mintOC - User contributions [en]

Source Inversion

2016-01-29T21:13:09Z

MirkoHahn:

The source inversion problem is a very basic problem in mixed-integer PDE constrained optimization. It has one PDE constraint equivalent to an inhomogenous Poisson problem with Robin-type boundary conditions. This type of problem arises naturally from the task of finding stationary solutions to the heat equation where the problem domain represents a medium being either heated or cooled and heat is lost to a surrounding medium at a rate proportional to the difference between the temperatures of the domain boundary and the surrounding medium.

The right hand side of Poisson's equation is sometimes referred to as the ''source term''. For the purposes of the source inversion problem, the source term is assumed to be a linear combination of a finite set of predefined terms with coefficients being chosen from <math>\lbrace 0, 1 \rbrace</math>. The objective is to find a source term such that the solution of the Poisson problem closely tracks a given function over the domain.

= Problem Statement =

For the purposes of this discussion, we will restrict ourselves to a one-dimensional setting and assume that the domain under consideration is the interval <math>[0,1]</math>. The Poisson problem with simple Robin-type boundary conditions then takes the following form:

:<math>
\begin{array}{rll}
- u'' &= f \qquad & \text{in } (0,1)\\
u'(0) &= u(0) & \\
u'(1) &= -u(0) &
\end{array}
</math>

Here, <math>f</math> denotes the source term. For a more general version of Robin-type boundary conditions, refer to the [[Controlled Heating|controlled heating problem]] which discusses the same problem in a multi-dimensional setting. Given a control grid <math>0 < \bar{x}_1 < \ldots < \bar{x}_{n_w} < 1</math>, the elementary source terms are given by:

:<math>
f_i(x) = a \cdot \exp \left( \frac{(x - \bar{x}_i)^2}{b} \right)
</math>

The parameters <math>a</math> and <math>b</math> are shared between the individual elementary source terms. Given a reference function <math>u_d</math>, the source inversion problem is given by:

:<math>
\begin{array}[t]{rl}
\min\limits_{u,w} & \int\limits_0^1 (u - u_d)^2 \, \mathrm{d}x \\
\text{s.t.} & \begin{array}[t]{rll}
-u'' &= \sum\limits_{i = 1}^{n_w} w_i f_i \qquad &\text{in } (0,1) \\[1.5ex]
u'(0) &= u(0) & \\
u'(1) &= -u(1) & \\[1.5ex]
\sum\limits_{i = 1}^{n_w} w_i &\leq N \\
w_i &\in \lbrace 0, 1 \rbrace \qquad & \forall i \in \lbrace 1, \ldots, n_w \rbrace
\end{array}
\end{array}
</math>

== Weak formulation ==

Some PDE discretization techniques (such as [[wikipedia:Finite Element Method|finite element methods]]) require the use of [[wikipedia:Weak solution|weak formulations]] of the original problem. The weak formulation of the Poisson problem with Robin-type boundary conditions as described above is obtained using [[wikipedia:Green's identities|Green's identities]]:

:<math>
\int\limits_0^1 \left\langle \nabla u, \nabla v \right\rangle \,\mathrm{d}x + u(1) \cdot v(1) - u(0) \cdot v(0) = \int\limits_0^1 f \cdot v \,\mathrm{d}x \qquad \forall v \in V
</math>

Here, <math>V</math> is a suitable space of test functions. The optimization problem then takes the following form:

:<math>
\begin{array}[t]{rl}
\min\limits_{u,w} & \int\limits_0^1 (u - u_d)^2 \, \mathrm{d}x \\
\text{s.t.} & \begin{array}[t]{rll}
\int\limits_0^1 \left\langle \nabla u, \nabla v \right\rangle \,\mathrm{d}x + u(1) \cdot v(1) - u(0) \cdot v(0) &= \sum\limits_{i = 0}^{n_w} \left( w_i \cdot \int\limits_0^1 f_i \cdot v \,\mathrm{d}x \right) \qquad &\forall v \in V \\[1.5ex]
\sum\limits_{i = 1}^{n_w} w_i &\leq N \\
w_i &\in \lbrace 0, 1 \rbrace \qquad & \forall i \in \lbrace 1, \ldots, n_w \rbrace
\end{array}
\end{array}
</math>

