Tensor train to solve stochastic PDEs

Numerical methods for solving stochastic partial
differential equations in the Tensor Train format
Alexander Litvinenko1
(joint work with Sergey Dolgov2,3, Boris Khoromskij3 and
Hermann G. Matthies4)
1 SRI UQ and Extreme Computing Research Center KAUST, 2
Max-Planck-Institut für Mathematik in den Naturwissenschaften,
Leipzig, MPI for dynamics of complex systems in 3 Magdeburg,
4 TU Braunschweig, Germany
Center for Uncertainty
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http://sri-uq.kaust.edu.sa/

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Motivation for UQ
Nowadays computational algorithms, run on
supercomputers, can simulate and resolve very
complex phenomena. But how reliable are these
predictions? Can we trust to these results?
Some parameters/coeﬃcients are unknown, lack of
data, very few measurements → uncertainty.
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Notation, problem setup
Consider
A(u; q) = f ⇒ u = S(f ; q),
where S is a solution operator.
Uncertain Input:
1. Parameter q := q(ω) (assume moments/cdf/pdf/quantiles of
q are given)
2. Boundary and initial conditions, right-hand side
3. Geometry of the domain
Uncertain solution:
1. mean value and variance of u
2. exceedance probabilities P(u > u∗)
3. probability density functions (pdf) of u.
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KAUST
Figure : KAUST campus, 5 years
old, approx. 7000 people (include
1400 kids), 100 nations.
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Children at KAUST
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Stochastic Numerics Group at KAUST
Figure : SRI UQ Group
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3rd UQ Workshop ”Advances in UQ Methods, Alg. & Appl.”
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PDE with uncertain diffusion coefficients
PART 1. Stochastic Forward Problems
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PDE with uncertain diffusion coefficients
Consider
− div(κ(x, ω) u(x, ω)) = f (x, ω) in G × Ω, G ⊂ R2,
u = 0 on ∂G,
(1)
where κ(x, ω) - uncertain diffusion coefficient. Since κ positive,
usually κ(x, ω) = eγ(x,ω).
For well-posedness see [Sarkis 09, Gittelson 10, H.J.Starkloff 11,
Ullmann 10].
Further we will assume that covκ(x, y) is given (or estimated from
the available data).
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Our previous work
After applying the stochastic Galerkin method, obtain:
Ku = f, where all ingredients are represented in a tensor format
Solve for u. Compute max{u}, var(u), level sets of u, pdf, cdf,
1. Efficient Analysis of High Dimensional Data in Tensor Formats,
[Espig, Hackbusch, A.L., Matthies and Zander, 2012]
Research rank of K (from which ingredients it depends)
2. Efficient low-rank approximation of the stochastic Galerkin
matrix in tensor formats, [Wähnert, Espig, Hackbusch, A.L., Matthies, 2013]
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Smooth transformation of Gaussian RF
Step 1: We assume κ = φ(γ) -a smooth transformation of the
Gaussian random ﬁeld γ(x, ω), e.g. φ(γ) = exp(γ).
[see PhD of E. Zander 2013, or PhD of A. Keese, 2005]
Step 2: Given the covariance matrix of κ(x, ω), we derive the
covariance matrix of γ(x, ω). After that the KLE may be
computed,
γ(x, ω) =
∞
m=1
gm(x)θm(ω),
D
covγ(x, y)gm(y)dy = λmgm(x),
(2)
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Full JM,p and sparse J sp
M,p multi-index sets
M-dimensional PCE approximation of κ writes (α = (α1, ..., αM))
κ(x, ω) ≈
α∈JM
κα(x)Hα(θ(ω)), Hα(θ) := hα1 (θ1) · · · hαM
(θM)
(3)
Definition
The full multi-index is defined by restricting each component
independently,
JM,p = {0, 1, . . . , p1}⊗· · ·⊗{0, 1, . . . , pM}, where p = (p1, . . . , pM)
is a shortcut for the tuple of order limits.
Definition
The sparse multi-index is defined by restricting the sum of
components,
J sp
M,p = {α = (α1, . . . , αM) : α ≥ 0, α1 + · · · + αM ≤ p} .
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TT compression of PCE coeffs
The Galerkin coefficients κα are evaluated as follows [Thm 3.10,
PhD of E. Zander 13],
κα(x) =
(α1 + · · · + αM)!
α1! · · · αM!
