Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs
Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D ⊂ℝ d are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L 2 ( D )-orthogonal bases,...
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| Published in | Foundations of computational mathematics Vol. 10; no. 6; pp. 615 - 646 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer-Verlag
01.12.2010
Springer Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1615-3375 1615-3383 |
| DOI | 10.1007/s10208-010-9072-2 |
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| Abstract | Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain
D
⊂ℝ
d
are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in
L
2
(
D
)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters
y
=
y
(
ω
)=(
y
i
(
ω
)). This yields an equivalent parametric deterministic PDE whose solution
u
(
x
,
y
) is a function of both the space variable
x
∈
D
and the in general countably many parameters
y
.
We establish new regularity theorems describing the smoothness properties of the solution
u
as a map from
y
∈
U
=(−1,1)
∞
to
. These results lead to analytic estimates on the
V
norms of the coefficients (which are functions of
x
) in a so-called “generalized polynomial chaos” (gpc) expansion of
u
.
Convergence estimates of approximations of
u
by best
N
-term truncated
V
valued polynomials in the variable
y
∈
U
are established. These estimates are of the form
N
−
r
, where the rate of convergence
r
depends only on the decay of the random input expansion. It is shown that
r
exceeds the benchmark rate 1/2 afforded by Monte Carlo simulations with
N
“samples” (i.e., deterministic solves) under mild smoothness conditions on the random diffusion coefficients.
A class of fully discrete approximations is obtained by Galerkin approximation from a hierarchic family
of finite element spaces in
D
of the coefficients in the
N
-term truncated gpc expansions of
u
(
x
,
y
). In contrast to previous works, the level
l
of spatial resolution is adapted to the gpc coefficient. New regularity theorems describing the smoothness properties of the solution
u
as a map from
y
∈
U
=(−1,1)
∞
to a smoothness space
W
⊂
V
are established leading to analytic estimates on the
W
norms of the gpc coefficients and on their space discretization error. The space
W
coincides with
in the case where
D
is a smooth or convex domain.
Our analysis shows that in realistic settings a convergence rate
in terms of the total number of degrees of freedom
N
dof
can be obtained. Here the rate
s
is determined by both the best
N
-term approximation rate
r
and the approximation order of the space discretization in
D
. |
|---|---|
| AbstractList | Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D super(d) are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L super(2)(D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y=y( omega )=(y sub(i) ( omega )). This yields an equivalent parametric deterministic PDE whose solution u(x,y) is a function of both the space variable x[isin]D and the in general countably many parameters y. We establish new regularity theorems describing the smoothness properties of the solution u as a map from y[isin]U=(-1,1) super( infinity ) to V = H 0 1 ( D ) . These results lead to analytic estimates on the V norms of the coefficients (which are functions of x) in a so-called "generalized polynomial chaos" (gpc) expansion of u. Convergence estimates of approximations of u by best N-term truncated V valued polynomials in the variable y[isin]U are established. These estimates are of the form N super(-r), where the rate of convergence r depends only on the decay of the random input expansion. It is shown that r exceeds the benchmark rate 1/2 afforded by Monte Carlo simulations with N "samples" (i.e., deterministic solves) under mild smoothness conditions on the random diffusion coefficients. A class of fully discrete approximations is obtained by Galerkin approximation from a hierarchic family { V sub(l) } super( infinity ) sub(l = 0) is a subset of V of finite element spaces in D of the coefficients in the N-term truncated gpc expansions of u(x,y). In contrast to previous works, the level l of spatial resolution is adapted to the gpc coefficient. New regularity theorems describing the smoothness properties of the solution u as a map from y[isin]U=(-1,1) super( infinity ) to a smoothness space W is a subset of V are established leading to analytic estimates on the W norms of the gpc coefficients and on their space discretization error. The space W coincides with H 2 ( D ) H 0 1 ( D ) in the case where D is a smooth or convex domain. Our analysis shows that in realistic settings a convergence rate N dof - s in terms of the total number of degrees of freedom N sub(dof) can be obtained. Here the rate s is determined by both the best N-term approximation rate r and the approximation order of the space discretization in D. Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D [subset] [R.sup.d] are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in [L.sup.2](D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y = y([omega]) = ([y.sub.i] ([omega])). This yields an equivalent parametric deterministic PDE whose solution u(x,y) is a function of both the space variable x [member of] D and the in general countably many parameters y. We establish new regularity theorems describing the smoothness properties of the solution u as a map from y [member of] U = [(-1, 1).sup.[infinity]] to V = [H.sup.1.sub.0](D). These results lead to analytic estimates on the V norms of the coefficients (which are functions of x) in a so-called "generalized polynomial chaos" (gpc) expansion of u. Convergence estimates of approximations of u by best N-term truncated V valued polynomials in the variable y [member of] U are established. These estimates are of the form [N.sup.-r], where the rate of convergence r depends only on the decay of the random input expansion. It is shown that r exceeds the benchmark rate 1/2 afforded by Monte Carlo simulations with N "samples" (i.e., deterministic solves) under mild smoothness conditions on the random diffusion coefficients. A class of fully discrete approximations is obtained by Galerkin approximation from a hierarchic family [{[V.sub.l]}.sup.[infinity].sub.l=0] [subset] V of finite element spaces in D of the coefficients in the N-term truncated gpc expansions of u(x, y). In contrast to previous works, the level l of spatial resolution is adapted to the gpc coefficient. New regularity theorems describing the smoothness properties of the solution u as a map from y [member of] U = [(-1, 1).sup.[infinity]] to a smoothness space W c V are established leading to analytic estimates on the W norms of the gpc coefficients and on their space discretization error. The space W coincides with [H.sup.2](D) [intersection] [H.sup.1.sub.0] (D) in the case where D is a smooth or convex domain. Our analysis shows that in realistic settings a convergence rate [N.sup.-s.sub.dof] in terms of the total number of degrees of freedom [N.sub.dof] can be obtained. Here the rate s is determined by both the best N-term approximation rate r and the approximation order of the space discretization in D. Keywords Stochastic and parametric elliptic equations * Wiener polynomial chaos * Approximation rates * Nonlinear approximation * Sparsity Mathematics Subject Classification (2000) 41A * 65N * 65C30 Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D^sup d^ are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L ^sup 2^(D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y=y(ω)=(y ^sub i^(ω)). This yields an equivalent parametric deterministic PDE whose solution u(x,y) is a function of both the space variable xD and the in general countably many parameters y. We establish new regularity theorems describing the smoothness properties of the solution u as a map from yU=(-1,1)^sup ∞^ to V = H 0 1 ( D ) . These results lead to analytic estimates on the V norms of the coefficients (which are functions of x) in a so-called "generalized polynomial chaos" (gpc) expansion of u. Convergence estimates of approximations of u by best N-term truncated V valued polynomials in the variable yU are established. These estimates are of the form N ^sup -r^, where the rate of convergence r depends only on the decay of the random input expansion. It is shown that r exceeds the benchmark rate 1/2 afforded by Monte Carlo simulations with N "samples" (i.e., deterministic solves) under mild smoothness conditions on the random diffusion coefficients. A class of fully discrete approximations is obtained by Galerkin approximation from a hierarchic family { V l } l = 0 ∞ V of finite element spaces in D of the coefficients in the N-term truncated gpc expansions of u(x,y). In contrast to previous works, the level l of spatial resolution is adapted to the gpc coefficient. New regularity theorems describing the smoothness properties of the solution u as a map from yU=(-1,1)^sup ∞^ to a smoothness space WV are established leading to analytic estimates on the W norms of the gpc coefficients and on their space discretization error. The space W coincides with H 2 ( D ) ∩ H 0 1 ( D ) in the case where D is a smooth or convex domain. Our analysis shows that in realistic settings a convergence rate N dof − s in terms of the total number of degrees of freedom N ^sub dof^ can be obtained. Here the rate s is determined by both the best N-term approximation rate r and the approximation order of the space discretization in D.[PUBLICATION ABSTRACT] Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D [subset] [R.sup.d] are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in [L.sup.2](D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y = y([omega]) = ([y.sub.i] ([omega])). This yields an equivalent parametric deterministic PDE whose solution u(x,y) is a function of both the space variable x [member of] D and the in general countably many parameters y. Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D ⊂ℝ d are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L 2 ( D )-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y = y ( ω )=( y i ( ω )). This yields an equivalent parametric deterministic PDE whose solution u ( x , y ) is a function of both the space variable x ∈ D and the in general countably many parameters y . We establish new regularity theorems describing the smoothness properties of the solution u as a map from y ∈ U =(−1,1) ∞ to . These results lead to analytic estimates on the V norms of the coefficients (which are functions of x ) in a so-called “generalized polynomial chaos” (gpc) expansion of u . Convergence estimates of approximations of u by best N -term truncated V valued polynomials in the variable y ∈ U are established. These estimates are of the form N − r , where the rate of convergence r depends only on the decay of the random input expansion. It is shown that r exceeds the benchmark rate 1/2 afforded by Monte Carlo simulations with N “samples” (i.e., deterministic solves) under mild smoothness conditions on the random diffusion coefficients. A class of fully discrete approximations is obtained by Galerkin approximation from a hierarchic family of finite element spaces in D of the coefficients in the N -term truncated gpc expansions of u ( x , y ). In contrast to previous works, the level l of spatial resolution is adapted to the gpc coefficient. New regularity theorems describing the smoothness properties of the solution u as a map from y ∈ U =(−1,1) ∞ to a smoothness space W ⊂ V are established leading to analytic estimates on the W norms of the gpc coefficients and on their space discretization error. The space W coincides with in the case where D is a smooth or convex domain. Our analysis shows that in realistic settings a convergence rate in terms of the total number of degrees of freedom N dof can be obtained. Here the rate s is determined by both the best N -term approximation rate r and the approximation order of the space discretization in D . |
| Audience | Academic |
| Author | Cohen, Albert Schwab, Christoph DeVore, Ronald |
| Author_xml | – sequence: 1 givenname: Albert surname: Cohen fullname: Cohen, Albert email: cohen@ann.jussieu.fr organization: Laboratoire Jacques-Louis Lions, UMR 7598, UPMC Univ. Paris 06, Laboratoire Jacques-Louis Lions, UMR 7598, CNRS – sequence: 2 givenname: Ronald surname: DeVore fullname: DeVore, Ronald organization: Department of Mathematics, Texas A& M University – sequence: 3 givenname: Christoph surname: Schwab fullname: Schwab, Christoph organization: Seminar for Applied Mathematics, ETH Zürich |
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| References | Ghanem, Spanos (CR9) 1997; 4 Kleiber, Hien (CR11) 1992 Babuška, Nobile, Tempone (CR2) 2007; 45 Ciarlet (CR3) 1978 CR6 Nobile, Tempone, Webster (CR14) 2008; 46 Schwab, Todor (CR17) 2006; 217 Nobile, Tempone, Webster (CR13) 2008; 46 Karniadakis, Xiu (CR10) 2002; 24 DeVore (CR7) 1998; 7 Todor, Schwab (CR20) 2007; 45 Cohen (CR4) 2003 Schoutens (CR16) 2000 Ledoux, Talagrand (CR12) 1991 Gantumur, Harbecht, Stevenson (CR8) 2007; 76 Todor (CR19) 2006; 44 Babuška, Tempone, Zouraris (CR1) 2004; 42 Smolyak (CR18) 1963; 4 Cohen, Dahmen, DeVore (CR5) 2001; 70 von Petersdorff, Schwab (CR21) 2006; 51 Wiener (CR22) 1938; 60 Rozza, Huynh, Patera (CR15) 2008; 15 A. Cohen (9072_CR4) 2003 I. Babuška (9072_CR1) 2004; 42 R. Todor (9072_CR19) 2006; 44 R. Ghanem (9072_CR9) 1997; 4 C. Schwab (9072_CR17) 2006; 217 R. DeVore (9072_CR7) 1998; 7 M. Kleiber (9072_CR11) 1992 R. Todor (9072_CR20) 2007; 45 T. Gantumur (9072_CR8) 2007; 76 T. Petersdorff von (9072_CR21) 2006; 51 N. Wiener (9072_CR22) 1938; 60 A. Cohen (9072_CR5) 2001; 70 9072_CR6 G. Rozza (9072_CR15) 2008; 15 G.E. Karniadakis (9072_CR10) 2002; 24 F. Nobile (9072_CR13) 2008; 46 W. Schoutens (9072_CR16) 2000 F. Nobile (9072_CR14) 2008; 46 S.A. Smolyak (9072_CR18) 1963; 4 P.G. Ciarlet (9072_CR3) 1978 I. Babuška (9072_CR2) 2007; 45 M. Ledoux (9072_CR12) 1991 |
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| Snippet | Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain
D
⊂ℝ
d
are introduced and their... Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D [subset] [R.sup.d] are introduced... Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D^sup d^ are introduced and their... Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D super(d) are introduced and their... |
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| SubjectTerms | Analysis Applications of Mathematics Approximation Computational mathematics Computer Science Computer simulation Convergence Economics Estimates Galerkin methods Linear and Multilinear Algebras Math Applications in Computer Science Mathematical analysis Mathematical models Mathematics Mathematics and Statistics Matrix Theory Monte Carlo method Numerical Analysis Smoothness Theorems |
| Title | Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs |
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