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Multiscale Problems: Numerical Analysisand Scientific Computing
with Applications in Energy & Environment
Robert Scheichl
Department of Mathematical SciencesUniversity of Bath
LMS Meeting on Prospects in MathematicsUniversity of Manchester, Wednesday 19th December 2012
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 1 / 42
What is Numerical Analysis & Scientific Computing
“Classical” Areas (incomplete list)
Numerical linear algebra (incl. iterative solvers) F. Tisseur
Discretisation schemes (temporal & spatial): design & analysis
Nonlinear problems and optimisation
Applications in CFD, structures, etc
“Novel” Areas (incomplete list)
Multiscale problems, quantitative mathematical biology
Uncertainty quantification, Bayesian inverse problems,data assimilation, stochastic differential equations
Networks, compressed sensing, random matrices
Massively parallel, GPU clusters
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 2 / 42
What is Numerical Analysis & Scientific Computing
“Classical” Areas (incomplete list)
Numerical linear algebra (incl. iterative solvers) F. Tisseur
Discretisation schemes (temporal & spatial): design & analysis
Nonlinear problems and optimisation
Applications in CFD, structures, etc
“Novel” Areas (incomplete list)
Multiscale problems, quantitative mathematical biology
Uncertainty quantification, Bayesian inverse problems,data assimilation, stochastic differential equations
Networks, compressed sensing, random matrices
Massively parallel, GPU clusters
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 2 / 42
Sustainable Energy & the Environment(my main application area)
Carbon-based (Oil, Gas, Coal, . . . )
→ Oil Reservoir & Sedimentary Basin Simulation
→ Carbon Capture and Storage Underground
Nuclear
→ Longterm Disposal of Radioactive Waste Underground
→ Reactor Safety: Neutron Diffusion/Transport Equations
Alternative (Wind, Solar, Wave, . . . )
→ Weather and Climate Prediction
→ Wave Energy
→ Fuel Cells (also porous medium flow)
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 3 / 42
Hierarchy of Scales
Subsurface
Other (e.g. atmospheric)
→ Range of length and time scales
→ Heterogeneities and anisotropies
→ Lack of and uncertainty in data
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 4 / 42
Hierarchy of Scales
Subsurface Other (e.g. atmospheric)
→ Range of length and time scales
→ Heterogeneities and anisotropies
→ Lack of and uncertainty in data
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 4 / 42
Challenges for Numerical Analysis & Scientific ComputingAccurate, reliable & efficient numerical tools to simulate subsurface& atmospheric flow to predict impact of current decisions & policies.
PDEs at heart (diffusion, convection,...)
Numerical simulation – discretisation (FEM, FVM, DG . . . )
Remarkably similar issues in all applications (cf. previous slide):
i.e. domain of interest several km ↔ rocks vary on mm-scale!
Modelling of all scales currently impossible(even on largest supercomputer)
=⇒ Hierarchical Multiscale Numerical Methods
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 5 / 42
Challenges for Numerical Analysis & Scientific ComputingAccurate, reliable & efficient numerical tools to simulate subsurface& atmospheric flow to predict impact of current decisions & policies.
PDEs at heart (diffusion, convection,...)
Numerical simulation – discretisation (FEM, FVM, DG . . . )
Remarkably similar issues in all applications (cf. previous slide):
i.e. domain of interest several km ↔ rocks vary on mm-scale!
Modelling of all scales currently impossible(even on largest supercomputer)
=⇒ Hierarchical Multiscale Numerical Methods
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 5 / 42
Hierarchical Multiscale Numerical MethodsMultilevel Iterative Methods(homogeneous coefficients in PDE)
Multigrid, AMG, DDM(with theory!)
(Theory?)
Multiscale/Upscaling Methods(heterogeneous coefficients in PDE)
↓
Homogenisation(with theory!)
↓
MsFEM, HMM
(Theory?)
Huge potential & many open questions
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 6 / 42
Hierarchical Multiscale Numerical MethodsMultilevel Iterative Methods(homogeneous coefficients in PDE)
Multigrid, AMG, DDM(with theory!)
(Theory?)
Multiscale/Upscaling Methods(heterogeneous coefficients in PDE)
↓
Homogenisation(with theory!)
↓
MsFEM, HMM
(Theory?)
Huge potential & many open questions
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 6 / 42
Hierarchical Multiscale Numerical MethodsMultilevel Iterative Methods(homogeneous coefficients in PDE)
Multigrid, AMG, DDM(with theory!)
(Theory?)
Multiscale/Upscaling Methods(heterogeneous coefficients in PDE)
↓
Homogenisation(with theory!)
↓
MsFEM, HMM
(Theory?)
Huge potential & many open questions
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 6 / 42
Hierarchical Multiscale Numerical MethodsMultilevel Iterative Methods(homogeneous coefficients in PDE)
Multigrid, AMG, DDM(with theory!)
(Theory?)
Multiscale/Upscaling Methods(heterogeneous coefficients in PDE)
↓
Homogenisation(with theory!)
↓
MsFEM, HMM(Theory?)
Huge potential & many open questions
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 6 / 42
Hierarchical Multiscale Numerical MethodsMultilevel Iterative Methods(homogeneous coefficients in PDE)
Multigrid, AMG, DDM(with theory!)
(Theory?)
Multiscale/Upscaling Methods(heterogeneous coefficients in PDE)
↓
Homogenisation(with theory!)
↓
MsFEM, HMM(Theory?)
Huge potential & many open questions
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 6 / 42
Extension of Hierarchical Approach to . . .
Systems of PDEs (Multiphase Flow, Multiphysics)
Oil Saturation
0.00
0.95
Vandji Sandstones Lower Cenomanian
TuronianUpper Cenomanian
e.g. CO2 capture & storage underground
Data Assimilation (Lack of Data, Inverse Problem)
e.g. find current state of atmosphere(given previous forecast and measurements)
Stochastic Modelling of Data Uncertainties (see below)
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 7 / 42
PhD Opportunities – Some SnapshotsBiased towards my research, but also some other topics
The multilevel Monte Carlo idea (vast gains!)