φα1+···+αM
M
m=1
gαm
m (x), (4)
where φ|α| := φα1+···+αM
is the Galerkin coefficient of the
transform function, and gαm
m (x) means just the αm-th power of the
KLE function value gm(x).
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Complexity reduction
Complexity reduction in Eq. (4) can be achieved with help of KLE
of κ(x, ω):
κ(x, ω) ≈ ¯κ(x) +
L
=1
√
µ v (x)η (ω) (5)
with the normalized spatial functions v (x).
Instead of using κα(x), (4), directly, we compute
˜κ (α) =
(α1 + · · · + αM)!
α1! · · · αM!
φα1+···+αM
D
M
m=1
gαm
m (x)v (x)dx.
Note that L N. Then we restore the approximate coeﬃcients
κα(x) ≈ ¯κ(x) +
L
=1
v (x)˜κ (α).
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Construction of the stochastic Galerkin operator
Given KLE of κ, assemble for i, j = 1, . . . , N, = 1, . . . , L:
K0(i, j) =
D
¯κ(x) ϕi (x)· ϕj (x)dx, K (i, j) =
D
v (x) ϕi (x)· ϕj (x)d
(6)
K
(ω)
(α, β) =
RM
Hα(θ)Hβ(θ)
ν∈JM
˜κ (ν)Hν(θ)ρ(θ)dθ
=
ν∈JM
∆α,β,ν ˜κ (ν),
∆α,β,ν = ∆α1,β1,ν1 · · · ∆αM ,βM ,νM
,
∆αm,βm,νm =
R
hαm (θ)hβm (θ)hνm (θ)ρ(θ)dθ,
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Stochastic Galerkin operator
Putting together previous formulas, obtain the stochastic Galerkin
operator,
K = K
(x)
0 ⊗ ∆0 +
L
=1
K
(x)
⊗ K
(ω)
, (7)
with K ∈ RN(p+1)M ×N(p+1)M
in case of full JM,p.
IDEA: If PCE coeﬃcients of κ are computed in the tensor product
format, the direct product in ∆ (15) allows to exploit the same
format for (7), and build the operator easily.
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Tensor Train
Two tensor Train examples
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Examples (B. Khoromskij’s lecture)
f (x1, ..., xd ) = w1(x1) + w2(x2) + ... + wd (xd )
= (w1(x1), 1)
1 0
w2(x2) 1
...
1 0
wd−1(xd−1) 1
1
wd (xd )
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Examples:
TT rank(f )=2
f = sin(x1 + x2 + ... + xd )
= (sin x1, cos x1)
cos x2 − sin x2
sin x2 cos x2
...
cos xd−1 − sin xd−1
sin xd−1 cos xd−1
cos xd
sin xd−1
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Low-rank response surface: PCE in the TT format
Calculation of
˜κ (α) =
(α1 + · · · + αM)!
α1! · · · αM!
φα1+···+αM
D
M
m=1
gαm
m (x)v (x)dx.
in TT format needs:
a procedure to compute each element of a tensor, e.g.
˜κα1,...,αM
.
build a TT approximation ˜κα ≈ κ(1)(α1) · · · κ(M)(αM) using a
feasible amount of elements (i.e. much less than (p + 1)M).
Such procedure exists, and relies on the cross interpolation of
matrices, generalized to a higher-dimensional case [Oseledets, Tyrtyshnikov
2010; Savostyanov 13; Grasedyck; Bebendorf].
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PCE coeﬃcients ˜κ (α) are :
˜κ (α) =
s1,...,sM−1
κ
(1)
,s1
(α1)κ
(2)
s1,s2 (α2) · · · κ
(M)
sM−1 (αM). (8)
Collect the spatial components into the “zeroth” TT block,
κ(0)
(x) = κ
(0)
(x)
L
=0
= ¯κ(x) v1(x) · · · vL(x) , (9)
then the PCE writes as the following TT format,
κ(x, α) =
,s1,...,sM−1
κ
(0)
(x)κ
(1)
,s1
(α1) · · · κ
(M)
sM−1 (αM). (10)
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Stochastic Galerkin matrix in TT format
Given κα(x), (10), we split the whole sum over ν in K,(7):
ν∈JM,p
∆α,β,ν ˜κ (ν) =
s1,...,sM−1
K
(1)
,s1
(α1, β1)K
(2)
s1,s2 (α2, β2) · · · K
(M)
sM−1 (αM,
K(m)
(αm, βm) =
pm
νm=0
∆αm,βm,νm κ(m)
(νm), m = 1, . . . , M.
