hot new directions for qmc research in step with...
TRANSCRIPT
Frances Kuo @ UNSW Australia p.1
Hot new directions forQMC research
in step with applications
Frances Kuo
University of New South Wales, Sydney, Australia
Frances Kuo @ UNSW Australia p.2
Outline
Motivating example – a flow through a porous medium
QMC theory
Setting 1
Setting 2
Setting 3
Applications of QMC
Application 1
Application 2
Application 3
Cost reduction strategies
Saving 1
Saving 2
Saving 3
Concluding remarks
Frances Kuo @ UNSW Australia p.3
Motivating exampleUncertainty in groundwater flow
eg. risk analysis of radwaste disposal or CO2 sequestration
Darcy’s law q + a ~∇p = κ
mass conservation law ∇ · q = 0in D ⊂ R
d, d = 1, 2, 3
together with boundary conditions
Uncertainty in a(xxx, ω) leads to uncertainty in q(xxx, ω) and p(xxx, ω)
Frances Kuo @ UNSW Australia p.4
Expected values of quantities of interestTo compute the expected value of some quantity of interest:
1. Generate a number of realizations of the random field(Some approximation may be required)
2. For each realization, solve the PDE using e.g. FEM / FVM / mFEM
3. Take the average of all solutions from different realizations
This describes Monte Carlo simulation.
Example : particle dispersion
p = 1 p = 0
∂p∂~n
= 0
∂p∂~n
= 0
◦release point →•
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Frances Kuo @ UNSW Australia p.5
Expected values of quantities of interestTo compute the expected value of some quantity of interest:
1. Generate a number of realizations of the random field(Some approximation may be required)
2. For each realization, solve the PDE using e.g. FEM / FVM / mFEM
3. Take the average of all solutions from different realizations
This describes Monte Carlo simulation.
NOTE: expected value = (high dimensional) integral
s = stochastic dimension
→ use quasi-Monte Carlo methods
103
104
105
106
10−5
10−4
10−3
10−2
10−1
error
n
n−1/2
n−1
or better QMC
MC
Frances Kuo @ UNSW Australia p.6
MC v.s. QMC in the unit cube∫
[0,1]sf(yyy) dyyy ≈ 1
n
n∑
i=1
f(ttti)
Monte Carlo method Quasi-Monte Carlo methodsttti random uniform ttti deterministic
n−1/2 convergence close to n−1 convergence or better
0 1
1
64 random points
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0 1
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First 64 points of a2D Sobol′ sequence
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A lattice rule with 64 points
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more effective for earlier variables and lower-order projectionsorder of variables irrelevant order of variables very important
use randomized QMC methods for error estimation
Frances Kuo @ UNSW Australia p.7
QMCTwo main families of QMC methods:
(t,m,s)-nets and (t,s)-sequenceslattice rules
0 1
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First 64 points of a2D Sobol′ sequence
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0 1
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A lattice rule with 64 points
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(0,6,2)-netHaving the right number ofpoints in various sub-cubes
A group under addition modulo Z
and includes the integer points
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Niederreiter book (1992)
Sloan and Joe book (1994)
Important developments:component-by-component (CBC ) construction , “fast” CBChigher order digital nets
Dick and Pillichshammer book (2010)
Dick, K., Sloan Acta Numerica (2013)
Nuyens and Cools (2006)Dick (2008)
Contribute to:Information-Based ComplexityTractability
Traub, Wasilkowski, Wozniakowski book(1988)
Novak and Wozniakowski books(2008, 2010, 2012)
Frances Kuo @ UNSW Australia p.8
Application of QMC to PDEs with random coefficients
[0] Graham, K., Nuyens, Scheichl, Sloan (J. Comput. Physics, 2011)
[1] K., Schwab, Sloan (SIAM J. Numer. Anal., 2012)
[2] K., Schwab, Sloan (J. FoCM, 2015)
[3] Graham, K., Nichols, Scheichl, Schwab, Sloan (Numer. Math., 2015)
[4] K., Scheichl, Schwab, Sloan, Ullmann (in review)
[5] Dick, K., Le Gia, Nuyens, Schwab (SIAM J. Numer. Anal., 2014)
[6] Dick, K., Le Gia, Schwab (SIAM J. Numer. Anal., to appear)
[7] Graham, K., Nuyens, Scheichl, Sloan (in progress)
Also Schwab (Proceedings of MCQMC 2012)
Le Gia (Proceedings of MCQMC 2012)
Dick, Le Gia, Schwab (in review, in review, in progress)
Harbrecht, Peters, Siebenmorgen (Math. Comp., 2015)
Gantner, Schwab (Proceedings of MCQMC 2014)
Ganesh, Hawkings (SIAM J. Sci. Comput., 2015)
Robbe, Nuyens, Vandewalle (in review)
Scheichl, Stuart, Teckentrup (in review)
Gilbert, Graham, K., Scheichl, Sloan (in progress)
Graham, Parkinson, Scheichl (in progress)etc.
