PDE-constrained Optimization Using PETSc/TAO...Using PETSc/TAO Presented to ATPESC 2019 Participants Alp Dener Postdoctoral Appointee, MCS, ANL Q Center, St. Charles, IL (USA) Date

ATPESC Numerical Software Track

PDE-constrained OptimizationUsing PETSc/TAO

Presented to ATPESC 2019 Participants

Alp DenerPostdoctoral Appointee, MCS, ANLQ Center, St. Charles, IL (USA)Date 08/06/2019

ATPESC 2017, July 30 – August 11, 20172

What is optimization?

• Optimization variables 𝑝 ∈ ℝ$– e.g.: boundary conditions, parameters, geometry

• Objective function 𝑓: ℝ$ → ℝ– e.g.: lift, drag, pressure, temperature, stress, strain

minimizep

f(p)<latexit sha1_base64="MetxMs3CmLkjgHxDqD5ro9CMW6c=">AAACK3icbVC7TsMwFHV4vykwslhUIFiqpCDBiGBhBIkCUlNVjnPTWthOsG8QJco38CWMrPARTCBWJn4Ct3TgdSRLR+ecq+t7okwKi77/4o2Mjo1PTE5Nz8zOzS8sVpaWz2yaGw4NnsrUXETMghQaGihQwkVmgKlIwnl0edj3z6/BWJHqU+xl0FKso0UiOEMntStbYa5j5wMWWVmECDdYKKGFErdQljS8yllMN5LNbKtdqfo1fwD6lwRDUiVDHLcrH2Gc8lyBRi6Ztc3Az7BVMIOCSyhnwtxCxvgl60DTUc0U2FYxOKmk606JaZIa9zTSgfp9omDK2p6KXFIx7NrfXl/8z2vmmOy1CqGzHEHzr0VJLimmtN8PjYUBjrLnCONGuL9S3mWGcXQt/dwiO6kLdFXdHePaCX538Zec1WvBdq1+slPdPxj2NEVWyRrZJAHZJfvkiByTBuHkjjyQR/Lk3XvP3qv39hUd8YYzK+QHvPdPnvWo6Q==</latexit>


What is optimization?

• Simplification: 𝑓(𝑝) is minimized where ∇+𝑓 𝑝 = 0

• Gradient-free: Heuristic search through 𝑝 space

• Gradient-based: Find search directions based on ∇+𝑓

minimizep

f(p)<latexit sha1_base64="MetxMs3CmLkjgHxDqD5ro9CMW6c=">AAACK3icbVC7TsMwFHV4vykwslhUIFiqpCDBiGBhBIkCUlNVjnPTWthOsG8QJco38CWMrPARTCBWJn4Ct3TgdSRLR+ecq+t7okwKi77/4o2Mjo1PTE5Nz8zOzS8sVpaWz2yaGw4NnsrUXETMghQaGihQwkVmgKlIwnl0edj3z6/BWJHqU+xl0FKso0UiOEMntStbYa5j5wMWWVmECDdYKKGFErdQljS8yllMN5LNbKtdqfo1fwD6lwRDUiVDHLcrH2Gc8lyBRi6Ztc3Az7BVMIOCSyhnwtxCxvgl60DTUc0U2FYxOKmk606JaZIa9zTSgfp9omDK2p6KXFIx7NrfXl/8z2vmmOy1CqGzHEHzr0VJLimmtN8PjYUBjrLnCONGuL9S3mWGcXQt/dwiO6kLdFXdHePaCX538Zec1WvBdq1+slPdPxj2NEVWyRrZJAHZJfvkiByTBuHkjjyQR/Lk3XvP3qv39hUd8YYzK+QHvPdPnvWo6Q==</latexit>


Why do we care?

Inputs

• Boundary Conditions

• Parameters• Geometry

Simulation

• Fluid Dynamics• Continuum

Mechanics• Heat Transfer

Outputs

• Lift• Drag• Stress• Temperature

We know a lot about how to solve the forward problem…


Why do we care?

Inputs

• Boundary Conditions

• Parameters• Geometry

Simulation

• Fluid Dynamics• Continuum

Mechanics• Heat Transfer

Outputs

• Lift• Drag• Stress• Temperature

We know a lot about how to solve the forward problem…

ControlDecisionDesign

Optimization Objective

…many scientific questions present as inverse problems!


Why do we care?