(11)
then the TT representation for the operator writes
K =
,s1,...,sM−1
K
(0)
⊗K
(1)
,s1
⊗· · ·⊗K
(M)
sM−1 ∈ R(N·#JM,p)×(N·#JM,p)
,
(12)
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Solving and Post-processing:
Solve the linear system Ku = f by alternating optimization
methods [Dolgov, Savostyanov 14] with a mean-ﬁeld
preconditioned. Obtain the solution u in the TT format.
u(x, α) =
s0,...,sM−1
u
(0)
s0 (x)u
(1)
s0,s1 (α1) · · · u
(M)
sM−1 (αM). (13)
u(x, θ) =
s0,...,sM−1
u
(0)
s0 (x)
p
α1=0
hα1 (θ1)u
(1)
s0,s1 (α1) · · · (14)


p
αM =0
hαM
(θM)u
(M)
sM−1 (αM)

 . (15)
Then compute: mean, co(variance), exceedance probabilities
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Numerics: Main steps
1. Use sglib (E. Zander, TU BS) for discretization and solution
with J sp
M,p.
2. Compute PCE (sglib) of the coeﬃcients κ(x, ω) in the TT
format by new block adaptive cross algorithm (TT toolbox)
3. Use TT-Toolbox for full JM,p,
4. amen cross.m for TT approximation of ˜κα,
5. Compute stochastic Galerkin matrix K in TT format,
6. Replace high-dimensional calculations by the TT-toolbox.
7. Compute solution of the linear system in TT (alternating
minimal energy, tAMEn)
8. Post-processing in TT format
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Numerical experiments, errors, accuracy
D = [−1, 1]2[0, 1]2. f = f (x) = 1, log-normal and beta
distributions for κ. 557, 2145, 8417 dofs.
Eκ =
1
Nmc
Nmc
z=1
N
i=1 (κ(xi , θz) − κ∗(xi , θz))2
N
i=1 κ2
∗(xi , θz)
where {θz}Nmc
z=1 are normally distributed random samples and
κ∗(xi , θz) = φ (γ(xi , θz)) is the reference coeﬃcient computed
without using the PCE for φ.
E¯u =
¯u − ¯u∗ L2(D)
¯u∗ L2(D)
, Evaru =
varu − varu∗ L2(D)
varu∗ L2(D)
.
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More numerics
We compute the maximizer of the mean solution,
xmax : ¯u(xmax) ≥ ¯u(x) ∀x ∈ D.
umax(θ) = u(xmax, θ), and û = ¯u(xmax).
Taking some τ > 1, we compute
P = P (umax(θ) > τû) =
RM
χumax(θ)>τû(θ)ρ(θ)dθ. (16)
By P∗ we will also denote the probability computed from the
Monte Carlo method, and estimate the error as EP = |P − P∗| /P∗.
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Sparse J sp
M,p or full JM,p ?
What is better sparse J sp
M,p or full JM,p multi-index set ?
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CPU times (sec.) versus p, log-normal distribution
TT (full index set JM,p) Sparse (index set J sp
M,p)
p Tκ Top Tu Tκ Top Tu
1 9.6 0.2 1.7 0.5 0.3 0.65
2 14.7 0.2 3 0.5 3.2 1.4
3 19.1 0.2 3.4 0.7 1028 18
4 24.4 0.2 4.2 2.2 — —
5 30.9 0.32 5.3 9.8 — —
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How does polynomial order influence the ranks ?
How does the max. polynomial order p influence the ranks ?
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Performance versus p, log-normal distribution
p CPU time, sec. rκ ru r ˆχ Eκ Eu P
TT Sparse ˆχ TT Sparse TT Sparse TT
1 11 1.4 0.2 32 42 1 4e-3 1.7e-1 1e-2 1e-1 0
2 18 5.1 0.3 32 49 1 1e-4 1.1e-1 5e-4 5e-2 0
3 23 1046 83 32 49 462 6e-5 2.e-3 3e-4 5e-4 2.8e-4
4 29 — 70 32 50 416 6e-5 — 1e-4 — 1.2e-4
5 37 — 103 32 49 410 6e-5 — 1e-4 — 6.2e-4
Take τ = 1.2:
P = P (umax(θ) > τû) =
RM
χumax(θ)>τû(θ)ρ(θ)dθ. (17)
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How does stochastic dimension M influence the ranks ?