= Parameters =

For testing purposes, the following parameters were chosen:

:<math>
\begin{array}{rl}
n_w &= 8 \\
\bar{x} &= \left( 0.0625, 0.1875, 0.3125, 0.4375, 0.5625, 0.6875, 0.8125, 0.9375 \right)^T \\
a &= 100 \\
b &= 0.02
\end{array}
</math>

The reference solution <math>u_d</math> is generated as a solution to the Poisson problem using the following source term:

:<math>
f_d(x) = 100 \cdot \left( \exp \left( \frac{(x - 0.5)^2}{0.02} \right) + \exp \left( \frac{(x - 3 / \pi)^2}{0.02} \right) \right)
</math>

= Reference solution =

The reference solution was generated using finite element discretizations with an equidistant mesh of <math>128</math> cells. Continuous Galerkin elements of degree <math>1</math> were employed. The linear FEM system was generated using FEniCS and used as part of the constraints in a QP that was solved using CasADi's IPOPT interface. The integer solution was determined using a simple Branch and Bound algorithm. The exact code used to solve the problem alongside detailed solution data can be found under [[Source Inversion (FEniCS/Casadi)]]. The optimal objective function value is <math>0.858837</math>.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
Image:source_inversion_solution.png| Solutions.
Image:source_inversion_controls.png| Controls.
</gallery>

= Source Code =

Model descriptions are available in

* [[:Category: Casadi | Casadi code]] using FEniCS at [[Source Inversion (FEniCS/Casadi)]]

[[Category:MIOCP]]
[[Category:PDE model]]
[[Category:Tracking objective]]

Category:Casadi

2016-01-29T21:12:44Z

MirkoHahn:

CasADi is an open-source software package primarily intended as a symbolic framework for automatic differentiation. It allows the construction, evaluation and automatic differentiation of complex symbolic expressions including invocations of various types of functions such as solvers for common types of optimization problems.

The software is released under the GNU Lesser General Public License and is available from the [http://www.casadi.org project website]. It can be used in software written in C++, Python or Matlab and is extensible through a plugin interface written in C++.

== References ==
<biblist />


[[Category: Implementation]]

Lotka Volterra fishing problem (Casadi)

2016-01-19T17:59:16Z

MirkoHahn: Created page with "This page contains a discretized version of the MIOCP Lotka Volterra fishing problem in [http://www.casadi.org Casadi] format. === Casadi === The model in Python code fo..."

This page contains a discretized version of the MIOCP [[Lotka Volterra fishing problem]] in [http://www.casadi.org Casadi] format.

=== Casadi ===

The model in Python code for a fixed control discretization grid using direct multiple shooting.
<source lang="Python">
# ------------------------------------------------------------------------------
# Lotka Volterra fishing problem with CasADi Python interface using direct
# multiple shooting with IPOPT and Sundials' integrator CVodes
#
# NOTE: CasADi currently has no support for mixed-integer problems. Only the
# relaxed solution is calculated
# ------------------------------------------------------------------------------

# Import
from casadi import *

# Introduce model parameters
t0 = 0
tf = 12
c0 = 0.4
c1 = 0.2
x0 = [0.5, 0.7, 0]

# Define length of time horizon as well as number and length of intervals
T = tf - t0
N = 100
dt = float(T) / N

# Define the DAE
x = SX.sym('x', 3)
w = SX.sym('w')
rhs = vertcat([
dt * ( x[0] - x[0] * x[1] - c0 * x[0] * w),
dt * (-x[1] + x[0] * x[1] - c1 * x[1] * w),
dt * ((x[0] - 1)**2 + (x[1] - 1)**2)
])
dae = SXFunction("dae", daeIn(x=x, p=w), daeOut(ode=rhs))