Outline

• Intro to PDE-constrained Optimization– Full-space formulation– Reduced-space formulation

• Introduction to TAO– Sample main program– User/problem function callback

• Sensitivity Analysis– Finite difference method– Adjoint method

• Hands-on Example: Boundary Control w/ 2D Laplace Equation


PDE-constrained Optimization

• Optimization variables 𝑝 ∈ ℝ$

• State variables 𝑢 ∈ ℝ/

• State equations 𝑅: ℝ$1/ → ℝ/

• “Full-space” formulation – optimization includes both optimization and state variables

minimizep,u

f(p, u)

subject to R(p, u) = 0<latexit sha1_base64="mGNk3+DRQJVjQtGbmfkQyc+lDe4=">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</latexit>


First-order Optimality Conditions

• Construct the Lagrangian where 𝜆 ∈ ℝ/

• Differentiate w.r.t. every input for first-order optimality

• Also known as the Karush-Kuhn-Tucker (KKT) conditions

L(p, u,�) = f(p, u) + �TR(p, u)<latexit sha1_base64="CweUwrxex4TBqKN6/xN9UdTRjbk=">AAACOXicbZDLSsNAFIYnXmu9RV26GSxCi6UkVagboejGhYsqvUFTy2Q6aYdOLsxMhBL6KD6JS7f6AC7diS59ASdpBNv6w8DPd87hzPntgFEhDeNNW1peWV1bz2xkN7e2d3b1vf2m8EOOSQP7zOdtGwnCqEcakkpG2gEnyLUZadmjq7jeeiBcUN+ry3FAui4aeNShGEmFenrFcpEcYsSim0k+KMKwCC2mxvuoAC+gk6ACPPmF9/W7KerpOaNkJIKLxkxNDqSq9fQvq+/j0CWexAwJ0TGNQHYjxCXFjEyyVihIgPAIDUhHWQ+5RHSj5MAJPFakDx2fq+dJmNC/ExFyhRi7tuqMzxHztRj+V+uE0jnvRtQLQkk8PF3khAxKH8ZpwT7lBEs2VgZhTtVfIR4ijrBUmc5uYQNfNQzdsjpGpWPOZ7FomuWSeVoq357lqpdpThlwCI5AHpigAqrgGtRAA2DwCJ7BC3jVnrR37UP7nLYuaenMAZiR9v0DpauqWA==</latexit>

rpL =@f

@p+ �T @R

@p= 0

ruL =@f

@u+ �T @R

@u= 0

r�L = R(u, p) = 0<latexit sha1_base64="3VjKXQj0UpdcCC2PbvDhovLgEeg=">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</latexit>


Solving the Problem

• Apply Newton’s method to the KKT conditionsFor k = 0, 1, 2, …

Convergence check (i.e., ∇+ℒ ≤ 𝜀+ and 𝑅 ≤ 𝜀6)

Solve

Step acceptance with globalization (e.g., line search)

2

64r2

ppL r2upL @R

@p

T

r2puL r2

uuL @R@u

T

@R@p

@R@u 0

3

75

k

0

@�p�u��

1

A =

0

@�rpL�ruL

�R(p, u)

1

A

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Solving the Problem

• Apply Newton’s method to the KKT conditionsFor k = 0, 1, 2, …

Convergence check (i.e., ∇+ℒ ≤ 𝜀+ and 𝑅 ≤ 𝜀6)

Solve

Step acceptance with globalization (e.g., line search)

2

64r2

ppL r2upL @R

@p

T

r2puL r2

uuL @R@u

T

@R@p

@R@u 0

3

75

k

0

@�p�u��

1

A =

0

@�rpL�ruL

�R(p, u)

1

A


Conjugate GradientQuasi-NewtonNewton-Krylov…


Solving the Problem

• Full-space formulation– PDE solution tightly coupled with optimization– Avoid the cost of a complete PDE solution at every optimization iteration– Large saddle-point problem – difficult to solve

2

64r2

ppL r2upL @R

@p

T

r2puL r2

uuL @R@u

T

@R@p

@R@u 0

3

75

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0

@�p�u��

1

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0

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1

A



An Alternative Approach

• Reduced-space formulation– PDE-constraint is eliminated via implicit function theorem– Optimization algorithm and PDE solver are independent– Each objective evaluation requires a full PDE solution

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f(p, u(p))<latexit sha1_base64="Q2NAkcEiP+Nlwr6pgNMqcpsshuM=">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</latexit>


Toolkit for Advanced Optimization (TAO)

• General-purpose continuous optimization toolbox for large-scale problems– Parallel (via PETSc data structures)– Gradient-based– Bound-constrained – Nonlinear constraint support under development

• Support for reduced-space PDE-constrained optimization• Distributed with PETSc (https://www.mcs.anl.gov/petsc/)

• Similar packages:– Rapid Optimization Library (https://trilinos.github.io/rol.html)– HiOP (https://github.com/LLNL/hiop)