How does the stochastic dimension M influence the TT ranks ?
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Performance versus M, log-normal distribution
M CPU time, sec. rκ ru r ˆχ Eκ Eu P
10 6 6 1.3 20 39 70 2e-4 1.7e-1 3e-4 1.5e-1 2.86e-4
15 12 92 23 27 42 381 8e-5 2e-3 3e-4 5e-4 3e-4
20 22 1e+3 67 32 50 422 6e-5 2e-3 3e-4 5e-4 2.96e-4
30 53 5e+4 137 39 50 452 6e-5 1e-1 3e-4 5.5e-2 2.78e-4
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How does covariance length influence the ranks ?
How does covariance length influence the ranks ?
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Performance versus cov. length, log-normal distribution
cov. CPU time, sec. rκ ru r ˆχ Eκ Eu P
length TT Sparse ˆχ TT Sparse TT Sparse TT
0.1 216 55800 0.9 70 50 1 2e-2 2e-2 1.8e-2 1.8e-2 0
0.3 317 52360 42 87 74 297 3e-3 3.5e-3 2.6e-3 2.6e-3 8e-31
0.5 195 51700 58 67 74 375 1.5e-4 2e-3 2.6e-4 3.1e-4 6e-33
1.0 57.3 55200 97 39 50 417 6.1e-5 9e-2 3.2e-4 5.6e-2 2.95e-04
1.5 32.4 49800 121 31 34 424 3.2e-5 2e-1 5e-4 1.7e-1 7.5e-04
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How does standard deviation σ influence the ranks ?
How does the standard deviation σ influence the TT ranks ?
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Performance versus σ, log-normal distribution
σ CPU time, sec. rκ ru rˆχ Eκ Eu P
0.2 16 1e+3 0.3 21 31 1 6e-5 5e-5 4e-5 1e-5 0
0.4 19 968 0.3 29 42 1 7e-5 8e-4 1e-4 2e-4 0
0.5 21 970 80 32 49 456 6e-5 2e-3 3e-4 5e-4 3e-4
0.6 24 962 25 34 57 272 9e-5 4e-3 6e-4 1e-3 2e-3
0.8 32 969 68 39 66 411 4e-4 8e-2 2e-3 3e-2 8e-2
1.0 51 1070 48 44 82 363 2e-3 4e-1 5e-3 3e-1 9e-2
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How does number of DoFs influence the ranks ?
How does number of DoFs influence the ranks ?
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Performance versus #DoFs, log-normal distribution
#DoFs CPU time, sec rκ ru r ˆχ Eκ Eu P
557 6 6 1.3 20 39 71 2e-4 1.7e-1 3e-4 1.5e-1 2.86e-4
2145 9 14 1.2 20 39 76 2e-4 2e-3 3e-4 5.7e-4 2.9e-4
8417 357 171 0.8 20 40 69 1.7e-4 2e-3 3e-4 5.6e-4 2.93e-4
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Comparison with the Monte Carlo
Comparison of the solution obtained via the (Stochastic Galerkin
+ TT) with the solution obtained via Monte Carlo (4000).
For the Monte Carlo test, we prepare the TT solution with
parameters p = 5 and M = 30.
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Veriﬁcation of the MC method (4000), log-normal distr.
Nmc TMC , sec. E¯u Evaru
P∗ EP TT results
102
0.6 9e-3 2e-1 0 ∞ Tsolve 97 sec.
103
6.2 2e-3 6e-2 0 ∞ Tˆχ 157 sec.
104
6.2·101
6e-4 7e-3 4e-4 5e-1 rκ 39
105
6.2·102
3e-4 3e-3 4e-4 5e-1 ru 50
106
6.3·103
1e-4 1e-3 5e-4 4e-1 rˆχ 432
P 6e-4
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Part II: diffusion coefficient has beta distrib.
κ(x, ω) = B−1
5,2


1 + erf γ(x,ω)
√
2
2

 + 1,
Ba,b(z) =
1
B(a, b)
z
0
ta−1
(1 − t)b−1
dt.
