Data-Driven FilteredReduced Order Modelsof Nonlinear Systems

T. Iliescu

Department of MathematicsICAM

Virginia Tech

MoRePaS 2018

NantesApril 11, 2018

National Science Foundation DMS-1522656 2015-2018

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 1 / 17

Outline

1 Problem

2 Solution

3 Connections

4 Conclusions

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 2 / 17

Outline

Collaborators

Muhammad Mohebujjaman (postdoc, Virginia Tech)

Xuping Xie (postdoc, Oak Ridge National Laboratory)

Leo Rebholz (professor, Clemson University)

Birgul Koc (student, Virginia Tech)

Changhong Mou (student, Virginia Tech)

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 3 / 17

Outline

Collaborators

Muhammad Mohebujjaman (postdoc, Virginia Tech)

Xuping Xie (postdoc, Oak Ridge National Laboratory)

Leo Rebholz (professor, Clemson University)

Birgul Koc (student, Virginia Tech)

Changhong Mou (student, Virginia Tech)

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 3 / 17

Outline

Collaborators

Muhammad Mohebujjaman (postdoc, Virginia Tech)

Xuping Xie (postdoc, Oak Ridge National Laboratory)

Leo Rebholz (professor, Clemson University)

Birgul Koc (student, Virginia Tech)

Changhong Mou (student, Virginia Tech)

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 3 / 17

Outline

Collaborators

Muhammad Mohebujjaman (postdoc, Virginia Tech)

Xuping Xie (postdoc, Oak Ridge National Laboratory)

Leo Rebholz (professor, Clemson University)

Birgul Koc (student, Virginia Tech)

Changhong Mou (student, Virginia Tech)

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 3 / 17

Outline

Collaborators

Muhammad Mohebujjaman (postdoc, Virginia Tech)

Xuping Xie (postdoc, Oak Ridge National Laboratory)

Leo Rebholz (professor, Clemson University)

Birgul Koc (student, Virginia Tech)

Changhong Mou (student, Virginia Tech)

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 3 / 17

Problem

General Problem

POD

RBM

DMD

systems theoretic ROMs (Serkan ,)

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 4 / 17

Problem

General Problem

POD

RBM

DMD

systems theoretic ROMs (Serkan ,)

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 4 / 17

Problem

General Problem

POD

RBM

DMD

systems theoretic ROMs (Serkan ,)

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 4 / 17

Problem

General Problem

POD

RBM

DMD

systems theoretic ROMs (Serkan ,)

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 4 / 17

Problem

What is the Problem?

(I) PDE•u = f(u) given

(II) FEM(

•uh ,vh

)=

(f(uh) ,vh

)given

(III) ROM

r given

{ϕ1, . . . ,ϕr} given

model ???

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 5 / 17

Problem

What is the Problem?

(I) PDE•u = f(u) given

(II) FEM(

•uh ,vh

)=

(f(uh) ,vh

)given

(III) ROM

r given

{ϕ1, . . . ,ϕr} given

model ???

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 5 / 17

Problem

What is the Problem?

(I) PDE•u = f(u) given

(II) FEM(

•uh ,vh

)=

(f(uh) ,vh

)given

(III) ROM

r given

{ϕ1, . . . ,ϕr} given

model ???

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 5 / 17

Problem

What is the Problem?

(I) PDE•u = f(u) given

(II) FEM(

•uh ,vh

)=

(f(uh) ,vh

)given

(III) ROM

r given

{ϕ1, . . . ,ϕr} given

model ???

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 5 / 17

Problem

What is the Problem?

(I) PDE•u = f(u) given

(II) FEM(

•uh ,vh

)=

(f(uh) ,vh

)given

(III) ROM

r given

{ϕ1, . . . ,ϕr} given

model ???

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 5 / 17

Problem

What is the Problem?

(I) PDE•u = f(u) given

(II) FEM(

•uh ,vh

)=

(f(uh) ,vh

)given

(III) ROM

r given

{ϕ1, . . . ,ϕr} given

model ???