Vec

KSP

SNES

TS

Mat

PC

DM

TAO

PETSc

https://www.mcs.anl.gov/petsc/

https://trilinos.github.io/rol.html

https://github.com/LLNL/hiop


TAO: The Basics

• Sample main programAppCtx user; Tao tao; Vec P;

PetscInitialize( &argc, &argv,(char *)0,help ); VecCreateMPI(PETSC_COMM_WORLD, user.n, user.N, &P); VecSet(P, 0.0);

TaoCreate(PETSC_COMM_WORLD, &tao); TaoSetType(tao, TAOBQNLS); /* BQNLS: quasi-Newton line search */TaoSetInitialVector(tao, P); TaoSetObjectiveAndGradientRoutine(tao, FormFunctionGradient, (void*) &user); TaoSetFromOptions(tao); TaoSolve(tao);

VecDestroy(&P); TaoDestroy(&tao); PetscFinalize();


TAO: The Basics

• User provides function for problem implementationAppCtx user; Tao tao; Vec P;

PetscInitialize( &argc, &argv,(char *)0,help ); VecCreateMPI(PETSC_COMM_WORLD, user.n, user.N, &P); VecSet(P, 0.0);

TaoCreate(PETSC_COMM_WORLD, &tao); TaoSetType(tao, TAOBQNLS); /* BQNLS: quasi-Newton line search */TaoSetInitialVector(tao, P); TaoSetObjectiveAndGradientRoutine(tao, FormFunctionGradient, (void*) &user); TaoSetFromOptions(tao); TaoSolve(tao);

VecDestroy(&P); TaoDestroy(&tao); PetscFinalize();


TAO: User Function

• User function computes objective and gradienttypedef struct {

/* user-created context for storing application data */} AppCtx;

PetscErrorCode FormFunctionGradient(Tao tao, Vec P, PetscReal *fcn, Vec G, void *ptr) { AppCtx *user = (AppCtx*)ptr; const PetscScalar *pp; PetscScalar *gg;

VecGetArrayRead(P, &pp); VecGetArray(G, &gg); /* USER TASK: Compute objective function and store in fcn *//* USER TASK: Compute compute gradient and store in gg */VecRestoreArrayRead(P, &pp); VecRestoreArray(G, &gg);

return 0; }


TAO: User Function

• State/PDE solution:Solve 𝑅 𝑝, 𝑢 = 0 for 𝑢(𝑝) at new 𝑝

• Objective evaluation: Compute 𝑓(𝑝, 𝑢 𝑝 )

• Sensitivity analysis:Compute G = ∇+𝑓 at 𝑝 and 𝑢(𝑝)


TAO: User Function

• State/PDE solution:Solve 𝑅 𝑝, 𝑢 = 0 for 𝑢(𝑝) at new 𝑝

• Objective evaluation: Compute 𝑓(𝑝, 𝑢 𝑝 )

• Sensitivity analysis:Compute G = ∇+𝑓 at 𝑝 and 𝑢(𝑝)

• Necessary for gradient-based optimization

• Converges to the local optimum faster than gradient-free methods (i.e., fewer PDE solutions)

• Types:• Finite difference• Discrete adjoint method


Sensitivity Analysis: Finite Difference

• Easy to implement– Only requires function evaluations

• Inefficient for large numbers of optimization variables• Step-size dilemma – truncation error vs. subtractive cancellation

Simulation𝑝9 + 𝛿 𝑓 + 𝛿

𝑝 𝑓−

+

𝑑𝑓𝑑𝑝9


Sensitivity Analysis: Discrete Adjoint Method

• Recall the first-order optimality conditions for PDE-constrained optimization

• Reduced-space formulation solves equations (2) and (3) fully at each optimization iteration and substitutes into equation (1)

rpL =@f

@p+ �T @R

@p= 0 (1)

ruL =@f

@u+ �T @R

@u= 0 (2)

r�L = R(u, p) = 0 (3)<latexit sha1_base64="dLQXPwLf7G8xO8cwlaekpdi+Em8=">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</latexit>



1. State/PDE Solution:Solve 𝑅 𝑝, 𝑢 = 0 for 𝑢 at new 𝑝

2. Adjoint Solution:

Solve >?>6

@𝜆 = − >A

>6 for 𝜆 at new 𝑝 and u(𝑝)

3. Gradient Assembly:Compute G = ∇+𝑓 =

>A>+ + 𝜆

@ >?>+



1. State/PDE Solution:Solve 𝑅 𝑝, 𝑢 = 0 for 𝑢 at new 𝑝

2. Adjoint Solution:

Solve >?>6

@𝜆 = − >A

>6 for 𝜆 at new 𝑝 and u(𝑝)

3. Gradient Assembly:Compute G = ∇+𝑓 =

>A>+ + 𝜆

@ >?>+

Solve ∇Cℒ = 0 (eqn. 3)