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We researched (for beta distribution)
1. Performance versus p
2. Performance versus stochastic dimension M
3. Performance versus cov. length
4. Performance versus #DoFs
5. Veriﬁcation of the Monte Carlo method
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Take to home
1. TT methods become preferable for high p, but otherwise the
full computation in a small sparse set may be incredibly fast.
This reﬂects well the “curse of order”, taking place for the
sparse set instead of the “curse of dimensionality” in the full
set: the cardinality of the sparse set grows exponentially with
p.
2. The TT approach scales linearly with p.
3. TT methods allow easy calculation of stochastic Galerkin
operator. With p < 10 TT storage of stoch. Galerkin operator
allows us to forget about the sparsity issues, since the number
of TT entries O(Mp2r2) is tractable.
4. Chebyshev, Laguerre, ... may be incorporated into the scheme
freely.
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Future plans for the next article
1. Compute Sobol indices in the TT format. Which uncertain
coefficients and which PCE terms are important ?
2. Solution of this linear elliptic SPDE is a ”working horse” for
the non-linear equation and the Newton method
3. Stochastic Galerkin in TT format above can be used as a
preconditioning it is very fast!) for more complicated
non-linear problems
4. Apply to more complicated diffusion coefficients (e.g. which
are not so good splittable)
5. To create analytic u, compute analytically RHS and solve the
problem again (to avoid to use MC as a reference)
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Approximate Bayesian Update
PART 2. Inverse Problems via approximate
Bayesian Update
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Setting for the identification process
General idea:
We observe / measure a system, whose structure we know in
principle.
The system behaviour depends on some quantities (parameters),
which we do not know ⇒ uncertainty.
We model (uncertainty in) our knowledge in a Bayesian setting:
as a probability distribution on the parameters.
We start with what we know a priori, then perform a measurement.
This gives new information, to update our knowledge
(identification).
Update in probabilistic setting works with conditional probabilities
⇒ Bayes’s theorem.
Repeated measurements lead to better identification.
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Mathematical setup
Consider
A(u; q) = f ⇒ u = S(f ; q),
where S is solution operator.
Operator depends on parameters q ∈ Q,
hence state u ∈ U is also function of q:
Measurement operator Y with values in Y:
y = Y (q; u) = Y (q, S(f ; q)).
Examples of measurements:
(ODE) u(t) = (x(t), y(t), z(t))T , y(t) = (x(t), y(t))T
(PDE) y(ω) = D0
u(ω, x)dx, y(ω) = D0
| grad u(ω, x)|2dx, u in
few points
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Inverse problem
For given f , measurement y is just a function of q.
This function is usually not invertible ⇒ ill-posed problem,
measurement y does not contain enough information.
In Bayesian framework state of knowledge modelled in a
probabilistic way,
parameters q are uncertain, and assumed as random.
Bayesian setting allows updating / sharpening of information
about q when measurement is performed.
The problem of updating distribution —state of knowledge of q
becomes well-posed.
Can be applied successively, each new measurement y and
forcing f —may also be uncertain—will provide new information.
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Conditional probability and expectation
With state u ∈ U ⊗ S a RV, the quantity to be measured
y(ω) = Y (q(ω), u(ω))) ∈ Y ⊗ S
is also uncertain, a random variable.
A new measurement z is performed, composed from the
“true” value y ∈ Y and a random error : z(ω) = y + (ω).
Classically, Bayes’s theorem gives conditional probability
P(Iq|Mz) =
P(Mz|Iq)
P(Mz)
P(Iq);
expectation with this posterior measure is conditional expectation.
Kolmogorov starts from conditional expectation E (·|Mz),
from this conditional probability via P(Iq|Mz) = E χIq |Mz .
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IDEA of the Bayesian Update (BU)
Let Y (x, θ), θ = (θ1, ..., θM, ...), is approximated:
Y (x, θ) =
β∈Jm,p
Hβ(θ)Yβ(x),
q(x, θ) =
β∈Jm,p
Hβ(θ)qβ(x),
Yβ(x) =
1
β! Θ
Hβ(θ)Y (x, θ) P(dθ).
Take qf (ω) = q0(ω).
Linear BU: qa = qf + K · (z − y)
Non-Linear BU: qa = qf + H1 · (z − y) + (z − y)T · H2 · (z − y).