New multigrid theory: weighted Poincare inequalities
Massively parallel implementations on CPU & GPU clusters
Other PhD opportunities (in Bath and elsewhere in UK)
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 8 / 42
The Multilevel Monte Carlo Idea
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 9 / 42
Monte Carlo for large scale problems (plain vanilla)
ZJ(ω) ∈ RJ Model(M)−→ XM(ω) ∈ RM Output−→ QM,J(ω) ∈ R or Rk
random input state vector quantity of interest
e.g. ZJ multivariate Gaussian; XM numerical solution of PDE;QM,J a nonlinear functional of XM
QM,J(ω) approximates inaccessible random variable Q(ω) s.t.
|E[QM,J − Q]| = O(M−α) +O(J−α′) as M , J→∞
Standard Monte Carlo estimator for E[Q]:
QMC :=1
N
N∑i=1
Q(i)M,J
where Q(i)M,JNi=1 are i.i.d. samples computed with Model(M)
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 10 / 42
Monte Carlo for large scale problems (plain vanilla)
ZJ(ω) ∈ RJ Model(M)−→ XM(ω) ∈ RM Output−→ QM,J(ω) ∈ R or Rk
random input state vector quantity of interest
e.g. ZJ multivariate Gaussian; XM numerical solution of PDE;QM,J a nonlinear functional of XM
QM,J(ω) approximates inaccessible random variable Q(ω) s.t.
|E[QM,J − Q]| = O(M−α) +O(J−α′) as M , J→∞
Standard Monte Carlo estimator for E[Q]:
QMC :=1
N
N∑i=1
Q(i)M,J
where Q(i)M,JNi=1 are i.i.d. samples computed with Model(M)
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 10 / 42
Monte Carlo for large scale problems (plain vanilla)
Convergence of plain vanilla MC (mean square error):
E[(QMC − E[Q]
)2]︸ ︷︷ ︸=: MSE
= V[QMC] +(E[QMC]− E[Q]
)2
=V[QM,J ]
N︸ ︷︷ ︸sampling error
+(E[QM,J − Q]
)2
︸ ︷︷ ︸model error (“bias”)
Typically (see below): α = 1/2 ⇒ MSE = O(N−1) +O(M−1)
Thus for MSE < TOL2 we get Cost = O(MN) = O(TOL−4)
(e.g. for TOL = 10−3 we need M ∼ N ∼ 106 and Cost = O(1012) !!)
Quickly becomes prohibitively expensive !
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 11 / 42
Monte Carlo for large scale problems (plain vanilla)
Convergence of plain vanilla MC (mean square error):
E[(QMC − E[Q]
)2]︸ ︷︷ ︸=: MSE
= V[QMC] +(E[QMC]− E[Q]
)2
=V[QM,J ]
N︸ ︷︷ ︸sampling error
+(E[QM,J − Q]
)2
︸ ︷︷ ︸model error (“bias”)
Typically (see below): α = 1/2 ⇒ MSE = O(N−1) +O(M−1)
Thus for MSE < TOL2 we get Cost = O(MN) = O(TOL−4)
(e.g. for TOL = 10−3 we need M ∼ N ∼ 106 and Cost = O(1012) !!)
Quickly becomes prohibitively expensive !
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 11 / 42
Uncertainty Quantification in Groundwater FlowBasic ideas extend directly to other examples
uncertain k →
Darcy’s Law: ~q + k ∇p = f
Incompressibility: ∇ · ~q = 0
+ Boundary Conditions
→ uncertain p, ~q
Society of Petroleum Engineers Benchmark SPE10R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 12 / 42
Uncertainty Quantification in Groundwater FlowBasic ideas extend directly to other examples
uncertain k →Darcy’s Law: ~q + k ∇p = f
Incompressibility: ∇ · ~q = 0
+ Boundary Conditions
→ uncertain p, ~q
Society of Petroleum Engineers Benchmark SPE10R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 12 / 42
Stochastic Modelling of Uncertainty
Typical simplified model (prior):
log k(x , ω) = (isotropic) Gaussian
meanfree with exponential covariance:
R(x , y) := σ2 exp
(−‖x − y‖
λ
)
Typical quantities of interest:
e.g. p(x∗) or effective keff := 1|D|
∫Dq1
Incorporating data (posterior):
Markov chain Monte Carlo
typical realisation
(λ = 164 , σ
2 = 8)
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 13 / 42
Stochastic Modelling of Uncertainty
Typical simplified model (prior):
log k(x , ω) = (isotropic) Gaussian
meanfree with exponential covariance:
R(x , y) := σ2 exp
(−‖x − y‖
λ
)
Typical quantities of interest:
e.g. p(x∗) or effective keff := 1|D|
∫Dq1
Incorporating data (posterior):
Markov chain Monte Carlo
typical realisation
(λ = 164 , σ
2 = 8)
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 13 / 42
Stochastic Modelling of Uncertainty
Typical simplified model (prior):
log k(x , ω) = (isotropic) Gaussian
meanfree with exponential covariance:
R(x , y) := σ2 exp
(−‖x − y‖
λ
)
Typical quantities of interest:
e.g. p(x∗) or effective keff := 1|D|
∫Dq1
Incorporating data (posterior):
Markov chain Monte Carlo
typical realisation
(λ = 164 , σ
2 = 8)
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 13 / 42
Key Computational Challengesin solving PDEs with highly heterogeneous random coefficients
−∇·(k(x , ω)∇p(x , ω)) = f (x , ω), x ∈ D ⊂ Rd , ω ∈ Ω (prob. space)
1 Sampling from random field k(x , ω)(Karhunen-Loeve expansion, factorisation, FFT-based, PDE-based, . . . )
2 High-dimensional integration over Ω(Monte Carlo & variants, stochastic Galerkin/collocation, . . . )
3 Solving large number of heterogeneous deterministic PDEs(Krylov, Multigrid, AMG, DD Methods, . . . )
Large stochastic dimension =⇒ dim(Ω) 100
Low spatial regularity (‘rough’) =⇒ fine mesh h 1
Large variance σ2 & exponentiation =⇒ high contrast kmaxkmin
> 106
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 14 / 42
Key Computational Challengesin solving PDEs with highly heterogeneous random coefficients
−∇·(k(x , ω)∇p(x , ω)) = f (x , ω), x ∈ D ⊂ Rd , ω ∈ Ω (prob. space)
1 Sampling from random field k(x , ω)(Karhunen-Loeve expansion, factorisation, FFT-based, PDE-based, . . . )
2 High-dimensional integration over Ω(Monte Carlo & variants, stochastic Galerkin/collocation, . . . )
3 Solving large number of heterogeneous deterministic PDEs(Krylov, Multigrid, AMG, DD Methods, . . . )
Large stochastic dimension =⇒ dim(Ω) 100
Low spatial regularity (‘rough’) =⇒ fine mesh h 1
Large variance σ2 & exponentiation =⇒ high contrast kmaxkmin
> 106
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 14 / 42
Key Computational Challengesin solving PDEs with highly heterogeneous random coefficients
−∇·(k(x , ω)∇p(x , ω)) = f (x , ω), x ∈ D ⊂ Rd , ω ∈ Ω (prob. space)
1 Sampling from random field k(x , ω)(Karhunen-Loeve expansion, factorisation, FFT-based, PDE-based, . . . )
2 High-dimensional integration over Ω(Monte Carlo & variants, stochastic Galerkin/collocation, . . . )
3 Solving large number of heterogeneous deterministic PDEs(Krylov, Multigrid, AMG, DD Methods, . . . )
Why is this problem so challenging?