Frances Kuo @ UNSW Australia p.9
Application of QMC to PDEs with random coefficients
Three different QMC theoretical settings:
[1,2] Weighted Sobolev space over [0, 1]s and “randomly shifted lattice rules”
[3,4] Weighted space setting in Rs and “randomly shifted lattice rules”
[5,6] Weighted space of smooth functions over [0, 1]s and (deterministic)“interlaced polynomial lattice rules”
Uniform Lognormal LognormalKL expansion Circulant embedding
Experiments only [0]*First order, single-level [1] [3]* [7]*First order, multi-level [2] [4]*Higher order, single-level [5]Higher order, multi-level [6]*
The * indicates there are accompanying numerical results
K., Nuyens (J. FoCM, to appear) – a survey of [1,2], [3,4], [5,6] in a unified view
Accompanying software packagehttp://people.cs.kuleuven.be/~dirk.nuyens/qmc4pde/
Frances Kuo @ UNSW Australia p.10
5-minute QMC primer (most important slide)
Common features among all three QMC theoretical settings:
Separation in error bound
(rms) QMC error ≤ (rms) worst case error γγγ × norm of integrand γγγ(rms) QMC error ≤ (rms) worst case error γγγ × norm of integrand γγγ
Weighted spaces
Pairing of QMC rule with function space
Optimal rate of convergence
Rate and constant independent of dimension
Fast component-by-component (CBC) construction
Extensible or embedded rules
Application of QMC theory
Estimate the norm (critical step)
Choose the weights
Weights as input to the CBC construction
Frances Kuo @ UNSW Australia p.11
Lattice rulesRank-1 lattice rules have points
ttti = frac
(i
nzzz
)
, i = 1, 2, . . . , n
zzz ∈ Zs – the generating vector, with all components coprime to n
frac(·) – means to take the fractional part of all components
0 1
1
A lattice rule with 64 points
64
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∼ quality determined by the choice of zzz ∼
n = 64 zzz = (1, 19) ttti = frac
(i
64(1, 19)
)
Frances Kuo @ UNSW Australia p.12
Randomly shifted lattice rulesShifted rank-1 lattice rules have points
ttti = frac
(i
nzzz +∆∆∆
)
, i = 1, 2, . . . , n
∆∆∆ ∈ [0, 1)s – the shift
shifted by
∆∆∆ = (0.1, 0.3)
0 1
1
A lattice rule with 64 points
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A shifted lattice rule with 64 points
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∼ use a number of random shifts for error estimation ∼
Frances Kuo @ UNSW Australia p.13
Component-by-component construction
Want to find zzz for which some error criterion is as small possible∼ Exhaustive search is practically impossible - too many choices! ∼
CBC algorithm [Korobov (1959); Sloan, Reztsov (2002); Sloan, K., Joe (2002),. . . ]
1. Set z1 = 1.2. With z1 fixed, choose z2 to minimize the error criterion in 2D.3. With z1, z2 fixed, choose z3 to minimize the error criterion in 3D.4. etc.
