Robust Simulation of Transient PDE Models under Uncertainty

Chenyu Wang, Matthew D. Stuber

2019 AIChE Annual MeetingOrlando, FL, November 14th

Robust Simulation

1

Robust simulation

ODE IVPs

Finite difference method

Differential inequalities

Taylor series expansion

Rigorous bounds

Rigorous bounds

Robust simulation is the ability to calculate rigorous bounds on the system’s states for all realizations of parametric uncertainty

[1] M. D. Stuber and P. I. Barton, “Robust simulation and design using semi-infinite programs with implicit functions,” International Journal of Reliability and Safety, vol. 5, no. 3-4, pp. 378-397, 2011.[2] J. K. Scott and P. I. Barton, “Bounds on the reachable sets of nonlinear control systems,” Automatica, vol. 49, no. 1, pp. 93-100, 2013.[3] G. F. Corliss, “Survey of interval algorithms for ordinary differential equations,” Applied Mathematics and Computation, vol. 31, pp. 112-120, 1989.

PDE IBVPs

Reacting flow systems

Thermofluidsystems

Drug delivery

Parametric Partial Differential Equations

• Provide valid and efficient bounds for parametric PDE system (IBVP)

2

• Assumptions:

― is locally Lipschitz continuous;

― For every , there exists a unique solution over the time domain .

• Objective: efficiently compute bounds on the reachable set with less conservatism.

• Reachable set: .

Interval Arithmetic and Affine Arithmetic

Interval Arithmetic (IA) Affine Arithmetic (AA)

•• Dependency problems: overestimation

• Keep track of the dependency;• Enhance the efficacy of interval operations

[4] J. Comba, J. Stolfi, “Affine arithmetic and its applications to computer graphics,” anais do vii sibgrapi, 9-18 (1993).[5] J. Stolfi, L. H. de Figueiredo, “An introduction to affine arithmetic,” Trends in Applied and ComputationalMathematics, 4 (3) (2003) 297–312.[6] L. H. De Figueiredo, J. Stolfi, “Affine arithmetic: concepts and applications,” Numerical Algorithms, 37 (1-4) (2004) 147–158. 3

Physical bounds G

Differential Inequalities

1 0 0( , ), {0,..., , }

k k k kh k x x f x x cp

1, , 0, 0,

1, , 0, 0,

([ , ], ),

([ , ], ),

L L L L U L Lk i k i i k k i iU L Uk i k i i k k i i

U U U U

x

x

x hf P c

x hf cP

x

x

x x

x x1, , 0, 0,

1, , 0, 0,

( [ , ], ),

( [ , ], ),

L L L L L

U U U U

U L Lk i k i i i k k i iU U L Uk i k i i i k k i i

x P

x

x hf x c

x hf x cP

x x

x x

0 0 0

( )( ( ), ( )), ( , ], [)

f

d tt t t

dtt t

t

xf x cpx

[ ] :([ , ]) { , }

([ , ]) { , }[ ] :

L L U L U Li i iU L U L U Ui i i

z x

xz

x x z xx

x x xxz

1, , 0, 0,

1, , 0, 0,

( ( ([ , ])), ),

( ( ([ , ])), ),

L L L L L U L Lk i k i i G i k k i iU U L Uk i k i i G i k k i i

U U U U

x P

x

x hf x c

x hf xP c

x x

x x

( ) ( ) , with

xnGX G X X X

X G∩ ⊂ ⊂ ∀ ⊂

∩ =∅/

Discrete-time DI

[7] X. Yang and J. K. Scott, “Efficient reachability bounds for discrete-time nonlinear systems by extending the continuous-time theory of differential inequalities," in 2018 Annual American Control Conference (ACC), pp. 6242-6247, IEEE, 2018.

( ) pnPt p

4

ODE IVPs

Explicit Euler

Interval extension Flattening operators Interval refinement operators

Bounding PDE

5

PDE IBVP

Bounds

Spatial FDM

DI

IA

AA

Discrete-time integration

? System of ODE IVPs

Continuous space state variable

Discrete space state variables

Continuous-time integration

?

