Model Order Reduction - Using Cubic Spline Curve … C-Splines FutureWork INTRODUCTION ... Model...

CR2GConstraint Research and Reading Group

MODEL ORDER REDUCTIONUsing Cubic Spline Curve-Fitting

Leobardo Valera, Martine CeberioMiguel ArgaezDepartment of Computer ScienceComputational Science ProgramThe University of Texas at El Paso

Introduction Discretization. . . . . . .Reduction C-Splines Future Work

INTRODUCTION

Given a partial differential equation, let's say, the Heat equation:

∂U

∂t= λ

∂2U

∂x2(1)

where :

→ x ∈ [0, L], where L is the length of the rod.→ t ≥ 0.→ U(x, 0) = f(x) ∀x ∈ [0, L]: Initial Condition.→ U(0, t) = 0 = U(L, t) ∀t > 0: Boundary condition.

2


DISCRETIZATION

If we discretize [0, L] taking ∆x = 10−2, in order to ensurequadratic convergence, we have to take ∆t ≤ 0.5× 10−4. Nowif 0 ≤ t ≤ 5, we have 100, 000 nodes in the temporal axis. Itmeans that we have to solve a linear system like:

Au = b (2)

100, 000 times.

As an example: If it takes 1 second to solve every linearsystem, it will take more than 27 hours to solve the wholeproblem.

It is imperative to Reduce the dimension of the problem.

3


REDUCTION

In our example, the solution of our system is u ∈ R100. Let'sassume that we know, for some reason, that u ∈ S, where S isa 10−dimentional subspace of R100 with base W , then:

Wy = u (3)

Now, sustituting (3) in (2), we obtain:

(AW )y = b (4)

4


REDUCTION

which is equivalent to solving:

(AW )T (AW )y = (AW )T b (5)

This is now a 10× 10 linear system and it is solved in 1/100 sec.And using (3) we get u. Thus

The whole problem is solved in 1, 000 sec.: much less than 1hour.

5


HOW TO FIND A BASE W

Proper Orthogonal Decomposition

The idea with this method is that the time response of a system,given a certain input, contain the behavior of the system.

We have the following algorithm to obtain W .

1. Solve the full-order model for several values of λ.2. For each λ, take one or more snapshots, which is the

solution of (2) in some values of t, and store suchsnapshots in a matrix called X, for example.

6


PROPER ORTHOGONAL DECOMPOSITION

7


... HOW TO FIND A BASE W ...

3. Compute the singular values decomposition (SVD) of X.[W,S, V ] = svd(X).

4. Find k such that σ =∑k

i=1 S(i)∑ni=1 S(i)

> 0.99

5. Consider only the first k columns and redefineW = W (:, [1 : k]).

Although the system (5) is solved very fast and the error of thesolution is very small, we had to solve the system (2) severaltimes in order to obtain the matrix of the snapshots X and wehad to compute X 's singular value decomposition whichrepresents a disadvantage.

8


... HOW TO FIND A BASE W . WAVELETS

Let's assume the matrix A is the following image:

100 200 300 400 500 600

100

200

300

400

500

600

and

Φ =

(W T

HT

)where W T is the lowpass HT is the high pass

9


... HOW TO FIND A BASE W ...

The product matrix AΦT will be;

100 200 300 400 500 600

100

200

300

400

500

600

AW AH

We save time computing W but when we use (3) to get theapproximation u, the error is greater than what the error weobtained using Proper Orthogonal Decomposition Method fromthe snapshots approach.

10


IS THERE ANYTHING ELSE WE CAN DO?

So far we have:

Good

Bad

Good

Bad

Time Error

→ POD is "bad" in time and "good" in error→ Wavelets is "good" in time but "bad" in error

11


YES, THERE IS SOMETHING ELSE WE CAN DO

We propose a "Midpoint"

Good

Bad

Good

Bad

Time Error

We propose a method in which we do not have to compute anybase and we get an acceptable error within a good computationtime. 12


METHOD USING C-SPLINES

Our method is inspired by super-resolution methods and isbased on Spline Interpolation. x = 0:10; y = sin(x); xx =0:.25:10; yy = spline(x,y,xx); plot(x,y,'o',xx,yy)

0 1 2 3 4 5 6 7 8 9 10−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

13


DESCRIPTION OF THE METHOD

Let's reconsider our original PDE problem (1)

∂U

∂t= λ

∂2U

∂x2

1. If we want to discretize the spatial domain with ∆x1 suchthat we get 400 nodes and ∆t < 0.5(∆x1)

2, we would haveto solve the system (2)

Au = b

where A is a 400× 400 matrix.

14


...DESCRIPTION OF THE METHOD

2. Instead of discretizing the spatial domain with ∆x1, we canuse ∆x2 = 4∆x1, for instance, and then we obtain 100nodes and we keep the same ∆t. We solve the system (2)

Au = b

where A is a 100× 100 matrix.3. After we obtain u, we interpolate it using spline in order to

get an approximation of u.

15


... DESCRIPTION. GRAPHIC VERSION

For each t, we have:

What we want

What we compute

What we getafter interpolating

x1 x2 x3 x4 x5 x6 x7 x8 x9 xn

u1 u2 u3 u4 u5 u6 u7 u8 u9 un· · ·

u1 u5 u9 u400· · ·

u1 u5 u9 u400· · ·u2 u3 u4 u6 u7 u8

· · ·

16


EXAMPLE AND NUMERICAL RESULTS

Suppose we want solve

∂U

∂t= λ

∂2U

∂x2

with x ∈ [0, 1], ∆x = 0.0025, t ∈ [0, 0.01] with ∆t = 2.50× 10−6

and λ = 1.