[K. (2003); Dick (2004)]
Optimal rate of convergence O(n−1+δ) in “weighted Sobolev space”,independently of s under an appropriate condition on the weights
∼ Averaging argument: there is always one choice as good as average! ∼
[Nuyens, Cools (2006)]
Cost of algorithm for “product weights” is O(n logn s) using FFT
Extensible/embedded variants[Hickernell, Hong, L’Ecuyer, Lemieux (2000); Hickernell, Niederreiter (2003)]
[Cools, K., Nuyens (2006)][Dick, Pillichshammer, Waterhouse (2007)]
http://people.cs.kuleuven.be/~dirk.nuyens/fast-cbc/http://people.cs.kuleuven.be/~dirk.nuyens/qmc-generators/
Frances Kuo @ UNSW Australia p.14
Fast CBC constructionn = 53
1
53
Natural ordering of the indices Generator ordering of the indices
Matrix-vector multiplication with a circulant matrix can be done using FFT
Frances Kuo @ UNSW Australia p.15
Setting 1: standard QMC for unit cube
Worst case error bound∣∣∣∣∣
∫
[0,1]sf(yyy) dyyy − 1
n
n∑
i=1
f(ttti)
∣∣∣∣∣≤ ewor(ttt1, . . . , tttn) ‖f‖
Weighted Sobolev space [Sloan, Wozniakowski (1998)]
∣∣∣∣∣
∫
[0,1]sf(yyy) dyyy − 1
n
n∑
i=1
f(ttti)
∣∣∣∣∣≤ ewor
γγγ (ttt1, . . . , tttn) ‖f‖γγγ
‖f‖2γγγ =
∑
u⊆{1:s}
1
γu
∫
[0,1]|u|
∣∣∣∣∣
∂|u|f
∂yyyu
(yyyu; 0)
∣∣∣∣∣
2
dyyyu
2s subsets “anchor” at 0 (also “unanchored”)“weights” Mixed first derivatives are square integrable
Small weight γu means that f depends weakly on the variables yyyu
Pair with randomly shifted lattice rules
Choose weights to minimize the error bound [K., Schwab, Sloan (2012)](
2
n
∑
u⊆{1:s}
γλuau
)1/(2λ)
︸ ︷︷ ︸
bound on worst case error (CBC)
(∑
u⊆{1:s}
bu
γu
)1/2
︸ ︷︷ ︸
bound on norm
⇒ γu =
(bu
au
)1/(1+λ)
Construct points (CBC) to minimize the worst case error
“POD weights”
Frances Kuo @ UNSW Australia p.16
Setting 2: QMC integration over Rs
Change of variables∫
Rs
f(yyy)s∏
j=1
φ(yj) dyyy =
∫
[0,1]sf(Φ−1(www)) dwww
Norm with weight function [Wasilkowski, Wozniakowski (2000)]
‖f‖2γγγ =
∑
u⊆{1:s}
1
γu
∫
R|u|
∣∣∣∣∣
∂|u|f
∂yyyu
(yyyu; 0)
∣∣∣∣∣
2∏
j∈u
ϕ2j (yj) dyyyu
also “unanchored” weight function
[K., Sloan, Wasilkowski, Waterhouse (2010); Nichols, K. (2014)]
Pair with randomly shifted lattice rules
CBC error bound for general weights γu
Convergence rate depends on the relationship between φ and ϕj
Fast CBC for POD weights
Important for applications : φ, ϕj , γu are up to us to choose/tune
Frances Kuo @ UNSW Australia p.17
Setting 3: smooth integrands in the cube
Norm involving higher order mixed derivatives
Pair with higher order digital nets [Dick (2008)]
Classical polynomial lattice rule [Niederreiter (1992)]
n = bm points with prime b
An irreducible polynomial with degree m
A generating vector of s polynomials with degree < m
Interlaced polynomial lattice rule [Goda, Dick (2012)]
Interlacing factor α
An irreducible polynomial with degree m
A generating vector of αs polynomials with degree < m