Convection-Diffusion System

Centered finite difference Forward finite difference Backward finite difference

t = 0.5

6

Convection-Diffusion System

Centered finite difference Forward finite difference Backward finite difference

y = 0.5

7

Convection-Reaction System

Centered finite difference Forward finite difference Backward finite difference

t = 0.5

8

Convection-Reaction System

Centered finite difference Forward finite difference Backward finite difference

y = 0.5

9

Diffusion-Reaction System

t = 0.5 y = 0.5

10

Coupled IBVPs

[8] J. Makungu, H. Haario, W. C. Mahera, “A generalized 1-dimensional particle transport method for convection diffusion reaction model,” Afrika Matematika, 23 (1) (2012) 21–39. 11

Coupled IBVPs

12

Transport Model in Tumor

22

2 ( )s spp pRα

∇ = −

udr

K dp= −

( )

( ) ( )

v

v p v

rSr pL pV

φ

φ

=∇⋅

= −

u

• Fluid Transport Model– Darcy’s law– Continuity equation

• Solute Transport Model2

22 2

1 ( 1 () ) sC r uC CD rt r r r r r

φ∂ ∂ ∂ ∂+ = +

∂ ∂ ∂ ∂Convection Diffusion

• Pore Theory2

8o

prLL

γµ

= oHDPL

γ= 1 Wσ = −

[9] J. D. Martin, M. Panagi, C. Wang, T. T. Khan, M. R. Martin, C. Voutouri, K. Toh, P. Papageorgis,F. Mpekris, C. Polydorou, et al., “Dexamethasone increases cisplatin-loaded nanocarrier delivery and efficacy in metastatic breast cancer by normalizing the tumor microenvironment,” ACS nano.

13

Transport Model in Tumor

[2.75E-7,2.85E-7]K ∈ [1.65E-6,1.75E-6]pL ∈

14

Dimensionless Radial Position r

CD

imen

sion

less

Con

cent

ratio

n

Concentration at 30 min post injection

Bounds DI

IA

FDMPDE

System of ODE IVPs

?

AAG

Concentration at medium radius

Transport Model in Tumor

[2.75E-7,2.85E-7]K ∈ [1.65E-6,1.75E-6]pL ∈

15C

Dim

ensi

onle

ss C

once

ntra

tion

Time (s)Bounds DI

IA

FDMPDE

System of ODE IVPs

?

AAG

Transport Model in Tumor

16

Dimensionless Radial Position r

CD

imen

sion

less

Con

cent

ratio

n

Concentration at 30 min post injectionMethod Time (s)

Exact solutions 0.156898524

FDM using IA and DI 4.366923814

FDM using IA and DI with G 212.759461536

FDM using AA and DI 4.033772619

FDM using AA and DI with G 201.610427039

0 0.02 0.04 0.06 0.08r

-0.4

-0.2

0

0.2

0.4

0.6

c

Summary

Backward finite difference

Conv-Diff Conv-Rxn Diff-Rxn Tumor Model

FD VIA=VAA VIA=VAA Unstable

BD VIA=VAA VIA=VAA VIA>VAA

CD VIA>VAA VIA>VAA VIA>VAA Unstable

Interval Extension 0 0.2 0.4 0.6 0.8 1

y

-6

-4

-2

0

2

4

6

x

10 148

Exact solutionsFDM using AA

0 0.2 0.4 0.6 0.8 1

y

-8

-6

-4

-2

0

2

4

6

8

x

10 217

Exact solutionsFDM using IA

17

Acknowledgements Group members

• Prof. Matthew D. Stuber, • Matthew Wilhelm, • William Hale• Other group members

Funding• University of Connecticut• National Science Foundation, Award No.: 1560072, 1706343, 1932723Any opinions, finding and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation

Any questions?

Backup Slides

Discrete-time Differential Inequalities

1 0 0( , ), {0,..., , }

k k k kh k x x f x x c

1, , 0, 0,

1, , 0, 0,

([ , ], ),

([ , ], ),

L L L L U L Lk i k i i k k i iU L Uk i k i i k k i i

U U U U

x

x

x hf x c

x hf x c

x x

x x

1, , 0, 0,

1, , 0, 0,

( [ , ], ),

( [ , ], ),

L L L L L U L Lk i k i i i k k i iU U L Uk i k i i i k k i i

U U U U

x

x

x hf x c

x hf x c

x x

x x

0 0 0

( )( ( ), ( )), ( , ], [)

f

d tt t t

dtt t

t

xf x x c

[ ] :([ , ]) { , }

([ , ]) { , }[ ] :

L L U L U Li i iU L U L U Ui i i

z x

xz

x x z xx

x x xxz

Explicit Euler

Interval extension

Flattening operators

• For continuous-time system, a trajectory cannot leave the enclosure without crossing its boundary

• Thus, at any , it is only necessary for to decrease faster than all trajectories that are already incident on the ith lower face of .