Large stochastic dimension =⇒ dim(Ω) 100
Low spatial regularity (‘rough’) =⇒ fine mesh h 1
Large variance σ2 & exponentiation =⇒ high contrast kmaxkmin
> 106
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 14 / 42
Key Computational Challengesin solving PDEs with highly heterogeneous random coefficients
−∇·(k(x , ω)∇p(x , ω)) = f (x , ω), x ∈ D ⊂ Rd , ω ∈ Ω (prob. space)
1 Sampling from random field k(x , ω)(Karhunen-Loeve expansion, factorisation, FFT-based, PDE-based, . . . )
2 High-dimensional integration over Ω(Monte Carlo & variants, stochastic Galerkin/collocation, . . . )
3 Solving large number of heterogeneous deterministic PDEs(Krylov, Multigrid, AMG, DD Methods, . . . )
Why is this problem so challenging?
Large stochastic dimension =⇒ dim(Ω) 100
Low spatial regularity (‘rough’) =⇒ fine mesh h 1
Large variance σ2 & exponentiation =⇒ high contrast kmaxkmin
> 106
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 14 / 42
Spatial discretisation with M = O(h−d) DOFs
Standard pw. linear FEs or cell-centred FVs:
−∇ · (k(x , ω)∇p(x , ω)) = f (x) −→ A(ω) XM(ω) = b(ω)
random elliptic PDE random M ×M linear system
Quantity of interest: Expected value E[Q] of Q := G(p)functional of the PDE solution p; in practice compute QM,J := GM(XM)
Recall (plain vanilla) Monte Carlo (MC) estimate for E[Q]:
QMC :=1
N
N∑i=1
Q(i)M,J , Q
(i)M,J i.i.d. samples with Model(M) .
Assume optimal multigrid solver ⇒ Cost(Q(i)M,J) = O(M)
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 15 / 42
Spatial discretisation with M = O(h−d) DOFs
Standard pw. linear FEs or cell-centred FVs:
−∇ · (k(x , ω)∇p(x , ω)) = f (x) −→ A(ω) XM(ω) = b(ω)
random elliptic PDE random M ×M linear system
Quantity of interest: Expected value E[Q] of Q := G(p)functional of the PDE solution p; in practice compute QM,J := GM(XM)
Recall (plain vanilla) Monte Carlo (MC) estimate for E[Q]:
QMC :=1
N
N∑i=1
Q(i)M,J , Q
(i)M,J i.i.d. samples with Model(M) .
Assume optimal multigrid solver ⇒ Cost(Q(i)M,J) = O(M)
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 15 / 42
What is cost to get MSE < TOL2?
Complexity Theorem for (plain vanilla) Monte Carlo
Assume that QM,J → Q with O(M−α) for some α > 0, then to obtainMSE < TOL2
Cost(QMC) = O(TOL−2− 1
α
)
Numerical Example (D = (0, 1)2,Q = keff , mixed FE & amg1r5)
Case 1: λ = 0.3, σ2 = 1
TOL M N Cost
0.01 1.7× 104 1.4× 104 21min
0.002 1.1× 106 3.5× 105 30days
Case 2: λ = 0.1, σ2 = 3
TOL M N Cost
0.01 2.6× 105 8.5× 103 4h
0.002 Prohibitively large!!
Here d = 2 & α ≈ 3/8 ⇒ Cost ≈ O(TOL−14/3) ≈ 25 × more work to halve error!
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 16 / 42
What is cost to get MSE < TOL2?
Complexity Theorem for (plain vanilla) Monte Carlo
Assume that QM,J → Q with O(M−α) for some α > 0, then to obtainMSE < TOL2
Cost(QMC) = O(TOL−2− 1
α
)Numerical Example (D = (0, 1)2,Q = keff , mixed FE & amg1r5)
Case 1: λ = 0.3, σ2 = 1
TOL M N Cost
0.01 1.7× 104 1.4× 104 21min
0.002 1.1× 106 3.5× 105 30days
Case 2: λ = 0.1, σ2 = 3
TOL M N Cost
0.01 2.6× 105 8.5× 103 4 h
0.002 Prohibitively large!!
Here d = 2 & α ≈ 3/8 ⇒ Cost ≈ O(TOL−14/3) ≈ 25 × more work to halve error!
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 16 / 42
Multilevel Monte Carlo [Heinrich, ’01], [Giles, ’07]
[Cliffe, Giles, RS, Teckentrup, ’11]Note that trivially
E[QL] = E[Q0] +∑L
`=1E[Q` − Q`−1]
(where for simplicity h` = h`−1/2, M` = 2dM`−1 and Q` := QM`,J`)
Idea: Define the following multilevel MC estimator for E[Q]
QMLL := QMC0 +
∑L
`=1Y MC` where Y` := Q` − Q`−1
Key Observation (much cheaper to compute corrections!!)
If Q` → Q then V[Y`]→ 0 as h` → 0
and the sampling error on level ` is V[Y`]/N` !
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 17 / 42
Multilevel Monte Carlo [Heinrich, ’01], [Giles, ’07]
[Cliffe, Giles, RS, Teckentrup, ’11]Note that trivially
E[QL] = E[Q0] +∑L
`=1E[Q` − Q`−1]
(where for simplicity h` = h`−1/2, M` = 2dM`−1 and Q` := QM`,J`)
Idea: Define the following multilevel MC estimator for E[Q]
QMLL := QMC0 +
∑L
`=1Y MC` where Y` := Q` − Q`−1
Key Observation (much cheaper to compute corrections!!)
If Q` → Q then V[Y`]→ 0 as h` → 0
and the sampling error on level ` is V[Y`]/N` !