Digit interlacing function Dα : [0, 1)α → [0, 1)
(0.x11x12x13 · · · )b, (0.x21x22x23 · · · )b, . . . , (0.xα1xα2xα3 · · · )b
becomes (0.x11x21 · · ·xα1x12x22 · · ·xα2x13x23 · · ·xα3
· · · )b
[Dick, K., Le Gia, Nuyens, Schwab (2014)]
Variants of fast CBC construction available “SPOD weights”
Frances Kuo @ UNSW Australia p.18
OverviewApplication 1
Option pricing
Application 2
GLMMmaximum likelihood
Application 3
PDEs with random coefficients
Setting 1domain [0, 1]s
randomly shifted lattice rules
mixed first derivatives
first order convergence
Setting 2domain R
s
randomly shifted lattice rules
mixed first derivatives
first order convergence
Setting 3domain [0, 1]s
interlaced polynomial lattice rules
mixed higher derivatives
higher order convergence
Frances Kuo @ UNSW Australia p.19
Application 1: QMC for option pricingPath-dependent option [Giles, K., Sloan, Waterhouse (2008)]
102
103
104
10−4
10−3
10−2
10−1
100
n
standard error
MC
QMC + PCA
naive QMC
∫
Rs
max
(
1
s
s∑
j=1
Sj(zzz) −K, 0
)
exp(−12zzzTΣ−1zzz)
√(2π)s det(Σ)
dzzz
=
∫
Rs
f(yyy)s∏
j=1
φnor(yj) dyyy
=
∫
[0,1]sg(www) dwww
Write Σ = AAT
Sub. zzz = Ayyy
Sub. yyy = Φ-1nor(www)
In the cube, g is unbounded and has a kink
[Griebel, K., Sloan (2010, 2013, 2016)]
ANOVA decomposition: f(yyy) =∑
u⊆{1:s} fu(yyyu)
All fu with u 6= {1 : s} are smooth under BB
[Griewank, Leövey, K., Sloan (in progress)]Smoothing by preintegration
(Pkf)(yyy) =
∫
R
f(yyy)φnor(yk) dyk is smooth for some choice of k[Ian Sloan – Mon 4:50pm Berg A ]
Frances Kuo @ UNSW Australia p.20
Application 2: QMC for maximum likelihood
Generalized linear mixed model [K., Dunsmuir, Sloan, Wand, Womersley (2008)]
∫
Rs
(s∏
j=1
exp(τj(β + zj) − eβ+zj )
τj!
)
exp(−12zzzTΣ−1zzz)
√(2π)s det(Σ)
dzzz
=
∫
Rs
f(yyy)s∏
j=1
φ(yj) dyyy
=
∫
[0,1]sg(www) dwww
Re-center and re-scale, and multiply and divide by φ
Sub. zzz = A∗yyy + zzz∗
Sub. yyy = Φ-1(www)cf. importance sampling
no re-centering (worst) no re-scaling (bad) φ normal (good) φ logistic (better) φ Student-t (best)
In the cube, g and/or its derivatives are unbounded
[K., Sloan, Wasilkowski, Waterhouse (2010)][Nichols, K. (2014)]New weighted space setting over Rs
[Sinescu, K., Sloan (2013)]Differentiate f to estimate the norm
Frances Kuo @ UNSW Australia p.21
Application 3: QMC for PDEs with random coeff.
−∇ · (a(xxx,yyy)∇u(xxx,yyy)) = κ(xxx), xxx ∈ D ⊂ Rd, d = 1, 2, 3
u(xxx,yyy) = 0, xxx ∈ ∂D
Uniform [Cohen, DeVore, Schwab (2011)]
a(xxx,yyy) = a(xxx) +∑
j≥1
yj ψj(xxx), yj ∼ U [−12, 12]
Goal:∫
(−12,12)NG(u(·, yyy)) dyyy G – bounded linear functional
Lognormala(xxx,yyy) = exp(Z(xxx,yyy))
Karhunen–Loève expansion :
Z(xxx,yyy) = Z(xxx) +∑
j≥1
yj√µj ξj(xxx), yj ∼ N (0, 1)
Goal:∫
RN∗
G(u(·, yyy))∏
j≥1
φnor(yj) dyyy
Frances Kuo @ UNSW Australia p.22
Application 3: QMC for PDEs with random coeff.