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Open questions
Open questions
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Open questions
Multivariate Cauchy distribution
The characteristic function ϕX(t) of the multivariate Cauchy
distribution is deﬁned as follow:
ϕX(t) = exp i(t1, t2) · (µ1, µ2)T
−
1
2
(t1, t2)
σ2
1 0
0 σ2
2
(t1, t2)T ,
(18)
ϕX(t) ≈
R
ν=1
ϕXν,1 (t1) · ϕXν,2 (t2). (19)
Again, from the inversion theorem, the probability density of X on
R2 can be computed from ϕX(t) as follow
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pX(y) =
1
(2π)2
R2
exp(−i y, t )ϕX(t)dt (20)
≈
1
(2π)2
R2
exp(−i(y1t1 + y2t2))
R
ν=1
ϕXν,1 (t1) · ϕXν,2 (t2)dt1dt2
(21)
≈
R
ν=1
1
(2π) R
exp(−iy1t1)ϕXν,1 (t1)dt1 ·
1
(2π) R
exp(−iy2t2)ϕXν,2 (t
(22)
≈
R
ν=1
pXν,1 (y1) · pXν,2 (y2), i.e. (23)
the probability density pX(y) is numerically splittable.
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4*
Elliptically contoured multivariate stable distribution
ϕX(t) = exp i(t1, t2) · (µ1, µ2)T
− (t1, t2)
σ2
1 0
0 σ2
2
(t1, t2)T
α/2
(24)
Now the question is to ﬁnd a separation of
(t1, t2)
σ2
1 0
0 σ2
2
(t1, t2)T
α/2
≈
R
ν=1
φν,1(t1) · φν,2(t2), (25)
with some tensor rank R.
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4*
Multivariate distribution
Assume that the characteristic function ϕX(t) of some multivariate
d-dimensional distribution is approximated as follow:
ϕX(t) ≈
R
=1
d
µ=1
ϕX ,µ
(tµ). (26)
pX(y) = const
Rd
exp(−i y, t )ϕX(t)dt (27)
≈ const
Rd
exp(−i
d
j=1
yj tj )
R
=1
d
µ=1
ϕX ,µ
(tµ)dt1...dtd (28)
≈
R
=1
const
d
µ=1 R
exp(−iy t )ϕX ,µ
(tµ)dt · (29)
≈
R
=1
d
µ=1
pX ,µ
(yµ). (30)
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4*
Actual computation of ϕX(t)
ϕX(τβ) = E (exp(i X(θ1, ..., θm), τβ ))
= · · ·
Θ
exp(i X(θ1, ..., θm), τβ )
M
m=1
pθm (θm)dθ1...dθM,
X(ω), τβ =
α∈J
ξα
Hα(θ), τβ ≈
d
=1 α∈J
ξα
Hα(θ)tβ ,
=
α∈J
d
=1
ξα
tβ , Hα(θ) =
α∈J
ξα
, τβ Hα(θ) (31)
Now compute the exp() function from the scalar product
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exp(i X(ω), τβ ) = exp(i
α∈J
ξα
, τβ Hα(θ)) (32)
=
α∈J
exp (i ξα
, τβ Hα(θ)) (33)
Now we apply integration:
ϕX(t) = E (exp(i X(ω), τβ ))
= · · ·
Θ α∈J
exp (i ξα
, τβ Hα(θ))
M
m=1
pθm (θm)dθ1...dθM
≈????
nq
=1
w
α∈J
exp (i ξα
, τβ Hα(θ ))
M
m=1
pθm (θm, )
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4*
Literature
1. Polynomial Chaos Expansion of random coefficients and the
solution of stochastic partial differential equations in the Tensor
Train format, S. Dolgov, B. N. Khoromskij, A. Litvinenko, H. G.
Matthies, 2015/3/11, arXiv:1503.03210
2. Efficient analysis of high dimensional data in tensor formats, M.
Espig, W. Hackbusch, A. Litvinenko, H.G. Matthies, E. Zander
Sparse Grids and Applications, 31-56, 40, 2013
3. Application of hierarchical matrices for computing the
Karhunen-Loeve expansion, B.N. Khoromskij, A. Litvinenko, H.G.
Matthies, Computing 84 (1-2), 49-67, 31, 2009
4. Efficient low-rank approximation of the stochastic Galerkin
matrix in tensor formats, M. Espig, W. Hackbusch, A. Litvinenko,
H.G. Matthies, P. Waehnert, Computers and Mathematics with
Applications 67 (4), 818-829
Quantification
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