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 17 / 42
Multilevel Monte Carlo [Heinrich, ’01], [Giles, ’07]
[Cliffe, Giles, RS, Teckentrup, ’11]Note that trivially
E[QL] = E[Q0] +∑L
`=1E[Q` − Q`−1]
(where for simplicity h` = h`−1/2, M` = 2dM`−1 and Q` := QM`,J`)
Idea: Define the following multilevel MC estimator for E[Q]
QMLL := QMC0 +
∑L
`=1Y MC` where Y` := Q` − Q`−1
Key Observation (much cheaper to compute corrections!!)
If Q` → Q then V[Y`]→ 0 as h` → 0
and the sampling error on level ` is V[Y`]/N` !
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 17 / 42
Complexity Theorem for Multilevel Monte Carlo
Assume Cost/sample O(M) and FE error O(M−α) (as above) as wellas
V[Q` − Q`−1] = O(M−β` ).
There exist L, N`L`=0 (computable on the fly) to obtain MSE < TOL2
with
Cost(QMLL ) = O(
TOL−2−max(0, 1−βα ))
+ possible log-factor
(Note. This is completely abstract! Applies also in other applications!)
If β ∼ 2α (as in example above) then
Cost(QMLL ) = O(
TOL−max(2, 1α))
= O (max(N0,ML)) !!
For α ≈ 3/8 (in example above): O(TOL−8/3) instead of O(TOL−14/3)
Asymptotically same cost as deterministic solver to the same accuracy !
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 18 / 42
Complexity Theorem for Multilevel Monte Carlo
Assume Cost/sample O(M) and FE error O(M−α) (as above) as wellas
V[Q` − Q`−1] = O(M−β` ).
There exist L, N`L`=0 (computable on the fly) to obtain MSE < TOL2
with
Cost(QMLL ) = O(
TOL−2−max(0, 1−βα ))
+ possible log-factor
(Note. This is completely abstract! Applies also in other applications!)
If β ∼ 2α (as in example above) then
Cost(QMLL ) = O(
TOL−max(2, 1α))
= O (max(N0,ML)) !!
For α ≈ 3/8 (in example above): O(TOL−8/3) instead of O(TOL−14/3)
Asymptotically same cost as deterministic solver to the same accuracy !
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 18 / 42
Complexity Theorem for Multilevel Monte Carlo
Assume Cost/sample O(M) and FE error O(M−α) (as above) as wellas
V[Q` − Q`−1] = O(M−β` ).
There exist L, N`L`=0 (computable on the fly) to obtain MSE < TOL2
with
Cost(QMLL ) = O(
TOL−2−max(0, 1−βα ))
+ possible log-factor
(Note. This is completely abstract! Applies also in other applications!)
If β ∼ 2α (as in example above) then
Cost(QMLL ) = O(
TOL−max(2, 1α))
= O (max(N0,ML)) !!
For α ≈ 3/8 (in example above): O(TOL−8/3) instead of O(TOL−14/3)
Asymptotically same cost as deterministic solver to the same accuracy !
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 18 / 42
Numerical ExampleD = (0, 1)2, σ2 = 1, λ = 0.1, h0 = 1
16 , dim(Ω) = 500, standard FE, UMFPACK
Q = keff , TOL = 0.001 Q = k2eff , h−1
L = 256
(second moment!)
Matlab implementation on 3GHz Intel Core 2 Duo E8400 proc, 3.2GByte RAM
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 19 / 42
Theory: Verifying Assumptions for Subsurface ApplicationMostly done by two PhD students A. Teckentrup (Bath) & J. Charrier (Rennes).
[Charrier, SINUM ’12]: analyses error in approximating input random field
[Charrier, RS, Teckentrup, SINUM ’13]: lognormal k (not uniformly
elliptic/bounded and low regularity k ∈ C t(D) with t < 1/2):
I Regularity result: all finite moments of ‖p‖H1+s bdd. (∀s < t)
I FE error analysis: all finite moments of H1-error are O(hs)
[Teckentrup, RS, Giles, Ullmann, NM ’13]: extension to (nonlinear)functionals, corner domains, discont. coeffs, level-dependent truncations
Multilevel idea and theory extends to Markov chain Monte Carlo[Ketelsen, Teckentrup, RS, Vassilevski, in preperation]
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 20 / 42
Theory: Verifying Assumptions for Subsurface ApplicationMostly done by two PhD students A. Teckentrup (Bath) & J. Charrier (Rennes).
[Charrier, SINUM ’12]: analyses error in approximating input random field
[Charrier, RS, Teckentrup, SINUM ’13]: lognormal k (not uniformly
elliptic/bounded and low regularity k ∈ C t(D) with t < 1/2):
I Regularity result: all finite moments of ‖p‖H1+s bdd. (∀s < t)
I FE error analysis: all finite moments of H1-error are O(hs)
[Teckentrup, RS, Giles, Ullmann, NM ’13]: extension to (nonlinear)functionals, corner domains, discont. coeffs, level-dependent truncations
Multilevel idea and theory extends to Markov chain Monte Carlo[Ketelsen, Teckentrup, RS, Vassilevski, in preperation]
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 20 / 42
Current & Future Projects (on Multilevel MC & MCMC)
UQ for groundwater flow: longterm radioactive waste disposal(current PhD & current PDRA w. Nottingham, Oxford, NDA & AMEC)
Turbulent atmospheric dispersion (volcanic ash, ...)(current PhD with Met Office)
Levy driven SDEs & Applications in Math Finance(current PDRA and open PhD 2013/14 with A. Kyprianou, Probability)
Carbon fibre composites: multiscale methods, UQ, optimisation(open PhD 2013/14 and possible PDRA with R. Butler, Material Sci.)
Bayesian inverse problems and applications in oil recovery(possible future PhD and PDRA with Schlumberger)
Data assimilation (particle filters, ...) in NWP and Climate(possible future PhD and PDRA with Imperial & Met Office)
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 21 / 42
Current & Future Projects (on Multilevel MC & MCMC)
UQ for groundwater flow: longterm radioactive waste disposal(current PhD & current PDRA w. Nottingham, Oxford, NDA & AMEC)
Turbulent atmospheric dispersion (volcanic ash, ...)(current PhD with Met Office)
Levy driven SDEs & Applications in Math Finance(current PDRA and open PhD 2013/14 with A. Kyprianou, Probability)
Carbon fibre composites: multiscale methods, UQ, optimisation(open PhD 2013/14 and possible PDRA with R. Butler, Material Sci.)