Error analysis [K., Nuyens (2016) – survey]
∫
(−12,12)NG(u(·, yyy)) dyyy ≈
1
n
n∑
i=1
G(ush(·, ttti))
(1) dimension truncation
(2) FE discretization
(3) QMC quadrature
Balance three sources of error
QMC error
Estimate the norm ‖G(ush)‖γγγ by differentiating the PDE
Choose the weights to minimize (bound on eworγγγ )×(bound on ‖G(us
h)‖γγγ )
Get POD weights (product and order dependent) ⇒ feed into fast CBC construction
∑
j≥1 ‖ψj‖p∞ < ∞ ⇒ O(n−min( 1
p− 1
2,1−δ)) convergence, independently of s
Improved to O(n− 1
p ) convergence, independently of s, with higher order QMC andSPOD weights (smoothness-driven product and order dependent)
Lognormal results depend on smoothness of covariance function
Pairing of QMC rule with function space is crucial
Want the best convergence rate under the least assumption
Want the convergence rate and implied constant to be independent of s
Frances Kuo @ UNSW Australia p.23
Application 3: QMC for PDEs with random coeff.
Extensions
Lognormal a(xxx,yyy) = exp(Z(xxx,yyy))
circulant embedding [Graham, K., Nuyens, Scheichl, Sloan (in progress)]
Z(xxxi, yyy) for i = 1, . . . ,M ⇒M ×M covariance matrixR
Assume stationary covariance function and regular gridExtendR to an s× s circulant matrixRext = AAT
Compute matrix-vector multiplicationAyyy using FFT
H-matrix [Feischl, K., Sloan (in progress)]
Does not require stationary covariance function or regular gridApproximateR by anH2-matrix of interpolation order p, Rp = BBT
Iterative method to computeByyy to some prescribed accuracy
[Michael Feischl – Fri 3:00pm Berg C ]
QMC for PDE eigenvalue problems (neutron transport)[Gilbert, Graham, K., Scheichl, Sloan (in progress)]
QMC for wave propagation models [Ganesh, K., Nuyens, Sloan (in progress)]
QMC for Bayesian inverse problems [several research groups]
Frances Kuo @ UNSW Australia p.24
OverviewApplication 1
Option pricing
Application 2
GLMMmaximum likelihood
Application 3
PDEs with random coefficients
uniform
lognormal
Setting 1domain [0, 1]s
randomly shifted lattice rules
mixed first derivatives
first order convergence
Setting 2domain R
s
randomly shifted lattice rules
mixed first derivatives
first order convergence
Setting 3domain [0, 1]s
interlaced polynomial lattice rules
mixed higher derivatives
higher order convergence
smoothing by preintegration
integral transformation
KL expansion
circulant embedding
H-matrix
Cost reductionSaving 1Saving 2Saving 3
Frances Kuo @ UNSW Australia p.25
Saving 1: multi-level and multi-indexMulti-level method [Heinrich (1998); Giles (2008); . . . ; Giles (2016)]
Telescoping sum
f =∞∑
ℓ=0
(fℓ − fℓ−1), f−1 := 0 AML(f) =L∑
ℓ=0
Qℓ(fℓ − fℓ−1)
|I(f) −AML(f)| ≤ |I(f) − IL(fL)|
︸ ︷︷ ︸
≤ 1
2ε
+L∑
ℓ=0
|(Iℓ −Qℓ)(fℓ − fℓ−1)|
︸ ︷︷ ︸
≤ 1
2ε
choose L specify parameters for Qℓ
Successive differences fℓ − fℓ−1 become smaller
Require less samples for higher dimensional sub-problems
Lagrange multiplier to minimize cost subject to a given error threshold
Multi-index method [Haji-Ali, Nobile, Tempone (2016)]
Apply multi-level simultaneously to more than one aspect of the problem
e.g., quadrature, spatial discretization, time discretization, etc.
