This document summarizes an approach for optimal control of coupled partial differential equation (PDE) networks using automated code generation. It discusses representing PDE networks as graphs, formulating the optimal control problem, deriving adjoint equations to compute gradients, discretizing control variables, and generating code to solve the direct and adjoint problems. Tools used include the DOT language for graph representation, SymPy for symbolic math, Cog for code generation, SfePy for PDE solvers, and SciPy for numerics.

1. Σ YSTEMS
Optimal control of coupled PDE networks
with automated code generation
Dimitrios Papadopoulos
Delta Pi Systems, Thessaloniki
ICNAAM 2012, Kos, Greece
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2. Overview
◮ Application of PDE Networks
◮ Representation of PDE Networks as a Graph
◮ Optimal Control of PDEs
◮ Adjoint Equations and Gradient
◮ Continuous vs. Discrete Adjoint Approach
◮ Discretization of the control variables
◮ Flowchart
◮ Tools of the Trade
◮ DOT Language and PyDot
◮ SymPy
◮ Cog
◮ SfePy
◮ SciPy (and NumPy)
◮ Conclusions
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3. Application of PDE Networks
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4. Representation of PDE Networks as a Graph
3
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∂y1
+ ∇ · (a · y1 ) − ∇ · (D∇yi ) = f (y1 , y2 , y3 ; u)
∂t
∂y2
+ ∇ · (a · y2 ) − ∇ · (D∇yi ) = f (y2 , y1 , y3 ; u)
∂t
∂y3
+ ∇ · (a · y3 ) − ∇ · (D∇yi ) = g(y3 , y1 ; u)
∂t
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5. Optimal Control of PDEs
Problem formulation
t1
min J(y, u) = φ(y, u) dxdt subject to:
u 0 Ω
∂yi
+ ∇ · (a · yi ) − ∇ · (D∇yi ) = fi , for i = 1, . . . , nn ,
∂t
where yi are the state variables, and y the state vector,
ui are the control variables, and u the control vector,
fi is a function of some yj and u,
nn is the number of nodes.
Lagrange function
nn t1
∂yi
min L(y, u, p) = J(y, u)+ ( −fi )pi +(ayi −D∇yi )·∇pi dxdt+BC
y,u
i=1 0 Ω ∂t
where pi are the Lagrange multipiers or adjoint variables Dimitris Papadopoulos — Delta Pi Systems
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6. Optimal Control of PDEs (cont.)
Adjoint Equations
n n
∂pi ∂fj
− − ∇ · (a · pi ) − ∇ · (D∇pi ) = φyi (y, u) + pi , for i = 1, . . . , nn ,
∂t j=1
∂yi
Gradient
n n t1
∂L ∂J ∂fi
= + pi dxdt
∂u ∂u i=1 0 Ω ∂u
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7. Continuous vs. Discrete Adjoint
nonlinear linear adjoint
PDE
discrete
equations
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8. Back to the basics
∂L ∂L
How do you calculate ∂u
or ∂y
?
◮ Calculate analytical derivatives and implement them by hand (too time
consuming and error prone)
◮ Symbolic differentiation (you have to add some code generation magic)
◮ Automatic (or algorithmic) differentiation (for each new problem you have
to pass your code through your AD-tool)
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9. Discretization of the control variables
◮ B-spline basis functions
1 if τi−1 ≤ t < τi
Ni0 (t) = ,
0 else
n t − τj−1 n−1 τj+n − t n−1
Nj (t) = Nj (t) + N (t), n>1
τj+n−1 − τj−1 τj+n − τj j+1
◮ B-spline curve formed by control points ci
L
n
u(t) = cj Nj (t)
j=0
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10. Parametrization - Finding a knot sequence
◮ uniform or equidistant
(ti − ti−1 )/(ti+1 − ti ) = 1
◮ chord length
∆ti ||∆di ||
=
∆ti+1 ||∆di+1 ||
◮ cetripetal
1/2
∆ti ||∆di ||
=
∆ti+1 ||∆di+1 ||
◮ Natural end conditions
¨
u(t0 ) = 0, ¨
u(tm ) = 0
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11. Tensor Product Patches
◮ B-spline surface
m n
um,n (t, x) = ci,j Nim (t)Nj (x)
n
i=0 j=0
where ci,j is a matrix representing all the control points of the (t, x) plane.