Full- Order Model (FOM): Time to solve the problem and getUfom: t = 10.16 sec.

17


... EXAMPLE AND NUMERICAL RESULTS

1. Time computing the snapshots and the reduced basedusing SVD: t = 267 sec.

2. Time computing the solution: t = 0.8 sec. The reducedbase W has 2 components and||Ufom − Upod||

||Ufom||= 1.92e− 12

18


FOM VS. POD

0 50 100 150 200 250 300 350 400 450−0.5

0

0.5

1

1.5

2Solution Computed with FOM and POD

Solution FOMSolution POD

19



1. Time computing the Wavelet Matrix Daubechies 2 withlevel 2: t = 0.047 sec.

2. Time computing the solution: t = 6.77 sec. The reduced

base W has 50 components and||Ufom − Uwav||

||Ufom||= 0.0046

20


FOM VS. WAVELETS

0 50 100 150 200 250 300 350 400 450−0.5

0

0.5

1

1.5

2

x

u

Solution Computed with FOM and Wavelets

Solution FOM

Solution Wavelets

65 70 75 80 85 90 95 100 105 110 1151.35

1.4

1.45

1.5

1.55

1.6

1.65

x

u


Solution FOM

Solution Wavelets

21



1. Time computing the solution using a coarse refinement:t = 0.69 sec.

2. Time interpolating using spline: t = 0.21 sec. and||Ufom − UCVM ||

||Ufom||= 4.09e− 05

22


FOM VS. USING SPLINES

0 50 100 150 200 250 300 350 400 4500.5

0

0.5

1

1.5

2

x

u

Solution using FOM Vs. Solution using Splines

Solution FOM

Solution Using Splines

100.5 100.6 100.7 100.8 100.9 101 101.1 101.2 101.3

1.397

1.398

1.399

1.4

1.401

x

u

Solution using FOM Vs. Solution Using Splines

Solution FOMSolution Splines

23


WAVELETS VS. USING SPLINES

65 70 75 80 85 90 95 100 105 110 1151.35

1.4

1.45

1.5

1.55

1.6

1.65

x

u


Solution FOM

Solution Wavelets

100.5 100.6 100.7 100.8 100.9 101 101.1 101.2 101.3

1.397

1.398

1.399

1.4

1.401

x

u

Solution using FOM Vs. Solution Using Splines

Solution FOMSolution Splines

24


SUMMARY

Method offline time online time ErrorFOM 10.16 secPOD 267 sec 0.8 sec 1.92e− 12

WAV 0.04 sec 6.77 sec 0.015

CVM 0.21 sec 0.69 sec 4.09e− 5

Good

Bad

Good

Bad

Time E

rror

25


FUTURE WORK

1. We plan to use adaptive discretization to avoid losingprecision in case the solution has drastic changes ofbehavior.

2. We also plan to address the problem of non-linear PDEsusing similar approaches; for example, for the Burgersequations.

26


REFERENCES

→ Wilhelmus H.A. Schilders, Henk A. van der Vorst, andJoost Rommes. 2008. Model Order Reduction. Theory,Research Aspects and Applications. Springer, BerlinHeidelberg.

→ Miguel Hernandez IV. 2013. Reduced-Order ModelingUsing Orthogonal and Bi-Orthogonal WaveletTransforms. Ph.D. Dissertation. Computational ScienceProgram, The University of Texas at El Paso.

→ Jean-Luc Starck, Fionn Murtagh, and Jalal M. Fadili. 2010.Sparse Image and Signal Processing. Wavelets,Curvelets, Morphological Diversity. Cambridg UniversityPress, 32 Avenue of the Americas, New York, NY.

27


ACKNOWLEDGMENT

1. Dr. Martine Ceberio.

2. Dr. Miguel Argaez.3. The Graduate Student Research Expo's Organizators.4. Computational Sciences Program.5. La universidad Metropolitana. Caracas, Venezuela

28


ACKNOWLEDGMENT

1. Dr. Martine Ceberio.2. Dr. Miguel Argaez.

3. The Graduate Student Research Expo's Organizators.4. Computational Sciences Program.5. La universidad Metropolitana. Caracas, Venezuela

28


ACKNOWLEDGMENT

1. Dr. Martine Ceberio.2. Dr. Miguel Argaez.3. The Graduate Student Research Expo's Organizators.

4. Computational Sciences Program.5. La universidad Metropolitana. Caracas, Venezuela

28


ACKNOWLEDGMENT

1. Dr. Martine Ceberio.2. Dr. Miguel Argaez.3. The Graduate Student Research Expo's Organizators.4. Computational Sciences Program.

5. La universidad Metropolitana. Caracas, Venezuela

28


ACKNOWLEDGMENT

1. Dr. Martine Ceberio.2. Dr. Miguel Argaez.3. The Graduate Student Research Expo's Organizators.4. Computational Sciences Program.5. La universidad Metropolitana. Caracas, Venezuela

28


THANKS!

Thanks!

29

Model Order Reduction - Using Cubic Spline Curve … C-Splines FutureWork INTRODUCTION ... Model...

Documents

Transcript of Model Order Reduction - Using Cubic Spline Curve … C-Splines FutureWork INTRODUCTION ... Model...