Bayesian inverse problems and applications in oil recovery(possible future PhD and PDRA with Schlumberger)
Data assimilation (particle filters, ...) in NWP and Climate(possible future PhD and PDRA with Imperial & Met Office)
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 21 / 42
New Multigrid Theory
Weighted Poincare Inequalities & Abstract Bramble-Hilbert Lemma
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 22 / 42
Model Problem (deterministic version of above)
How realistic is assumption cost/sample = O(h−d)?
Elliptic PDE in bounded domain D ⊂ Rd , d = 2, 3
−∇ · (k∇p) = f
Highly variable coefficient k(x)
FE discretisation (cts. p.w. linears V h): AX = b
Two possible aims:
I h-optimal, k-robust solver (k resolved on fine mesh T h)
I H-optimal, k-robust approximation in coarse space VH
(k not resolved on T H – “Upscaling”)
Key Question (for both): Coefficient-robust coarsening
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 23 / 42
Model Problem (deterministic version of above)
How realistic is assumption cost/sample = O(h−d)?
Elliptic PDE in bounded domain D ⊂ Rd , d = 2, 3
−∇ · (k∇p) = f
Highly variable coefficient k(x)
FE discretisation (cts. p.w. linears V h): AX = b
Two possible aims:
I h-optimal, k-robust solver (k resolved on fine mesh T h)
I H-optimal, k-robust approximation in coarse space VH
(k not resolved on T H – “Upscaling”)
Key Question (for both): Coefficient-robust coarsening
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 23 / 42
Model Problem (deterministic version of above)
How realistic is assumption cost/sample = O(h−d)?
Elliptic PDE in bounded domain D ⊂ Rd , d = 2, 3
−∇ · (k∇p) = f
Highly variable coefficient k(x)
FE discretisation (cts. p.w. linears V h): AX = b
Two possible aims:
I h-optimal, k-robust solver (k resolved on fine mesh T h)
I H-optimal, k-robust approximation in coarse space VH
(k not resolved on T H – “Upscaling”)
Key Question (for both): Coefficient-robust coarsening
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 23 / 42
Model Problem (deterministic version of above)
How realistic is assumption cost/sample = O(h−d)?
Elliptic PDE in bounded domain D ⊂ Rd , d = 2, 3
−∇ · (k∇p) = f
Highly variable coefficient k(x)
FE discretisation (cts. p.w. linears V h): AX = b
Two possible aims:
I h-optimal, k-robust solver (k resolved on fine mesh T h) Now
I H-optimal, k-robust approximation in coarse space VH
(k not resolved on T H – “Upscaling”) Hot Topic
Key Question (for both): Coefficient-robust coarsening
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 23 / 42
Other Applications: Simulation in Heterogeneous Media
Elasticity, e.g. in wood or bone
Electrostatics (Al2O3–TiO2) or Nonlinear Magnetostatics
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 24 / 42
Difficulties
Resolving k requires very fine mesh: h diam(D)
A very large (M = O(h−d)) and very ill-conditioned:
κ(A) . maxx ,y∈D
k(x)
k(y)h−2
Complicated variation of k(x) on many scales(hard to resolve by “geometric” coarse mesh)
Multilevel iterative methods optimal w.r.t. mesh size h, i.e.
Total Cost ≈ O(M)
but: How does constant depend on variations in k?
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 25 / 42
Difficulties
Resolving k requires very fine mesh: h diam(D)
A very large (M = O(h−d)) and very ill-conditioned:
κ(A) . maxx ,y∈D
k(x)
k(y)h−2
Complicated variation of k(x) on many scales(hard to resolve by “geometric” coarse mesh)
Multilevel iterative methods optimal w.r.t. mesh size h, i.e.
Total Cost ≈ O(M)
but: How does constant depend on variations in k?
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 25 / 42
Classical theory
in standard H1 and H1/2 Sobolev norms
based on standard Poincare inequalities
New theory
directly in the (coefficient-dependent) energy norm
based on (coefficient) weighted Poincare type inequalities
I [Pechstein, RS, DD19 Proc 2009, IMAJNA 2012]
and an abstract Bramble-Hilbert Lemma
I [RS, Vassilevski, Zikatanov, MMS 2011]
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 26 / 42
Classical theory
in standard H1 and H1/2 Sobolev norms
based on standard Poincare inequalities
New theory
directly in the (coefficient-dependent) energy norm
based on (coefficient) weighted Poincare type inequalities
I [Pechstein, RS, DD19 Proc 2009, IMAJNA 2012]
and an abstract Bramble-Hilbert Lemma
I [RS, Vassilevski, Zikatanov, MMS 2011]
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 26 / 42
Subspace Correction Methods (e.g. multigrid)FE Problem (in variational form): Find uh ∈ Vh s.t.
a(uh, vh) ≡∫Dα∇uh · ∇vh = (f , vh) for all vh ∈ Vh.
Iterate by solving (exactly or approx.) in subspaces V0,V1, ...VL ⊂ Vh
in parallel (additive) or successively (multiplicative)
Two-level overlapping Schwarz
Ω`L`=1 overlapping cover of D
V` = Vh(Ω`) (i.e. restricted to D`)
Ω2Ω Ω31
χ1 2
χ3
χ
Geometric Multigrid
T `L`=1 nested triangulations of D
V` = p.w. lin. FE space on T `
V1= V2= + = all3V
V0 = spanΦ0j ∈ Vh : j = 1, . . . ,M0, e.g. p.w. lin. FE space on T 0 = TH
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 27 / 42
Subspace Correction Methods (e.g. multigrid)FE Problem (in variational form): Find uh ∈ Vh s.t.
a(uh, vh) ≡∫Dα∇uh · ∇vh = (f , vh) for all vh ∈ Vh.
Iterate by solving (exactly or approx.) in subspaces V0,V1, ...VL ⊂ Vh
in parallel (additive) or successively (multiplicative)
Two-level overlapping Schwarz
Ω`L`=1 overlapping cover of D
V` = Vh(Ω`) (i.e. restricted to D`)
Ω2Ω Ω31
χ1 2
χ3
χ
Geometric Multigrid
T `L`=1 nested triangulations of D
V` = p.w. lin. FE space on T `
V1= V2= + = all3V
V0 = spanΦ0j ∈ Vh : j = 1, . . . ,M0, e.g. p.w. lin. FE space on T 0 = TH
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 27 / 42
Subspace Correction Methods (e.g. multigrid)FE Problem (in variational form): Find uh ∈ Vh s.t.
a(uh, vh) ≡∫Dα∇uh · ∇vh = (f , vh) for all vh ∈ Vh.