◮ A surface with the topology of a sphere is not representable as a tensor
product surface, without degeneracies (but we don’t really care)
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12. Flowchart
parse DOT graph
build symbolic PDEs
initialize control points
build symbolic adjoint PDEs
k 1 k k+1
build symbolic gradient
solve direct problem
generate source code for solution of PDEs
objective function compute gradient
generate source code for solution of adjoint PDEs
no
k
generate source code for gradient J <ε solve adjoint problem
yes
solution
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13. Tools of the Trade
Extended DOT Language
Direct Adjoint
SymPy
Cog Cog
SfePy SfePy
SciPy
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14. DOT Language
A DSL for graph representation
digraph network {
3->1 [type=1];
3->2 [type=1];
1->2 [style=dashed, type=2];
2->1 [style=dashed, type=2];
1->3 [style=dotted, type=3];
3 [shape=box];}
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15. Why Python?
◮ programmer’s time vs. computing time
◮ need for manipulation of the objects at runtime
◮ need for source code generation at runtime
◮ lisp, smalltalk, perl, python, ruby, javascript vs. C, C++, java
◮ Paul Graham’s classical closure example of the accumulator:
def foo(n):
s = [n]
def bar(i):
s[0] += i
return s[0]
return bar
◮ many tools and numerical libraries available
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16. PyDot
A Python library for the DOT language
import pydot
graph = pydot.graph_from_dot_file(’test.dot’)
shape_dict = {}
function_dict = {}
for node in graph.get_nodes():
my_name = node.get_name()
shape_dict[my_name] = node.get(’shape’)
function_dict[my_name] = node.get(’function’)
type_dict = {}
style_dict = {}
for edge in graph.get_edges():
my_edge = (edge.get_source(), edge.get_destination())
type_dict[my_edge] = edge.get(’type’)
style_dict[my_edge] = print edge.get(’style’)
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17. SymPy
A Python library for symbolic mathematics
from sympy import Eq, Function, Symbol
from sympy import Derivative as D, Integral
from sympy.abc import x, t, a
y_list = [] # same for u_list
for node in graph.get_nodes():
my_name = node.get_name()
y_list.append(’y_’+str(my_name)) # same for u_list
y = map(Function, y_list) # same for u_list
f = y[1]+u # for example
eq = Eq(D(y[1](x,t),t)+a*D(y[1](x,t),x)-D(D(y[1](x,t),x),x),f)
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18. SymPy (cont.)
Derivative of the Lagrangian
1. Build list of symbols for functions,
2. build a list of substitutions for the functions and the reverse,
3. substitute for the functions,
4. run derivatives with respect to the symbols,
5. substitute functions for the symbols,
6. take any derivatives with respect to the function variables.
Integral(Derivative(F.subs(u(x),foo),foo).doit().subs(foo,←֓
u(x))*v(x),x)
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19. Cog
A code generation tool
...
# [[[cog
# import cog
# functions = [direct, adjoint, gradient]
# for func in functions:
# cog.outl("def %s():" % func)
# cog.outl(" pass")
# ]]]
# [[[end]]]
...
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20. SfePy
A software for solving systems of coupled PDEs by the FEM
◮ SfePy: Simple finite element in python
◮ Other alternatives: FENICS or your favorite FE solver (needs to be general
enough)
◮ Future work: bring metaprogramming to the next level, i.e.,
◮ use FENICS, which generates C++ at runtime from the problem
formulation which can also be run parallel
graph symbolic intermediated parallel
representation representation python code C++ code
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21. SciPy (and NumPy)
A library for doing everything else
import numpy as np
from scipy.optimize import fmin_bfgs
from newly_created_file import objective, gradient
c0 = np.array([0.0, 0.0, 0.0, 0.0])
res = fmin_bfgs(objective, c0, gradient, args, gtol, norm, ...)
and the rest is a piece of cake!
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22. Conclusions
Summary
◮ Framework for optimal control of coupled PDE networks
◮ Automated code generation
◮ Graph representation of coupled PDEs
◮ Symbolic differentiation for the adjoint equations and the gradient
◮ Solution with the finite element method
Outlook
◮ Extention to more general equations,
◮ with 2D or 3D geometries,
◮ and constraints on the state or control variables.
◮ Adaptive refinement of the control variables.
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23. Thank you for your attention!
Delta Pi Systems
Thessaloniki, Greece
http://www.delta-pi-systems.eu
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