,P)) and 0 ∈ Y be fixed. We consider
T (x, z(ω)) = 0 (ω ∈ Ω).
In addition, we consider the approximate random operator equations
Tn(x, zn(ω)) = 0 (ω ∈ Ωn;n ∈ N),
where for each n ∈ N, Xn ⊂ X , Zn ⊂ Z, Tn : Xn × Zn → Y
be Borel measurable and zn be a Zn-valued random variable (on
(Ωn,Fn,Pn)).
Let (xn) be a sequence of random solutions to the approximate
random operator equations.
Motivation: Approximation procedures for solving random equa-
tions require a ’discretization’ of T and an approximation (’sam-
pling’, ’estimation’) of z.
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Weak convergence in P(X): (µn) converges weakly to µ iff
limn→∞
∫X
f (x)µn(dx) =
∫X
f (x)µ(dx) ∀f ∈ Cb(X,R).
The topology of weak convergence is metrizable if X is separable.
Weak compactness is characterized by uniform tightness due to
Prokhorov’s theorem.
Problem: Find conditions on (Tn) and T that imply weak conver-
gence of (L(xn)) if (L(zn)) converges weakly to L(z).
A sequence (Tn) converges discretely to T iff
(i) d(x,Xn) = infy∈Xn dX(x, y)→ 0 (∀x ∈ X),
d(z, Zn) = infv∈Zn dZ(z, v)→ 0 (∀z ∈ Z).
(ii) For all (x, z) ∈ X×Z and sequences (xn, zn) ∈ Xn×Zn such
that xn → x in X and zn → z in Z it holds
T (xn, zn)→ T (x, z) (in Y ).
(Stummel, Reinhardt, Vainikko)
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Theorem:Let the following conditions be satisfied:
(a)⋃n∈N[Tn(·, zn)]−1(K)∩B is relatively compact in X for each
zn ∈ Zn, n ∈ N, bounded B ⊂ X and compact K ⊂ Y .
(b) Tn(x, ·) : x ∈ B ∩ Xn, n ∈ N is equicontinuous on K for
each bounded B ⊂ X and compact K ⊂ Z.
(c) (Tn) converges discretely to T .
(d) (L(zn)) converges weakly to L(z).
(e) The set L(xn) : n ∈ N is stochastically bounded, i.e., for
each ε > 0 there exists a bounded Borel set Bε ⊂ X such that
infn∈NL(xn)(Bε) ≥ 1− ε .
Then the set L(xn) : n ∈ N is relatively compact with respect to
the weak topology and each weak limit of a subsequence is a weak
solution of the random operator equation T (x, z(ω)) = 0 (ω ∈ Ω).
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Elliptic PDEs with random coefficients
We consider the random elliptic PDE
−∇(a(ξ, z(ξ, ω))∇x(ξ)) = f (ξ) (ξ ∈ D), x(ξ) = 0 (ξ ∈ ∂D).
where D ⊂ Rm is a bounded polyhedron, z a Z-valued random
variable, where Z is a bounded subset of L∞(D), X = H10(D)
and Y = H−1(D) = X∗.
Variational formulation:
X = H10(D) ⊂ H = L2(D) ⊂X∗= H−1(D), ‖x‖X = ‖∇x‖Lm2∫
D
a(ξ, z)〈∇x(ξ),∇v(ξ)〉dξ =
∫D
f (ξ)v(ξ)dξ (∀v ∈ X).
Let a be continuous, Cmin ≤ a(ξ, z) ≤ Cmax, ∀(ξ, z) ∈ D × Z.
Let Xn be a finite element subspace of X such that d(x,Xn)→ 0
∀x ∈ X . Assume (L(zn)) converges weakly to L(z).
Then the sequence (L(xn)) converges weakly to a weak solution.
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Approximate the integral
I(G(z)) =
∫Z
G(z)dz = lims→∞
∫[0,1]s
G(z1, . . . , zs, 0, . . .)dz1 · · · dzs
by a lattice rule Qs,N(4, ·) with random shift 4 in [0, 1]s and N
points in s dimensions, and for each shifted lattice point solve the
approximate elliptic problem∫D
a(ξ, z(ω))〈∇xh(ξ),∇v(ξ)〉dξ =
∫D
f (ξ)v(ξ)dξ (∀v ∈ Xh).
in a finite element subspace Xh (on D ⊂ Rm) with Mh = O(h−m)
degrees of freedom with linear cost O(Mh).
Theorem: (Kuo/Schwab/Sloan 12)
The convergence rate of the scheme is(E[|I(G(z))−Qs,N(4;G(xh))|2
])12 ≤ C
(s−1 + N−1+δ + hτ
),
where 0 ≤ τ = t + t′ ≤ 2, N is prime, δ ∈ (0, 12], f ∈ H−1+t(D),
G ∈ H−1+t′(D), p = 23 and
∑j∈N ‖ψj‖
pL∞(D) <∞.
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Conclusions and thanks
• Recently the numerical analysis of random differential equa-
tions became a very active field of research. In particular, com-
binations of generalized Wiener or Karhunen-Loeve expansions
with multi-level Monte Carlo and Quasi-Monte Carlo methods
became popular.
• There is a pre-history of existence and approximation results
for random operator equations.
• Heinz Engl contributed basic results to both existence and ap-
proximation approaches of random equations.
• Many thanks Heinz for several years of valuable collaboration.
• Many thanks Heinz for all your support and for the lovely
visits to Austria.
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Congratulations Heinz to your 60th birthday !
Thank you for your attention !
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ReferencesA. T. Bharucha-Reid: Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc. 82(1976), 641–657.
A. T. Bharucha-Reid (ed.): Approximate Solution of Random Equations, North-Holland, New York-Oxford, 1979.
A. T. Bharucha-Reid and W. Romisch: Projective schemes for random operator equations. Weakcompactness of approximate solution measures, J. Integr. Equat. 8 (1985), 95–111.
J. Dick, F. Y. Kuo, and I. H. Sloan: High dimensional integration – the Quasi-Monte Carlo way,Acta Numerica (2014), 1–153.
J. Dick and F. Pillichshammer: Digital Nets and Sequences, Cambridge University Press, 2010.
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H. W. Engl and W. Romisch: Approximate solutions of nonlinear random operator equations: Con-vergence in distribution, Pac. J. Math. 120 (1985), 55–77.
H. W. Engl and W. Romisch: Weak covergence of approximate solutions of stochastic equationswith applications to random differential and integral equations, Numer. Funct. Anal. Optim. 9(1987), 61–104.
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O. Fiedler and W. Romisch: Weak convergence of approximate solutions of random equations,Numer. Funct. Anal. Optim. 13 (1992), 495–511.
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