Iterate by solving (exactly or approx.) in subspaces V0,V1, ...VL ⊂ Vh
in parallel (additive) or successively (multiplicative)
Two-level overlapping Schwarz
Ω`L`=1 overlapping cover of D
V` = Vh(Ω`) (i.e. restricted to D`)
Ω2Ω Ω31
χ1 2
χ3
χ
Geometric Multigrid
T `L`=1 nested triangulations of D
V` = p.w. lin. FE space on T `
V1= V2= + = all3V
V0 = spanΦ0j ∈ Vh : j = 1, . . . ,M0, e.g. p.w. lin. FE space on T 0 = TH
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 27 / 42
Subspace Correction Methods (e.g. multigrid)FE Problem (in variational form): Find uh ∈ Vh s.t.
a(uh, vh) ≡∫Dα∇uh · ∇vh = (f , vh) for all vh ∈ Vh.
Iterate by solving (exactly or approx.) in subspaces V0,V1, ...VL ⊂ Vh
in parallel (additive) or successively (multiplicative)
Two-level overlapping Schwarz
Ω`L`=1 overlapping cover of D
V` = Vh(Ω`) (i.e. restricted to D`)
Ω2Ω Ω31
χ1 2
χ3
χ
Geometric Multigrid
T `L`=1 nested triangulations of D
V` = p.w. lin. FE space on T `
V1= V2= + = all3V
V0 = spanΦ0j ∈ Vh : j = 1, . . . ,M0, e.g. p.w. lin. FE space on T 0 = TH
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 27 / 42
Theorem (Multigrid Convergence) [RS, Vassilevski, Zikatanov, SINUM ’11]
For all K ∈ TH , let CPK > 0 be such that the following weighted
Poincare inequality holds (with a slight variation near Dirichlet boundaries):
infγ∈R
∫ωK
k(x)(v − γ)2 ≤ CPK diam(ωK )2
∫ωK
k(x)|∇v |2 ∀v ∈ Vh
where ωK :=⋃K ′ : K ∩ K ′ 6= ∅ (the “neighbourhood” of K ).
Then the geometric multigrid algorithm converges with rate
ρMG ≤ 1− 1
κ
where κ . c L maxK∈TH CPK and c is independent of k(x), L and h.
There is no a priori assumption that coefficient is resolved on grids!
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 28 / 42
Poincare’s inequality
Domain ω ⊂ Rd (open, bounded, connected set). ∃C > 0 s.t.
infγ∈R‖u − γ‖2
L2(ω) ≤ C diam (ω)2 |∇u|2L2(ω) ∀u ∈ H1(ω).
C depends only on shape of ω, not on diam (ω)
Infimum attained at
γ∗ = uω :=1
|ω|
∫ω
u dx
Inequality also works for
γ = uX :=1
|X |
∫X
u dx
where X ⊂ ω is a (d − 1)-dimensional manifold
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 29 / 42
Poincare’s inequality
Domain ω ⊂ Rd (open, bounded, connected set). ∃C > 0 s.t.
infγ∈R‖u − γ‖2
L2(ω) ≤ C diam (ω)2 |∇u|2L2(ω) ∀u ∈ H1(ω).
C depends only on shape of ω, not on diam (ω)
Infimum attained at
γ∗ = uω :=1
|ω|
∫ω
u dx
Inequality also works for
γ = uX :=1
|X |
∫X
u dx
where X ⊂ ω is a (d − 1)-dimensional manifold
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 29 / 42
Weighted Poincare type inequalityFor k ∈ L∞(ω) uniformly positive, we define
‖v‖2L2(ω),k :=
∫ω
k |v |2dx and |v |2H1(ω),k :=
∫ω
k |∇v |2dx
Clearly,
‖u − uX‖2L2(ω),k ≤ C max
x ,y∈ω
k(x)
k(y)diam (ω)2 |u|2H1(ω),k
Question
Can we find CP independent of variation & contrast in k such that
infγ∈R‖u − γ‖2
L2(ω),k ≤ CP |u|2H1(ω),k
for some class of weights k : ω → R+ ?
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 30 / 42
Weighted Poincare type inequalityFor k ∈ L∞(ω) uniformly positive, we define
‖v‖2L2(ω),k :=
∫ω
k |v |2dx and |v |2H1(ω),k :=
∫ω
k |∇v |2dx
Clearly,
‖u − uX‖2L2(ω),k ≤ C max
x ,y∈ω
k(x)
k(y)diam (ω)2 |u|2H1(ω),k
Question
Can we find CP independent of variation & contrast in k such that
infγ∈R‖u − γ‖2
L2(ω),k ≤ CP |u|2H1(ω),k
for some class of weights k : ω → R+ ?
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 30 / 42
Model Case #1
Assume ω = ω1 ∪ ω2 (ωk “well-shaped”)
with interface Γ12 := ∂ω1 ∩ ∂ω2
and k|ωk= αk = const
ΩΓ
1
12
2Ω
Apply standard Poincare inequality on ω1 and ω2, i.e.
‖u − uΓ12‖2L2(ωk ) ≤ C diam (ωk)2 |u|2H1(ωk ) ∀ u ∈ H1(ωk)
Then multiplying by αk and adding implies
‖u − uΓ12‖2L2(ω),k ≤ C diam (ω)2 |u|2H1(ω),k
with C depending on (the shape of) ωk and Γ12 but not on k !
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 31 / 42
Model Case #1
Assume ω = ω1 ∪ ω2 (ωk “well-shaped”)
with interface Γ12 := ∂ω1 ∩ ∂ω2
and k|ωk= αk = const
ΩΓ
1
12
2Ω
Apply standard Poincare inequality on ω1 and ω2, i.e.
‖u − uΓ12‖2L2(ωk ) ≤ C diam (ωk)2 |u|2H1(ωk ) ∀ u ∈ H1(ωk)
Then multiplying by αk and adding implies
‖u − uΓ12‖2L2(ω),k ≤ C diam (ω)2 |u|2H1(ω),k
with C depending on (the shape of) ωk and Γ12 but not on k !