I_opt = {
'linear_solver': 'csparse',
't0': 0.0,
'tf': 1.0
}
I = Integrator("integrate", "cvodes", dae, I_opt)

# Define number of state variables, controls and variables per interval
nx = x.size1() # Number of states
nu = w.size1() # Number of controls
ns = nx + nu # Number of variables per shooting interval

# Create symbolic expressions for actual NLP variables and define DMS structure
X = MX.sym('X', (N + 1) * nx + N * nu) # Optimization variables
G = [] # Nonlinear constraints
F = X[N*ns+2] # Mayer objective
for i in range(0, N):
X0 = X[i*ns:i*ns+nx] # Initial state for interval
U = X[i*ns+nx:(i+1)*ns] # Control for interval
X1 = X[(i+1)*ns:(i+1)*ns+nx] # Terminal state for interval

Y = I({'x0': X0, 'p': U})
G = vertcat([G, Y['xf'] - X1])

# Define NLP
nlp = MXFunction('nlp', nlpIn(x=X), nlpOut(f=F, g=G))

# Create feasible initial solution by simulating with constant zero control
Xstart = DMatrix.zeros(X.size1(), X.size2())
Xstart[0:nx] = x0
for i in range(0, N):
X0 = Xstart[i*ns:i*ns+nx]
Xstart[i*ns+nx:(i+1)*ns] = 0.0
Xstart[(i+1)*ns:(i+1)*ns+nx] = I({'x0': X0, 'p': 0.0})['xf']

# Create and execute nonlinear solver
solver = NlpSolver('solver', 'ipopt', nlp)
result = solver({
'x0': Xstart,
'lbg': DMatrix.zeros(G.size1()),
'ubg': DMatrix.zeros(G.size1()),
'lbx': x0 + [0.0] + [0.0, 0.0, 0.0, 0.0] * (N - 1) + [0.0, 0.0, 0.0],
'ubx': x0 + [1.0] + [inf, inf, inf, 1.0] * (N - 1) + [inf, inf, inf]
})

# Extract solution and objective
x = result['x']
f = result['f']

# Disassemble solution into easily plottable lists
t = [i*dt for i in range(0, N + 1)]
x0 = [x[i*ns] for i in range(0, N + 1)]
x1 = [x[i*ns+1] for i in range(0, N + 1)]
u = [x[i*ns+nx] for i in range(0, N )]
</source>

== Results ==

=== IPOPT ===
The relaxed solution calculated by IPOPT (Ipopt 3.12, Linux x86_64, default settings, 4 GHz quadcore, Linux 4.2.0-23-generic) has an objective function value of <math>x_2(t_f) = 1.34428</math>. IPOPT requires 22 iterations (6.062 seconds of processing time). The following is a Gnuplot compatible tabular version of the solution:

<pre>
# t, x_0, x_1, u
0, 0.5, 0.7, 5.18436e-09
0.12, 0.519594, 0.659989, 5.42459e-09
0.24, 0.542429, 0.623852, 5.79496e-09
0.36, 0.568601, 0.591429, 6.30563e-09
0.48, 0.59823, 0.562571, 6.97415e-09
0.6, 0.631456, 0.537143, 7.827e-09
0.72, 0.66843, 0.515028, 8.90236e-09
0.84, 0.709312, 0.496134, 1.02545e-08
0.96, 0.75426, 0.480401, 1.19611e-08
1.08, 0.803421, 0.4678, 1.41346e-08
1.2, 0.856919, 0.458344, 1.69417e-08
1.32, 0.914842, 0.452091, 2.06377e-08
1.44, 0.97722, 0.449154, 2.56285e-08
1.56, 1.044, 0.449708, 3.2592e-08
1.68, 1.11503, 0.454003, 4.27322e-08
1.8, 1.18998, 0.462372, 5.83658e-08
1.92, 1.26837, 0.475251, 8.44574e-08
2.04, 1.34942, 0.493189, 1.33425e-07
2.16, 1.43209, 0.516862, 2.44977e-07
2.28, 1.51492, 0.547087, 6.24464e-07
2.4, 1.59605, 0.584816, 0.715281
2.52, 1.61776, 0.61833, 0.999999
2.64, 1.61116, 0.6499, 0.999999
2.76, 1.59844, 0.68229, 1
2.88, 1.57963, 0.714939, 1
3, 1.55497, 0.747196, 1
3.12, 1.52487, 0.778343, 0.999999
3.24, 1.48993, 0.807625, 0.999999
3.36, 1.4509, 0.83429, 0.999999
3.48, 1.40865, 0.857635, 0.999998
3.6, 1.36412, 0.877047, 0.999997
3.72, 1.31827, 0.892039, 0.999991
3.84, 1.27203, 0.90228, 0.999646
3.96, 1.22628, 0.907616, 0.536112
4.08, 1.20755, 0.919653, 0.581667
4.2, 1.18502, 0.928525, 0.503883
4.32, 1.16606, 0.936899, 0.464153
4.44, 1.14851, 0.944178, 0.418428
4.56, 1.13278, 0.950658, 0.378617
4.68, 1.11859, 0.956378, 0.341493
4.8, 1.10582, 0.961433, 0.3073
4.92, 1.09438, 0.965904, 0.276462
5.04, 1.08411, 0.969849, 0.248219
5.16, 1.07492, 0.973333, 0.222481
5.28, 1.0667, 0.976412, 0.19935
5.4, 1.05936, 0.97913, 0.178262
5.52, 1.0528, 0.981533, 0.159402
5.64, 1.04696, 0.983656, 0.142349
5.76, 1.04175, 0.985532, 0.127041
5.88, 1.0371, 0.987191, 0.1133
6, 1.03297, 0.988657, 0.100985
6.12, 1.02929, 0.989954, 0.0899604
6.24, 1.02601, 0.991101, 0.0800981
6.36, 1.0231, 0.992117, 0.0712934
6.48, 1.02051, 0.993015, 0.0634308
6.6, 1.01821, 0.993811, 0.0564127
6.72, 1.01617, 0.994515, 0.0501573
6.84, 1.01435, 0.995139, 0.0445824
6.96, 1.01274, 0.995691, 0.0396177
7.08, 1.01131, 0.99618, 0.0351979
7.2, 1.01003, 0.996613, 0.0312714
7.32, 1.0089, 0.996997, 0.027765
7.44, 1.0079, 0.997337, 0.0246626
7.56, 1.00701, 0.997639, 0.0218975
7.68, 1.00622, 0.997905, 0.0194423
7.8, 1.00552, 0.998142, 0.0172603
7.92, 1.0049, 0.998351, 0.0153224
8.04, 1.00435, 0.998537, 0.0136015
8.16, 1.00386, 0.998701, 0.0120735
8.28, 1.00342, 0.998847, 0.0107171
8.4, 1.00303, 0.998976, 0.00951328
8.52, 1.00269, 0.99909, 0.00844491
8.64, 1.00239, 0.999191, 0.00749702
8.76, 1.00212, 0.999281, 0.00665598
8.88, 1.00188, 0.99936, 0.00590975
9, 1.00167, 0.99943, 0.00524776
9.12, 1.00148, 0.999492, 0.00466052
9.24, 1.00131, 0.999547, 0.00413914
9.36, 1.00117, 0.999596, 0.00367206
9.48, 1.00103, 0.999637, 0.00326159
9.6, 1.00092, 0.999675, 0.00290302
9.72, 1.00082, 0.999707, 0.00258504
9.84, 1.00072, 0.999736, 0.00230262
9.96, 1.00064, 0.999762, 0.00205186
10.08, 1.00057, 0.999785, 0.00182951
10.2, 1.00051, 0.999805, 0.00163302
10.32, 1.00045, 0.999822, 0.00146039
10.44, 1.0004, 0.999838, 0.00131011
10.56, 1.00036, 0.999851, 0.00118097
10.68, 1.00032, 0.999863, 0.00107188
10.8, 1.00028, 0.999872, 0.000981659
10.92, 1.00025, 0.99988, 0.00090944
11.04, 1.00022, 0.999886, 0.000854893
11.16, 1.00019, 0.99989, 0.000818284
11.28, 1.00017, 0.999892, 0.000801181
11.4, 1.00014, 0.999891, 0.000807844
11.52, 1.00012, 0.999886, 0.000848734
11.64, 1.00009, 0.999878, 0.000951928
11.76, 1.00006, 0.999864, 0.00121852
11.88, 1.00002, 0.999838, 0.00267036
12, 0.999915, 0.999769, 0
</pre>

