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C. Picard, M. Garbey, July 20th 2006 Extrapolation method and optimum design of solutions - p. 1

Extrapolation method and optimum design ofsolutions

Christophe Picard, Marc GarbeyDept. of Computer Science, University of Houston, USA

http://cs.uh.edu

Solution Verification

Problem

Comment

Optimized Extrapolation

Technique

General Idea

Practical Consequences

Three level methods (1)


A Posteriori Error

Algorithm/Result with detail

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Conclusion



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Problem

Boundary value problem (Ω) is a polygonal domain andn = 2 or 3 :

L[u(x)] = f(x), x ∈ Ω ⊂ IRn, u = g on ∂Ω.

Assume that the PDE problem is well posed and has a uniquesolution. We consider an approximation on a family of meshesM(h) parameterized by h > 0 a small parameter.We denote symbolically the corresponding family of linearsystems

AhUh = Fh.

Let ph denotes the projection of the continuous solution u ontothe mesh M(h). We assume a priori that (||.|| is a givendiscrete norm):

||Uh − ph(u)|| → 0, as h → 0,

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Comment

||Uh − ph(u)|| → 0, as h → 0,

Manufactured solution

f(x) = L[u(x)]

Polynomial solution to verify the code.

Test each term of the equation.

Useful for parallel codes

Use of symbolic manipulation languages

Constraint by conservation of physical quantities

Principle of nearby exact solution (C.J.Roy andM.M.Hopkins).

Possible use of image analysis on experimental data.

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Optimized Extrapolation Technique

• Let M(h1) and M(h2) be two 6= meshes used to build twoapproximations U1 and U2 of the PDE problem.• A consistent linear extrapolation writes

αU1 + (1 − α)U2,

where α is a weight function.• In classical Richardson Extrapolation (RE) α is a constant.• In our optimized extrapolation method α is a function solutionof the following optimization problem:

Pα: Find α ∈ Λ(Ω) ⊂ L∞ such thatG(α U1 + (1 − α) U2) isminimum.

• For computational efficiency, Λ(Ω) should be a finite vectorspace of very small dimension compared to the dimension ofAh.

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General Idea One can choose to work with a posteriori FE error estimates:

Equilibrated residual method for FE .- see Ainsworth& Oden and ref.

A posteriori Finite-Element free constant outputbounds - see Patera and ref.

From now on, and to make our technique general, we will work wi thdiscrete value functions and discrete norms: Why is it possi ble?

Our ambition: a numerical estimate on ||Uj − U∞||, j = 1, 2,without computing U∞.

M(h∞) should capture a priori all the scales needed. In practice h∞ << h1, h2.

The solution Uj can be verified, assuming convergence ofthe approximation method, i.e U∞ → u, as h∞ → 0.

Reuse extensive knowledge of Physicist and AsymptoticAnalysis.

Reuse Stability Theory from Linear Algebra.

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Practical Consequences• Both coarse grid solutions U1 and U2 must be projected ontoM(h∞). Notation: U1 and U2.

• The objective function is a discrete norm of the residual:

G(Uα) = ||Ah∞Uα − Fh∞

||, where Uα = α U1 + (1 − α) U2.

The Optimized Extrapolated Solution (OES) if it exists, isdenoted Ve = αeU1 + (1 − αe)U2.

• The choice of the discrete norm depends on the property ofthe solution.

• One can choose to work in a subspace:

Estimate on a functional of the solution.

Estimate in subdomain.

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020

4060

80100

0

5

10

15

20−5

0

5

10

neighborhood of the sliding wall

contour plot of estimated convergence order for vorticity

y indices

Cancelation phenomenon: surface plot of space dependentconvergence order approximation for ω − ψ code with Re=100.

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Three level methods (2)• There are subset of Ω where U1 − U2 << hk, with k expectedorder of convergence.

• Optimized two-level extrapolation problem is ill posed.

Three levels OES Find α, β ∈ Λ(Ω) ⊂ L∞ such that

G(α U1 + β U2 + (1 − α− β) U3) is minimum.

•If all Uj , j = 1..3, coincide at the same space location there iseither no local convergence or all solutions Uj are exact.

• Robustness of the method should come from the fact that wedo not suppose a priori any asymptotic formula on theconvergence rate of the numerical method as opposed to RE.

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Stability estimate (1)



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General Principle (1)• Let us assume that Ve exists and has been computed.