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 5 / 17

Solution

Standard Solution Discretization Levels

(I) PDE dim =∞

(II) FEM dim = N = O(106)

(III) ROM dim = r = O(10)

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 6 / 17

Solution

Standard Solution Discretization Levels

(I) PDE dim =∞

(II) FEM dim = N = O(106)

(III) ROM dim = r = O(10)

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 6 / 17

Solution

Standard Solution Discretization Levels

(I) PDE dim =∞

(II) FEM dim = N = O(106)

(III) ROM dim = r = O(10)

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 6 / 17

Solution

Standard Solution Model

ur =r∑

j=1

aj ϕj

(•

ur ,ϕi

)=

(f(ur) ,ϕi

)i = 1, . . . , r

Galerkin ROM 1

1NSE•a = A a + a> B a

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 7 / 17

Solution

Standard Solution Model

ur =r∑

j=1

aj ϕj

(•

ur ,ϕi

)=

(f(ur) ,ϕi

)i = 1, . . . , r

Galerkin ROM 1

1NSE•a = A a + a> B a

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 7 / 17

Solution

Standard Solution Model

ur =r∑

j=1

aj ϕj

(•

ur ,ϕi

)=

(f(ur) ,ϕi

)i = 1, . . . , r

Galerkin ROM 1

1NSE•a = A a + a> B a

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 7 / 17

Solution

New Solution Discretization Levels

(I) PDE dim =∞

(II) FEM dim = N = O(106)

(III) FEM-ROM dim = d = O(103)

(IV) ROM dim = r = O(10)

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 8 / 17

Solution

New Solution Discretization Levels

(I) PDE dim =∞

(II) FEM dim = N = O(106)

(III) FEM-ROM dim = d = O(103)

(IV) ROM dim = r = O(10)

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 8 / 17

Solution

New Solution Discretization Levels

(I) PDE dim =∞

(II) FEM dim = N = O(106)

(III) FEM-ROM dim = d = O(103)

(IV) ROM dim = r = O(10)

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 8 / 17

Solution

New Solution Discretization Levels

(I) PDE dim =∞

(II) FEM dim = N = O(106)

(III) FEM-ROM dim = d = O(103)

(IV) ROM dim = r = O(10)

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 8 / 17

Solution

New Solution Model

ud =d∑

j=1

aj ϕj

(•

ud ,ϕi

)=

(f(ud) ,ϕi

)i = 1, . . . , r

(•

ur ,ϕi

)=

(f(ur) ,ϕi

)+

[(f(ud) ,ϕi

)−(

f(ur) ,ϕi

)]i = 1, . . . , r

Galerkin ROM + correction

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 9 / 17

Solution

New Solution Model

ud =d∑

j=1

aj ϕj

(•

ud ,ϕi

)=

(f(ud) ,ϕi

)i = 1, . . . , r

(•

ur ,ϕi

)=

(f(ur) ,ϕi

)+

[(f(ud) ,ϕi

)−(

f(ur) ,ϕi

)]i = 1, . . . , r

Galerkin ROM + correction

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 9 / 17

Solution

New Solution Model

ud =d∑

j=1

aj ϕj

(•

ud ,ϕi

)=

(f(ud) ,ϕi

)i = 1, . . . , r

(•

ur ,ϕi

)=

(f(ur) ,ϕi

)+

[(f(ud) ,ϕi

)−(

f(ur) ,ϕi

)]i = 1, . . . , r

Galerkin ROM + correction

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 9 / 17

Solution

New Solution Model

ud =d∑

j=1

aj ϕj

(•

ud ,ϕi

)=

(f(ud) ,ϕi

)i = 1, . . . , r

(•

ur ,ϕi

)=

(f(ur) ,ϕi

)+

[(f(ud) ,ϕi

)−(

f(ur) ,ϕi

)]i = 1, . . . , r

Galerkin ROM + correction

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 9 / 17

Solution

New Solution Data-Driven Model

[(f(ud) ,ϕi

)−(

f(ur) ,ϕi

)]data-driven≈

(g(ur) ,ϕi

)

(•

ur ,ϕi

)=

(f(ur) ,ϕi

)+

(g(ur) ,ϕi

)i = 1, . . . , r

Data-Driven Filtered ROM (DDF-ROM) 2

2NSE•a =

(A + Ã

)a + a>

(B + B̃

)a

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 10 / 17

Solution

New Solution Data-Driven Model

[(f(ud) ,ϕi

)−(

f(ur) ,ϕi

)]data-driven≈

(g(ur) ,ϕi

)