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 31 / 42
Model Case #2
Assume ω = ω1 ∪ ω2 ∪ ω3 (ωk “well-shaped”)
s.t. k|ωk= αk = const and α3 ≥ α2 ≥ α1
Define manifold X ∗ := ∂ω1 ∩ ∂ω3 Ω
X*
1Ω
2
Ω3
Treat ω1 and ω3 as before, and
‖u − uX∗‖2
L2(ω2),k = α2 ‖u − uX∗‖2
L2(ω2∪ω3)
≤ α2 C diam (ω)2 |u|2H1(ω2∪ω3)
≤ C diam (ω)2∫
ω2
α2 |∇u|dx +
∫ω3
α2︸︷︷︸≤α3
|∇u|dx
≤ C diam (ω)2 |u|2H1(ω2∪ω3),k
Again C depends on (the shape of) ωk and X ∗, but not on k !
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 32 / 42
Model Case #2
Assume ω = ω1 ∪ ω2 ∪ ω3 (ωk “well-shaped”)
s.t. k|ωk= αk = const and α3 ≥ α2 ≥ α1
Define manifold X ∗ := ∂ω1 ∩ ∂ω3 Ω
X*
1Ω
2
Ω3
Treat ω1 and ω3 as before, and
‖u − uX∗‖2
L2(ω2),k = α2 ‖u − uX∗‖2
L2(ω2∪ω3)
≤ α2 C diam (ω)2 |u|2H1(ω2∪ω3)
≤ C diam (ω)2∫
ω2
α2 |∇u|dx +
∫ω3
α2︸︷︷︸≤α3
|∇u|dx
≤ C diam (ω)2 |u|2H1(ω2∪ω3),k
Again C depends on (the shape of) ωk and X ∗, but not on k !R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 32 / 42
Model Case #2
Assume ω = ω1 ∪ ω2 ∪ ω3 (ωk “well-shaped”)
s.t. k|ωk= αk = const and α3 ≥ α2 ≥ α1
Define manifold X ∗ := ∂ω1 ∩ ∂ω3 Ω
X*
1Ω
2
Ω3
However, if α1, α2 α3 then such an inequality cannot exist:
Ω Ω Ω
ε 1
1 3 2
1
α
u
0 1 x1
Counter example:α1 = α2 = 1 and α3 = ε 1
‖u‖2L2(ω),k ∼ 1
|u|2H1(ω),k ∼ ε
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 32 / 42
Robust Weighted Poincare Inequality [Pechstein, RS, IMAJNA’12]
Let xmax ∈ ω be the point where k(x) attains its maximum on ω. Ifthere exists a path P from every point x ∈ ω to xmax such that knever decreases along P (quasi-monotonicity), then there exists aconstant CP > 0 independent of h, k(x) and diam(ω) such that
infγ∈R
∫ω
k(x)(v − γ)2 ≤ CP diam(ω)2
∫ω
k(x)|∇v |2 ∀v ∈ Vh .
Examples
*
*
X
X
min
H
ηmin
η
CP,m & k2
k1
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 33 / 42
Robust Weighted Poincare Inequality [Pechstein, RS, IMAJNA’12]
Let xmax ∈ ω be the point where k(x) attains its maximum on ω. Ifthere exists a path P from every point x ∈ ω to xmax such that knever decreases along P (quasi-monotonicity), then there exists aconstant CP > 0 independent of h, k(x) and diam(ω) such that
infγ∈R
∫ω
k(x)(v − γ)2 ≤ CP diam(ω)2
∫ω
k(x)|∇v |2 ∀v ∈ Vh .
Examples
*
*
X
X
min
H
ηmin
η
CP,m & k2
k1
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 33 / 42
Numerical Example (Geometric Multigrid)
D = (0, 1)2, uniform grids T `L`=0 with L = 4 and hL = 1/384.
Two “islands” not alligned with T0 and T1 where k(x) = α
(k(x) = 1 elsewhere. In right table islands closer to each other!)
CPK bounded for all ωK CP
K not bdd. on some ωK
α λ−11 #MG Its (tol = 10−8) λ−1
1 #MG Its (tol = 10−8)101 1.69 10 1.72 10102 2.75 14 3.87 19103 3.32 12 14.5 23104 3.42 10 115.5 70105 3.42 10 1125 76
Guiding principle
TH sufficiently fine (locally) s.t. k(x) quasi-monotone on all ωK
Otherwise coefficient dependent coarse space (e.g. local eigensolves)
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 34 / 42
Numerical Example (Geometric Multigrid)
D = (0, 1)2, uniform grids T `L`=0 with L = 4 and hL = 1/384.
Two “islands” not alligned with T0 and T1 where k(x) = α
(k(x) = 1 elsewhere. In right table islands closer to each other!)
CPK bounded for all ωK CP
K not bdd. on some ωK
α λ−11 #MG Its (tol = 10−8) λ−1
1 #MG Its (tol = 10−8)101 1.69 10 1.72 10102 2.75 14 3.87 19103 3.32 12 14.5 23104 3.42 10 115.5 70105 3.42 10 1125 76
Guiding principle
TH sufficiently fine (locally) s.t. k(x) quasi-monotone on all ωK
Otherwise coefficient dependent coarse space (e.g. local eigensolves)
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 34 / 42
Numerical Example (Geometric Multigrid)
D = (0, 1)2, uniform grids T `L`=0 with L = 4 and hL = 1/384.
Two “islands” not alligned with T0 and T1 where k(x) = α
(k(x) = 1 elsewhere. In right table islands closer to each other!)
CPK bounded for all ωK CP
K not bdd. on some ωK
α λ−11 #MG Its (tol = 10−8) λ−1
1 #MG Its (tol = 10−8)101 1.69 10 1.72 10102 2.75 14 3.87 19103 3.32 12 14.5 23104 3.42 10 115.5 70105 3.42 10 1125 76
Guiding principle (Possible PhD Projects!)
TH sufficiently fine (locally) s.t. k(x) quasi-monotone on all ωK
Otherwise coefficient dependent coarse space (e.g. local eigensolves)
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 34 / 42
Massively Parallel Multilevel MethodsCPU and GPU Clusters (current PDRA with Met Office)
Please ask me afterwards or email me if you are interested.
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 35 / 42
Massively Parallel Multilevel MethodsCPU and GPU Clusters (current PDRA with Met Office)
Please ask me afterwards or email me if you are interested.