[[Category:Casadi]]

Source Inversion

2016-01-12T21:57:42Z

MirkoHahn: /* Reference solution */

The source inversion problem is a very basic problem in mixed-integer PDE constrained optimization. It has one PDE constraint equivalent to an inhomogenous Poisson problem with Robin-type boundary conditions. This type of problem arises naturally from the task of finding stationary solutions to the heat equation where the problem domain represents a medium being either heated or cooled and heat is lost to a surrounding medium at a rate proportional to the difference between the temperatures of the domain boundary and the surrounding medium.

The right hand side of Poisson's equation is sometimes referred to as the ''source term''. For the purposes of the source inversion problem, the source term is assumed to be a linear combination of a finite set of predefined terms with coefficients being chosen from <math>\lbrace 0, 1 \rbrace</math>. The objective is to find a source term such that the solution of the Poisson problem closely tracks a given function over the domain.

= Problem Statement =

For the purposes of this discussion, we will restrict ourselves to a one-dimensional setting and assume that the domain under consideration is the interval $[0,1]$. The Poisson problem with simple Robin-type boundary conditions then takes the following form:

:<math>
\begin{array}{rll}
- u'' &= f \qquad & \text{in } (0,1)\\
u'(0) &= u(0) & \\
u'(1) &= -u(0) &
\end{array}
</math>

Here, <math>f</math> denotes the source term. For a more general version of Robin-type boundary conditions, refer to the [[Controlled Heating|controlled heating problem]] which discusses the same problem in a multi-dimensional setting. Given a control grid <math>0 < \bar{x}_1 < \ldots < \bar{x}_{n_w} < 1</math>, the elementary source terms are given by:

:<math>
f_i(x) = a \cdot \exp \left( \frac{(x - \bar{x}_i)^2}{b} \right)
</math>

The parameters <math>a</math> and <math>b</math> are shared between the individual elementary source terms. Given a reference function <math>u_d</math>, the source inversion problem is given by:

:<math>
\begin{array}[t]{rl}
\min\limits_{u,w} & \int\limits_0^1 (u - u_d)^2 \, \mathrm{d}x \\
\text{s.t.} & \begin{array}[t]{rll}
-u'' &= \sum\limits_{i = 1}^{n_w} w_i f_i \qquad &\text{in } (0,1) \\[1.5ex]
u'(0) &= u(0) & \\
u'(1) &= -u(1) & \\[1.5ex]
\sum\limits_{i = 1}^{n_w} w_i &\leq N \\
w_i &\in \lbrace 0, 1 \rbrace \qquad & \forall i \in \lbrace 1, \ldots, n_w \rbrace
\end{array}
\end{array}
</math>

== Weak formulation ==

Some PDE discretization techniques (such as [[wikipedia:Finite Element Method|finite element methods]]) require the use of [[wikipedia:Weak solution|weak formulations]] of the original problem. The weak formulation of the Poisson problem with Robin-type boundary conditions as described above is obtained using [[wikipedia:Green's identities|Green's identities]]:

:<math>
\int\limits_0^1 \left\langle \nabla u, \nabla v \right\rangle \,\mathrm{d}x + u(1) \cdot v(1) - u(0) \cdot v(0) = \int\limits_0^1 f \cdot v \,\mathrm{d}x \qquad \forall v \in V
</math>