• Let Uj be one of the coarse grid approximations; We look fora global a posteriori estimate of the error

||Uj − ph(u)||.

• Recovery method:

IF ||Ve − ph(u)||2 << ||Uj − ph(u)||2,

THEN ||Uj − Ve||2 ∼ ||Uj − ph(u)||2

• Heuristic: provides a good lower bound on the error in ournumerical experiments with steady incompressible NavierStokes (NS).

• But there is no guarantee that a smaller residual for Ve thanfor U2 on the fine grid M(h∞) leads to a smaller error.

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• From a stability estimate with the discrete operator:

||Ve − U∞|| < µ∞ G(Ve), where µ ≥ ||(Ah∞)−1||.

we conclude

||U2 − U∞||2 < µ G(Ve) + ||Ve − U2||2.

• Uses extrapolation on µ1, µ2, µ3 to get ≈ µ∞.

• L2 norm: the estimate on µ uses a standard eigenvalueiterative procedure to get the smallest eigenvalue.

• L1 norm: see N J.Higham papers.

• Additional Test: Verify that the upper bound on ||U∞ − U2||increases toward an asymptotic limit as M(h∞) gets finer.Feasible test because the fine grid solution is never computedin OES.

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• Let (E, ||.||E) and (F, ||.||F ) be two normed linear space,N ∈ L(E,F ) be the operator corresponding to the CFDproblem.

• Assuming the problem N(u) = s is well posed for s ∈ B(S, d),and N(Uh) ∈ B(S, d), for some discrete solution Uh.

• Defining ρ the residual and e the error, an upper bound of theerror is given by

||e||E ≤ ||ρ||F (||∇sN−1(S + ρ)||E +

K

2||ρ||F ).

• Let bEi , i = 1..N, be a basis of E, and ε ∈ such thatε = o(1).

• Let (V ∓

i )i=1..N , be the family of solutions of the followingproblems N(Uh ∓ εVi) = S + ρ∓ εbi .

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• We get from finite differences the approximation

Ch∞= ||∇SN

−1(S + ρ)|| ≈ ||(1

2(V +

j − V −

j ))j=1..N || +O(ε2).

• Fundamental tool: compact representation of the solution viaa projection: trigonometric polynomial or wavelet or spectralelements for example.

• One get the error estimate in this smaller space only.

• The size of the space is fixed adaptively according to thequality of the approximation of trigonometricpolynomial/wavelets of the FE solution!.

• Both stability estimate and OES construction relies onseveral hundred of residual/smoothing computation that areextremely fast and have embarassing parallelism → distributedcomputing !!!

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div(ρ∇u) = f

Error Estimate in L2 norm


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Algorithm/Result with detail codeknowledge

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div(ρ∇u) = f



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General Principle(1)• Assume that the operator is linear and the objective functionis the discrete L2 norm of the residual.

• Let ei, i = 1..m be a set of basis function of Λ(Ω).

The solution process can be decomposed into three steps.

• Step 1: interpolation of the coarse grid solution fromM(hj), j = 1..p, to M(h∞) and postprocessing to smooth outthe "spurious modes".

• Step 2: evaluate the residual

R[ei (Uj − Uj+1)], i = 1..m, j = 1..p− 1,

and R[Up] on the fine grid M(h∞).

• Step 3: solution of the least square linear algebra problemthat has m unknowns for each weight coefficient α and β.

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div(ρ∇u) = f



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• Compact representation of unknown weight functions: m ismuch lower than the number of grid points on any coarse gridused.

• Estimate on the number of iterates to regularized Uj , j = 1..p

• Generalization to non-linear elliptic problems via a Newtonlike loop.

• Difficulties: A posteriori Error estimate depends then on thefunction used to linearized the operator.

• Generalization to L1 and L∞ with appropriate minimizationprocedure.

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div(ρ∇u) = f



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div(ρ∇u) = f

020

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80100

120140

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150−0.005

0

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0.025

solution on the fine grid

ρ ≈ 100 in the disc, one elsewhere. Domain has a L-shape. coarse grid solutions: h1 = 1/14, h2 = 1/20, h3 = 1/26.

fine grid: h0 = 1/128.

