Multiscale Problems: Numerical Analysis and Scienti c ... · Multiscale Problems: Numerical Analysis and Scienti c Computing with Applications in Energy & Environment Robert Scheichl

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


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


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)


Hierarchy of Scales

Subsurface

Other (e.g. atmospheric)

→ Range of length and time scales

→ Heterogeneities and anisotropies

→ Lack of and uncertainty in data


Hierarchy of Scales

Subsurface Other (e.g. atmospheric)

→ Range of length and time scales

→ Heterogeneities and anisotropies

→ Lack of and uncertainty in data


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


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


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




(Theory?)


↓


↓

MsFEM, HMM

(Theory?)





(Theory?)


↓


↓

MsFEM, HMM

(Theory?)





(Theory?)


↓


↓

MsFEM, HMM(Theory?)





(Theory?)


↓


↓

MsFEM, HMM(Theory?)



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)


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)


The Multilevel Monte Carlo Idea


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)



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)



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 !



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 !


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(x , y) := σ2 exp

(−‖x − y‖

λ

)



∫Dq1



typical realisation

(λ = 164 , σ

2 = 8)






R(x , y) := σ2 exp

(−‖x − y‖

λ

)



∫Dq1



typical realisation

(λ = 164 , σ

2 = 8)


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










> 106







Why is this problem so challenging?




> 106







Why is this problem so challenging?




> 106


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)


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)


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!


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!


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` !




E[QL] = E[Q0] +∑L

`=1E[Q` − Q`−1]



QMLL := QMC0 +

∑L

`=1Y MC` where Y` := Q` − Q`−1


If Q` → Q then V[Y`]→ 0 as h` → 0





E[QL] = E[Q0] +∑L

`=1E[Q` − Q`−1]



QMLL := QMC0 +

∑L

`=1Y MC` where Y` := Q` − Q`−1


If Q` → Q then V[Y`]→ 0 as h` → 0



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 !




V[Q` − Q`−1] = O(M−β` ).


with

Cost(QMLL ) = O(





Cost(QMLL ) = O(

TOL−max(2, 1α))

= O (max(N0,ML)) !!






V[Q` − Q`−1] = O(M−β` ).


with

Cost(QMLL ) = O(





Cost(QMLL ) = O(

TOL−max(2, 1α))

= O (max(N0,ML)) !!




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


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]


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]


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)


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)


New Multigrid Theory

Weighted Poincare Inequalities & Abstract Bramble-Hilbert Lemma


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





−∇ · (k∇p) = f



Two possible aims:









−∇ · (k∇p) = f



Two possible aims:









−∇ · (k∇p) = f



Two possible aims:

I h-optimal, k-robust solver (k resolved on fine mesh T h) Now


(k not resolved on T H – “Upscaling”) Hot Topic



Other Applications: Simulation in Heterogeneous Media

Elasticity, e.g. in wood or bone

Electrostatics (Al2O3–TiO2) or Nonlinear Magnetostatics


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?


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?


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]


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]


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









Ω2Ω Ω31

χ1 2

χ3

χ

Geometric Multigrid



V1= V2= + = all3V










Ω2Ω Ω31

χ1 2

χ3

χ

Geometric Multigrid



V1= V2= + = all3V










Ω2Ω Ω31

χ1 2

χ3

χ

Geometric Multigrid



V1= V2= + = all3V



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!


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


Poincare’s inequality

Domain ω ⊂ Rd (open, bounded, connected set). ∃C > 0 s.t.


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


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


L2(ω),k ≤ CP |u|2H1(ω),k

for some class of weights k : ω → R+ ?


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


L2(ω),k ≤ CP |u|2H1(ω),k

for some class of weights k : ω → R+ ?


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 !


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 !


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 !


Model Case #2




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




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


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


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


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)








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










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!)




Massively Parallel Multilevel MethodsCPU and GPU Clusters (current PDRA with Met Office)

Please ask me afterwards or email me if you are interested.


Massively Parallel Multilevel MethodsCPU and GPU Clusters (current PDRA with Met Office)

Please ask me afterwards or email me if you are interested.


Other PhD Opportunitiesin Bath and Elsewhere in the UK


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.





2 length scales: wavelength 2π/k and size of obstacleI very difficult problem when wavelength size of obstacle


1 2 3 4 5 6

-1.0

-0.5

0.5

1.0








1 2 3 4 5 6

-1.0

-0.5

0.5

1.0








1 2 3 4 5 6

-1.0

-0.5

0.5

1.0








1 2 3 4 5 6

-1.0

-0.5

0.5

1.0



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!


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

0

1

−0.3

−0.2

−0.1

0

0.1

50100

150200

250

0

200

400

600

0

1

2

3

4

x 104

x1

x2

x3

10−5

10−6

10−7

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”


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:

[email protected]


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:

[email protected]


Documents

Multiscale Problems: Numerical Analysis and Scienti c ... · Multiscale Problems: Numerical Analysis and Scienti c Computing with Applications in Energy & Environment Robert Scheichl