Frances Kuo @ UNSW Australia p.26
Saving 2: MDMMultivariate decomposition method [. . . ; Wasilkowski (2013); . . . ]
[K., Nuyens, Plaskota, Sloan, Wasilkowski (in progress)]Decomposition
f =∑
u⊂{1,2,...},|u|<∞
fu AMDM(f) =∑
u∈A
Qu(fu)
|I(f) −AMDM(f)| ≤∑
u/∈A
|Iu(fu)|
︸ ︷︷ ︸
≤ 1
2ε
+∑
u∈A
|(Iu −Qu)(fu)|
︸ ︷︷ ︸
≤ 1
2ε
choose A specify parameters for Qu
Assume bounds on the norms ‖fu‖Fu≤ Bu are given
Qu can be QMC or other quadratures including Smolyak
Cardinality of active set A might be huge, but cardinality of u ∈ A is small
Implementation [Gilbert, K., Nuyens, Wasilkowski (in progress)]
Anchored decomposition fu(xxxu) =∑
v⊆u(−1)|u|−|v|f(xxxv;aaa)
Explore nestedness/embedding in the quadratures
Efficient data structure to reuse function evaluations
Application to PDEs with random coefficients[Alexander Gilbert – Mon 2:25pm Berg A ]
Frances Kuo @ UNSW Australia p.27
Saving 3: QMC fast matrix-vector mult.QMC fast matrix-vector multiplication [Dick, K., Le Gia, Schwab (2015)]
Want to compute yyyiA for all i = 1, . . . , n
Row vectors yyyi = ϕ(ttti) where ttti are QMC points
Write Y ′ =
yyy1...
yyyn−1
= CP
C(n−1)×(n−1) is circulant
P has a single 1 in each columnand 0 everywhere else
Can compute Y ′aaa in O(n logn) operations for any column aaa of A
When does/doesn’t this work?deterministic lattice points, deterministic polynomial lattice pointsinverse cumulative normal, tent transformdoes not work with random shifting, scrambling, interlacing
Where does this appy?Generating normally distributed points with a general covariance matrixPDEs with uniform random coefficientsPDEs with lognormal random coefficients involving FE quadrature
Frances Kuo @ UNSW Australia p.28
Summary and concluding remarksApplication 1
Option pricing
Application 2
GLMMmaximum likelihood
Application 3
PDEs with random coefficients
uniform
lognormal
Setting 1domain [0, 1]s
randomly shifted lattice rules
mixed first derivatives
first order convergence
Setting 2domain R
s
randomly shifted lattice rules
mixed first derivatives
first order convergence
Setting 3domain [0, 1]s
interlaced polynomial lattice rules
mixed higher derivatives
higher order convergence
smoothing by preintegration
integral transformation
KL expansion
circulant embedding
H-matrix
Cost reductionSaving 1 multi-level and multi-index methods
Saving 2 multivariate decomposition method (MDM)
Saving 3 QMC fast matrix-vector multiplication
Frances Kuo @ UNSW Australia p.29
Setting 4: higher order QMC for Rs???
Dick, Leobacher, Irrgeher, Pillichshammer
[Christian Irrgeher – Mon 5:20pm Berg A ]
Cools, Nguyen, Nuyens [Dong Nguyen – MCM Linz 2015 ]
Stretch the unit cube
Truncate the integrand
Apply higher order QMC without Φ-1 mapping
s dependence???
Frances Kuo @ UNSW Australia p.30
Thanks to my collaborators
AU
Austria
Belgium
Ecuador
Germany
NZ
Poland
Switzerland
UK
USA