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0 0.005 0.012

2.5

3

3.5

4

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5

5.5x 10

−4

Recovery method

for two levelsfor three levelsfor Aitken acc.for true errorfine grid accuracy

0 0.01 0.020

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L2 residual for LSE

L2 two levelsL2 three levels

0 0.01 0.02−4

−3.5

−3

−2.5

−2

−1.5

−1

Upper bound on the error

for three levelsfor Aitken acc.for true errorfine grid accuracy

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Benchmark Problems

Adina Software

Results(1)

Results(2)

Results : stability and error for

backstep flow


the battery

Software Architecture

Conclusion


Algorithm/Result with No detail codeknowledge

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Benchmark Problems

Adina Software

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Let us denote N [u] = 0 the supposedly well posed PDEproblem to be solved, and its unsteady companion problem,∂tu = N [u].The algorithm is as follow:

• Step 1 Call coarse Mesh : We generate the (coarse) meshes G1

and G2. If hi is the average space step for the grid Gi weshould have h2 < h1 but this is not necessary.

• Step 2 Call fine Mesh : We generate a fine mesh G∞ that issupposed to solve all the scales of the problem. G∞ might be astructured mesh or not. We must have h∞ << h1, h2.

• Step 3 Call Solver : We solve the problem on G1 and G2,possibly in parallel. The solutions are denoted respectively u1

and u2 on G1 and G2.

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Benchmark Problems

Adina Software

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• Step 4 Call Projection : We project these coarse solutions u1

and u2 onto G∞. We denote these projections u∞1 and u∞2 .

• Step 5 Create sample : We create sample solutionsu∞α = [αu∞1 + (1 − α)u∞2 ]. We smooth out the spurious highfrequency components of the build solution with few explicittime steps of ∂tu = N(u) starting from the initial condition: u∞α .The choice of the Optimum Design Space in which α is takenis one the main item of our research.

• Step 6 We compute the best α that minimizes the L2 norm ofthe residual. We may use a surface response technique.

http://cs.uh.edu


A Posteriori Error


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Benchmark Problems

Adina Software

Results(1)

Results(2)


backstep flow


the battery


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Benchmark Problems

• Laminar flow over a backward facing stepThe flow problem has the following characteristics:

Density 1 Length of the pipe 10

Viscosity 0.01 Inflow diameter 1

Maximum Inflow Velocity U=1 Outflow diameter 2

Reynold Number 500

• Heat Transfer in a composite material (Sandia Nat. Lab.)

∂

∂xi

(kij(T ))∂T

∂xi

+Q(T ) = 0 in Ω × (0, T )

The boundary conditions are of robin type to describeradiations effects. Remark : The problem is very stiff and thesolution is near discontinuous between material.

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Benchmark Problems

Adina Software

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backstep flow


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Adina Software

In this simulation, the number of elements are respectively10347 on the fine grid G∞, 1260 on the coarse grid G1, and

2630 on the coarse grid G2.

Contour of velocity magnitude on fine grid: Adina R&DThe steady solutions are obtained using a transient scheme for

the incompressible Navier-Stokes equation.

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Results(1)

0

2e-05

4e-05

6e-05

8e-05

0.0001

0.00012

0.00014

0.00016

0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004

Res

idual

Error

LSE: error and residual for Adina R&D in L2 norm

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Results(2)

Performance of LSE and Richardson Extrapolation

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backstep flow


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Results : stability and error for backstep flow

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

110000

120000

400 225 100 64 36 25 16 9 4 1

Err

or

Size of the compact representation space

Evaluation of the stability constant

Figure 1: Evaluation of the sta-bility constant

1e-05

1e-04

0.001

0.01

0.1

1

400 225 100 64 36 25 16 9 4 1

Err

or


Evolution of error to fine grid solution

Figure 2: Evaluation of the er-ror upper bound

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Results : stability and error for the battery

20000

22000

24000

26000

28000

30000

32000

34000

36000

38000

40000

400 225 100 64 36 25 16 9 4 1

Err

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Evaluation of the stability constant

Figure 3: Evaluation of the sta-bility constant

Figure 4: Evaluation of the er-ror upper bound

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Postprocessing Code Code Preparation

Computational Code Computational Nexus User Interface

Master Node Tasks

Slave Node Tasks

Master Node Tasks

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Conclusions


Conclusion

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Conclusions


Conclusions

A new extrapolation methods for PDEs.

A better tool for solution verification than RE when theconvergence order is space dependent or far from theasymptotic rate of convergence.

Toward a Solution Verification Method with Hands offCoding.

Toward a system that is user friendly and scalable.

Time dependent problem currently under investigation.

http://cs.uh.edu

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