(•

ur ,ϕi

)=

(f(ur) ,ϕi

)+

(g(ur) ,ϕi

)i = 1, . . . , r

Data-Driven Filtered ROM (DDF-ROM) 2

2NSE•a =

(A + Ã

)a + a>

(B + B̃

)a

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 10 / 17

Solution

New Solution Data-Driven Model

[(f(ud) ,ϕi

)−(

f(ur) ,ϕi

)]data-driven≈

(g(ur) ,ϕi

)

(•

ur ,ϕi

)=

(f(ur) ,ϕi

)+

(g(ur) ,ϕi

)i = 1, . . . , r

Data-Driven Filtered ROM (DDF-ROM) 2

2NSE•a =

(A + Ã

)a + a>

(B + B̃

)a

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 10 / 17

Solution

DDF-ROM Reproductive

Re = 1000

0.5 1 1.5 2 2.5 3

t

0.605

0.61

0.615

0.62

En

erg

y

r=4

0.5 1 1.5 2 2.5 3

t

2.5

3

3.5

4

4.5

Dra

g

N=4

0.5 1 1.5 2 2.5 3

t

-3

-2

-1

0

1

2

3

Lift

N=4

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 11 / 17

Solution

DDF-ROM + Constraints Predictive

Re = 1000

50% data

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t

0.599

0.6

0.601

0.602

0.603

0.604

0.605

0.606

En

erg

y

r=3

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 12 / 17

Connections

Correction = Numerical Dissipation

eddy viscosity ROMs

mixing length: Lumley et al

Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al

variational multiscale: Iollo et al, Wang et al , . . .

dynamic subgrid-scale: Wang et al

numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .

Schneier Friday talk

Stabile Thursday talk

Ali, Hijazi Wednesday poster

Taddei Wednesday poster

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17

Connections

Correction = Numerical Dissipation

eddy viscosity ROMs

mixing length: Lumley et al

Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al

variational multiscale: Iollo et al, Wang et al , . . .

dynamic subgrid-scale: Wang et al

numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .

Schneier Friday talk

Stabile Thursday talk

Ali, Hijazi Wednesday poster

Taddei Wednesday poster

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17

Connections

Correction = Numerical Dissipation

eddy viscosity ROMs

mixing length: Lumley et al

Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al

variational multiscale: Iollo et al, Wang et al , . . .

dynamic subgrid-scale: Wang et al

numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .

Schneier Friday talk

Stabile Thursday talk

Ali, Hijazi Wednesday poster

Taddei Wednesday poster

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17

Connections

Correction = Numerical Dissipation

eddy viscosity ROMs

mixing length: Lumley et al

Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al

variational multiscale: Iollo et al, Wang et al , . . .

dynamic subgrid-scale: Wang et al

numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .

Schneier Friday talk

Stabile Thursday talk

Ali, Hijazi Wednesday poster

Taddei Wednesday poster

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17

Connections

Correction = Numerical Dissipation

eddy viscosity ROMs

mixing length: Lumley et al

Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al

variational multiscale: Iollo et al, Wang et al , . . .

dynamic subgrid-scale: Wang et al

numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .

Schneier Friday talk

Stabile Thursday talk

Ali, Hijazi Wednesday poster

Taddei Wednesday poster

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17

Connections

Correction = Numerical Dissipation

eddy viscosity ROMs

mixing length: Lumley et al

Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al

variational multiscale: Iollo et al, Wang et al , . . .

dynamic subgrid-scale: Wang et al

numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .

Schneier Friday talk

Stabile Thursday talk

Ali, Hijazi Wednesday poster

Taddei Wednesday poster

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17

Connections

Correction = Numerical Dissipation

eddy viscosity ROMs

mixing length: Lumley et al

Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al

variational multiscale: Iollo et al, Wang et al , . . .

dynamic subgrid-scale: Wang et al

numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .

Schneier Friday talk

Stabile Thursday talk

Ali, Hijazi Wednesday poster

Taddei Wednesday poster

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17

Connections

Correction = Numerical Dissipation

eddy viscosity ROMs

mixing length: Lumley et al

Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al

variational multiscale: Iollo et al, Wang et al , . . .

dynamic subgrid-scale: Wang et al

numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .