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 35 / 42
Other PhD Opportunitiesin Bath and Elsewhere in the UK
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 36 / 42
Robust methods for high frequency wave scatteringCASE studentship at Bath with Schlumberger, Supervisors: IG Graham, E Spence
Forward problem: given obstacle and incident, find scattered
Simplest model: the Helmholtz equation ∆u + k2u = 0, k > 0
2 length scales: wavelength 2π/k and size of obstacle
I very difficult problem when wavelength size of obstacle
Fundamental difficulty with linear wave propagation:
1 2 3 4 5 6
-1.0
-0.5
0.5
1.0
Need to ensure that number of DOFs M ∼ kd as k increases.Very quickly becomes computationally intractable.
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 37 / 42
Robust methods for high frequency wave scatteringCASE studentship at Bath with Schlumberger, Supervisors: IG Graham, E Spence
Forward problem: given obstacle and incident, find scattered
Simplest model: the Helmholtz equation ∆u + k2u = 0, k > 0
2 length scales: wavelength 2π/k and size of obstacleI very difficult problem when wavelength size of obstacle
Fundamental difficulty with linear wave propagation:
1 2 3 4 5 6
-1.0
-0.5
0.5
1.0
Need to ensure that number of DOFs M ∼ kd as k increases.Very quickly becomes computationally intractable.
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 37 / 42
Robust methods for high frequency wave scatteringCASE studentship at Bath with Schlumberger, Supervisors: IG Graham, E Spence
Forward problem: given obstacle and incident, find scattered
Simplest model: the Helmholtz equation ∆u + k2u = 0, k > 0
2 length scales: wavelength 2π/k and size of obstacleI very difficult problem when wavelength size of obstacle
Fundamental difficulty with linear wave propagation:
1 2 3 4 5 6
-1.0
-0.5
0.5
1.0
Need to ensure that number of DOFs M ∼ kd as k increases.Very quickly becomes computationally intractable.
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 37 / 42
Robust methods for high frequency wave scatteringCASE studentship at Bath with Schlumberger, Supervisors: IG Graham, E Spence
Forward problem: given obstacle and incident, find scattered
Simplest model: the Helmholtz equation ∆u + k2u = 0, k > 0
2 length scales: wavelength 2π/k and size of obstacleI very difficult problem when wavelength size of obstacle
Fundamental difficulty with linear wave propagation:
1 2 3 4 5 6
-1.0
-0.5
0.5
1.0
Need to ensure that number of DOFs M ∼ kd as k increases.Very quickly becomes computationally intractable.
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 37 / 42
Robust methods for high frequency wave scatteringCASE studentship at Bath with Schlumberger, Supervisors: IG Graham, E Spence
Forward problem: given obstacle and incident, find scattered
Simplest model: the Helmholtz equation ∆u + k2u = 0, k > 0
2 length scales: wavelength 2π/k and size of obstacleI very difficult problem when wavelength size of obstacle
Fundamental difficulty with linear wave propagation:
1 2 3 4 5 6
-1.0
-0.5
0.5
1.0
Need to ensure that number of DOFs M ∼ kd as k increases.Very quickly becomes computationally intractable.
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 37 / 42
This PhD project (Bath)
Idea:
using results from asymptotic analysis/PDE theory designalgorithms that reduce the cost of solving the linear system
implement the resulting new methods on model problems
An opportunity to do something that is:
mathematically interesting
of universal importance in understanding wave propagation
....and thus of great practical interest!
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 38 / 42
Manchester: Numerical Analysis PhD OpportunitiesS Cotter, S Guttel, N Higham, M Lotz, C Powell, D Silvester
Numerical Linear Algebra generalised and quadratic eigenvalue
problems, matrix functions, stability analysis
Software Development e.g. for NAG, MATLAB, LAPACK and BLAS
Optimisation sparse data recovery and compressed sensing,
computational complexity, conditioning of optimisation algorithms
Finite Element Approximation Approximation theory, error
analysis, fast solversand application, e.g., in fluid flows.
Stochastic DEs & UQ modelling, approximation theory, error
analysis and fast solvers for problems with random inputs.
−1 0 1 2 3 4 5−1
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0
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Figure: Navier-Stokes flow (left). PDF for a biochemical network (right)R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 39 / 42
Sussex: B During, O. Lakkis, C Makridakis, V StylesNumerical Analysis and Scientific Computation Group
Strong interaction with Math Bio and Analysis & PDEs groups
Current PhD’s:I Elham Khairi “Numerical optimisation problems in solar power
generation plants” (B During, O Lakkis).I Muhammad Yau “Computational study of pattern formation
in epidemiological models” (K Blyuss, O Lakkis).I Andy Chung “Mathematical and computational models for
cardiovascular diagnosis” (O Lakkis, A Madzvamuse).
Available PhD or Postdoc projects:I “Computational methods for nonlinear stochastic models with
applications to environmental risk assessment”
I “Identification methods in cardiovascular diagnosis”
I “Galerkin methods for fully nonlinear PDE’s with application togeometric motions, mass transportation and optimisation”
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 40 / 42
Durham: PhD Topics on Numerical Solution of PDEs
Max Jensen
Numerical approximation offully-nonlinear equations –advancing tools for the analysis of
FE methods in regard to (mesh)
geometry, anisotropic PDEs or
gradient convergence.
(Hamilton-Jacobi-Bellman,
Monge-Ampere, . . . )
Financing offshore windfarms – modelling and simulation
of financial products to attract
diverse investor base for large-scale
renewable energy activities. Jointly
with European Investment Bank.
Anthony Yeates
Quantifying 3-d magneticreconnection – developing
numerical methods to analyse the
topological structure of magnetic
fields (from lab data or from
large-scale numerical simulations).
Numerical modelling of thesolar dynamo – assisting in the
development of large-scale
numerical simulations of a
kinematic dynamo model, and
applying it to current problems in
the solar dynamo such as
long-term predictability.R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 41 / 42
Main Places for NA and Sci Comp in the UK
Bath Leicester Warwick + othersManchester Sussex CambridgeOxford Durham LeedsStrathclyde Heriot-Watt NottinghamReading Edinburgh UCL
If you have any questions or want some more advice just email me:
R.Scheichl@bath.ac.uk
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 42 / 42
Main Places for NA and Sci Comp in the UK
Bath Leicester Warwick + othersManchester Sussex CambridgeOxford Durham LeedsStrathclyde Heriot-Watt NottinghamReading Edinburgh UCL
If you have any questions or want some more advice just email me:
R.Scheichl@bath.ac.uk
R. Scheichl (Bath) Multiscale Num Anal & Sci Comp Manchester, Wed 19/12/12 42 / 42
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