0( ; , )tx c

[ ( ), ( )]UL t tx x

0[ ],

ftt t ( )L

ix t

0( ; , )

ix t c

[ ( ), ( )]UL t tx x

1, , 0, 0,

1, , 0, 0,

( ( ([ , ])), ),

( ( ([ , ])), ),

L L L L L U L Lk i k i i G i k k i iU U L Uk i k i i G i k

Ui

U Ui

Uk

x

x

x hf x c

x hf x c

x x

x x

Prior enclosure G

Example: [ ] {[1 5],[2, 4],, , [1,7]}L U x x

2

2

{[1 5],[2,2],[1,7]}

{[1 5],[4, 4],[1,

([ , ]) ,

( 7]}[ , ]) ,

L L U

U L U

x x

x x

I.C. and B.C.

Coupled IBVPs

[8] J. Makungu, H. Haario, W. C. Mahera, “A generalized 1-dimensional particle transport method for convection diffusion reaction model,” Afrika Matematika, 23 (1) (2012) 21–39. 11

Coupled IBVPs

[8] J. Makungu, H. Haario, W. C. Mahera, “A generalized 1-dimensional particle transport method for convection diffusion reaction model,” Afrika Matematika, 23 (1) (2012) 21–39. 11

Discrete-time Differential Inequalities

1 0 0( , ), {0,..., , }

k k k kh k x x f x x c

1, , 0, 0,

1, , 0, 0,

([ , ], ),

([ , ], ),

L L L L U L Lk i k i i k k i iU L Uk i k i i k k i i

U U U U

x

x

x hf x c

x hf x c

x x

x x

1, , 0, 0,

1, , 0, 0,

( [ , ], ),

( [ , ], ),

L L L L L U L Lk i k i i i k k i iU U L Uk i k i i i k k i i

U U U U

x

x

x hf x c

x hf x c

x x

x x

3

Discrete-time differential inequalities

Tight enclosure

Discrete-time Differential inequalities

0 0 0

( )( ( ), ( )), ( , ], [)

f

d tt t t

dtt t

t

xf x x c

[ ] :([ , ]) { , }

([ , ]) { , }[ ] :

L L U L U Li i iU L U L U Ui i i

z x

xz

x x z xx

x x xxz

Explicit Euler

Interval extension

Flattening operators

Global optimization

[9] X. Yang and J. K. Scott, “Efficient reachability bounds for discrete-time nonlinear systems by extending the continuous-time theory of differential inequalities," in 2018 Annual American Control Conference (ACC), pp. 6242-6247, IEEE, 2018.

Discrete-time DI vs Continuous-time DI

Discrete-time methods Time per step Time

exact 0.0001506623 0.035599025

Interval 0.001027332844 0.241191195

SDI 0.000212136452 0.048447647

IGB 0.0005043008 0.122923914

BIG 0.000196413496 0.052653721

MG 0.000389502404 0.096926013

Continuous-time methods

Time Steps Time per step Literature

exact 0.167765429 44 0.0038128507 0.004

Interval 0.28842996 63 0.0045782533

SDI 0.374489244 1160 0.0003228356 0.014

MG 0.097186758 64 0.0015185431 0.055

Hukuhara = Affine[ , ], [ , ]L U ULX x x Y y y= =

, ,2 2

,2 2

L U U L L U U L

X X Y Y

X X

Y Y

x x x x y y y ym m

r

r r

mr

XY m

=

=

+ − + −= = = =

+

+

when ( ) ( )

(

(

) ( )

) ( )2 2 2 2

1 1) ( )2 21 1) [ ,

( [ 1,1]

(2 2

X Y

X X Y Y

X Y X YL U L U U L U L

L L U U U U L L

L L U U U U L L U U L

r rm r r

x

r rX Y m

x y y x x y y

x y x y x y x y

x y x y x y x

m

y x y x y

m+ +− + −

+ + − −− + −

− + − + − − +

− + − + − + +

≥

− = −=

=

+

=

−= −

−

−

]

[ , ]

L

L L U Ux y x y− −=

when ( ) ( )

(

(

) ( )

) ( )2 2 2 2

1 1) ( )2 21 1) [ ,

( [ 1,1]

(2 2

X Y

X X Y Y

X Y X YL U L U U L U L

L L U U U U L L

L L U U U U L L U U L

r rm r r

x

r rX Y m

x y y x x y y

x y x y x y x y

x y x y x y x

m

y x y x y

m+ +− + −

+ + − −− + −

− + − + − − +

− + − + − −

<

− = −=

=

= −

= + − + + −

]

[ , ]

L

U U L Lx y x y−= −