Schneier Friday talk

Stabile Thursday talk

Ali, Hijazi Wednesday poster

Taddei Wednesday poster

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17

Connections

Correction = Numerical Dissipation

eddy viscosity ROMs

mixing length: Lumley et al

Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al

variational multiscale: Iollo et al, Wang et al , . . .

dynamic subgrid-scale: Wang et al

numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .

Schneier Friday talk

Stabile Thursday talk

Ali, Hijazi Wednesday poster

Taddei Wednesday poster

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17

Connections

Correction = Numerical Dissipation

eddy viscosity ROMs

mixing length: Lumley et al

Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al

variational multiscale: Iollo et al, Wang et al , . . .

dynamic subgrid-scale: Wang et al

numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .

Schneier Friday talk

Stabile Thursday talk

Ali, Hijazi Wednesday poster

Taddei Wednesday poster

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17

Connections

Correction = Numerical Dissipation

eddy viscosity ROMs

mixing length: Lumley et al

Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al

variational multiscale: Iollo et al, Wang et al , . . .

dynamic subgrid-scale: Wang et al

numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .

Schneier Friday talk

Stabile Thursday talk

Ali, Hijazi Wednesday poster

Taddei Wednesday poster

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17

Connections

Galerkin ROM Stable, Not Accurate

stable

‖unr ‖2 + ν ∆tM∑

n=1

‖∇unr ‖2 ≤ data

not accurate

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 14 / 17

Connections

Galerkin ROM Stable, Not Accurate

stable

‖unr ‖2 + ν ∆tM∑

n=1

‖∇unr ‖2 ≤ data

not accurate

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 14 / 17

Connections

Galerkin ROM Stable, Not Accurate

stable

‖unr ‖2 + ν ∆tM∑

n=1

‖∇unr ‖2 ≤ data

not accurate

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 14 / 17

Connections

Connections

Large Eddy Simulationu u

h

Variational Multiscale

Nonlinear Galerkin

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 15 / 17

Connections

Connections

Large Eddy Simulationu u

h

Variational Multiscale

Nonlinear Galerkin

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 15 / 17

Connections

Connections

Large Eddy Simulationu u

h

Variational Multiscale

Nonlinear Galerkin

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 15 / 17

Connections

Related Work

no ROM

20th century Adrian, Moser

21st century Iaccarino, Duraisamy, Ling, Xiao

ROM

21st century Chorin, Duraisamy, Codina, Majda, San, Liu

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 16 / 17

Connections

Related Work

no ROM

20th century Adrian, Moser

21st century Iaccarino, Duraisamy, Ling, Xiao

ROM

21st century Chorin, Duraisamy, Codina, Majda, San, Liu

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 16 / 17

Connections

Related Work

no ROM

20th century Adrian, Moser

21st century Iaccarino, Duraisamy, Ling, Xiao

ROM

21st century Chorin, Duraisamy, Codina, Majda, San, Liu

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 16 / 17

Connections

Related Work

no ROM

20th century Adrian, Moser

21st century Iaccarino, Duraisamy, Ling, Xiao

ROM

21st century Chorin, Duraisamy, Codina, Majda, San, Liu

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 16 / 17

Connections

Related Work

no ROM

20th century Adrian, Moser

21st century Iaccarino, Duraisamy, Ling, Xiao

ROM

21st century Chorin, Duraisamy, Codina, Majda, San, Liu

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 16 / 17

Conclusions

Conclusions

Galerkin ROM + correction

correction data-driven

Re = 100, Re = 500, Re = 1000

reproductive X

predictive (cross-validation) X

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 17 / 17

Conclusions

Conclusions

Galerkin ROM + correction

correction data-driven

Re = 100, Re = 500, Re = 1000

reproductive X

predictive (cross-validation) X

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 17 / 17

Conclusions

Conclusions

Galerkin ROM + correction

correction data-driven

Re = 100, Re = 500, Re = 1000

reproductive X

predictive (cross-validation) X

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 17 / 17

Conclusions

Conclusions

Galerkin ROM + correction

correction data-driven

Re = 100, Re = 500, Re = 1000

reproductive X

predictive (cross-validation) X

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 17 / 17

Conclusions

Conclusions

Galerkin ROM + correction

correction data-driven

Re = 100, Re = 500, Re = 1000

reproductive X

predictive (cross-validation) X

T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 17 / 17

ProblemSolutionConnectionsConclusions