Solve ∇6ℒ = 0 (eqn. 2)

Evaluate ∇+ℒ (eqn. 1)



• Computational cost is independent of the number of optimization variables– One linear system solution for each scalar function

• Requires transpose operators for the PDE Jacobians (w.r.t. both state and optimization variables)

• PETSc/TS (Time Steppers) package implements checkpointing for backwards-in-time adjoint solutions– Not covered in this tutorial


Hands-on Example: Boundary Control w/ 2D Laplace Eqn

• Control left, top and bottom Dirichlet bounds to recover target solution at the right boundary

minimizep

1

2

Z 1

0(u(1, y)� utarget)

2dy

governed by@2u

@x2+

@2u

@y2= 0, 8 x, y 2 (0, 1)

@u

@x

��u(1,y)

= 0, 8 y 2 (0, 1)

p =⇥u(x, 0) u(x, 1) u(0, y)

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• PDE solution, objective function and gradient evaluations implemented with AMReX(https://amrex-codes.github.io/amrex/)

• Target solution set to 𝑢DEFGHD = 4(𝑦 − 0.5) M−0.5

• Lapace equation is self-adjoint– PDE Jacobian is symmetric

• Command line options:– Problem size: -nx 128– Switch on finite difference gradient: -fd

https://amrex-codes.github.io/amrex/



Initial Final



$ ./main2d.gnu.MPI.ex inputs –tao_monitor –tao_ls_type armijo –tao_fmin 1e-6 –tao_gatol 1e-12

AMReX (19.08) initialized0 TAO, Function value: 0.00564792, Residual: 4.57574e-081 TAO, Function value: 0.00564792, Residual: 4.57574e-082 TAO, Function value: 0.000351831, Residual: 6.72001e-093 TAO, Function value: 0.000104118, Residual: 4.5399e-094 TAO, Function value: 4.39909e-06, Residual: 5.62804e-105 TAO, Function value: 3.20011e-06, Residual: 3.18434e-106 TAO, Function value: 2.90562e-06, Residual: 3.67294e-107 TAO, Function value: 2.82687e-06, Residual: 3.54986e-108 TAO, Function value: 2.74912e-06, Residual: 3.45265e-109 TAO, Function value: 2.74912e-06, Residual: 3.45265e-10

10 TAO, Function value: 2.5271e-06, Residual: 1.99102e-1011 TAO, Function value: 2.41122e-06, Residual: 1.43435e-1012 TAO, Function value: 2.10344e-06, Residual: 8.34685e-1113 TAO, Function value: 1.78759e-06, Residual: 1.11393e-1014 TAO, Function value: 1.30814e-06, Residual: 8.4585e-1115 TAO, Function value: 1.10837e-06, Residual: 9.14759e-1116 TAO, Function value: 1.08865e-06, Residual: 9.52258e-1117 TAO, Function value: 8.87623e-07, Residual: 1.29118e-10

TaoSolve() duration: 1316289 microseconds[The Pinned Arena] space (MB) used spread across MPI: [8 ... 8]AMReX (19.08) finalized



• Tutorial goals:– Compare computational cost of finite difference gradient to the adjoint method

– Verify that the cost of the adjoint method is (mostly) independent of the number of optimization variables

– Solve the problem with different TAO algorithms

– Interpret the TAO monitor output and assess convergence


Take Away Messages

• PDE-constrained optimization does not need to be intimidating!– PETSc/TAO provides interfaces and algorithms that work with reduced-space formulations– PETSc/TAO can compute gradients and Hessians automatically with finite differencing– PETSc data structures are easy to couple with most PDE solvers (e.g., AMReX)

• The adjoint method is ideal for sensitivity analysis– Some PDE solvers already have necessary building blocks– Possible to take implementation shortcuts in self-adjoint problems

• PETSc/TAO offers parallel optimization algorithms for large-scale problems– Optimization data structures are duplicated from user-generated PETSc vectors– User has full control over parallel distribution and vector type


Acknowledgements

AMReX: https://amrex-codes.github.io/amrex/

PETSc/TAO: https://www.mcs.anl.gov/petsc/

Special thanks to Don Willcox for support in developing the hands-on example code.

Support for this work was provided through Scientific Discovery through Advanced Computing (SciDAC) program and the Exascale Computing Project funded by U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research.

https://amrex-codes.github.io/amrex/

https://www.mcs.anl.gov/petsc/

Documents

PDE-constrained Optimization Using PETSc/TAO...Using PETSc/TAO Presented to ATPESC 2019 Participants Alp Dener Postdoctoral Appointee, MCS, ANL Q Center, St. Charles, IL (USA) Date