Here, <math>V</math> is a suitable space of test functions. The optimization problem then takes the following form:

:<math>
\begin{array}[t]{rl}
\min\limits_{u,w} & \int\limits_0^1 (u - u_d)^2 \, \mathrm{d}x \\
\text{s.t.} & \begin{array}[t]{rll}
\int\limits_0^1 \left\langle \nabla u, \nabla v \right\rangle \,\mathrm{d}x + u(1) \cdot v(1) - u(0) \cdot v(0) &= \sum\limits_{i = 0}^{n_w} \left( w_i \cdot \int\limits_0^1 f_i \cdot v \,\mathrm{d}x \right) \qquad &\forall v \in V \\[1.5ex]
\sum\limits_{i = 1}^{n_w} w_i &\leq N \\
w_i &\in \lbrace 0, 1 \rbrace \qquad & \forall i \in \lbrace 1, \ldots, n_w \rbrace
\end{array}
\end{array}
</math>

= Parameters =

For testing purposes, the following parameters were chosen:

:<math>
\begin{array}{rl}
n_w &= 8 \\
\bar{x} &= \left( 0.0625, 0.1875, 0.3125, 0.4375, 0.5625, 0.6875, 0.8125, 0.9375 \right)^T \\
a &= 100 \\
b &= 0.02
\end{array}
</math>

The reference solution <math>u_d</math> is generated as a solution to the Poisson problem using the following source term:

:<math>
f_d(x) = 100 \cdot \left( \exp \left( \frac{(x - 0.5)^2}{0.02} \right) + \exp \left( \frac{(x - 3 / \pi)^2}{0.02} \right) \right)
</math>

= Reference solution =

The reference solution was generated using finite element discretizations with an equidistant mesh of <math>128</math> cells. Continuous Galerkin elements of degree <math>1</math> were employed. The linear FEM system was generated using FEniCS and used as part of the constraints in a QP that was solved using CasADi's IPOPT interface. The integer solution was determined using a simple Branch and Bound algorithm. The exact code used to solve the problem alongside detailed solution data can be found under [[Source Inversion (FEniCS/Casadi)]]. The optimal objective function value is <math>0.858837</math>.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
Image:source_inversion_solution.png| Solutions.
Image:source_inversion_controls.png| Controls.
</gallery>

= Source Code =

Model descriptions are available in

* [[:Category: Casadi | Casadi code]] using FEniCS at [[Source Inversion (FEniCS/Casadi)]]

[[Category:MIPDECO]]
[[Category:PDE model]]

Controlled Heating

2016-01-12T21:56:59Z

MirkoHahn: /* Reference solution */

The controlled heating problem is a simple problem from the domain of continuous PDE constrained optimal control. The goal is to parametrize the [[wikipedia:Heat equation|heat equation]] under Robin-type boundary conditions such that the stationary solution tracks a predefined reference function. This is equivalent to solving [[wikipedia:Poisson equation|Poisson's problem]] under Robin-type boundary equations. The problem is therefore closely related to the [[Source Inversion|source inversion problem]]. As opposed to the source inversion problem however, the controlled heating problem does not require the source term to be a linear combination of elementary source terms. The source term can be an arbitrary function which is usually chosen from the same function space as the temperature function.

= Problem Statement =

Let <math>\Omega \subset \mathbb{R}^n</math> be a contiguous domain and let <math>\nu \colon \partial \Omega \to \mathbb{R}^n</math> be an outer unit normal of <math>\Omega</math>. The Poisson problem with simple Robin-type boundary conditions then takes the following form:

:<math>
\begin{array}{rll}
- \Delta u &= f \qquad & \text{in } \Omega \\
\frac{\partial u}{\partial \nu} &= -u \qquad & \text{on } \partial \Omega
\end{array}
</math>

Here, <math>f</math> denotes the source term. We demand that <math>C_1 \leq f(x) \leq C_2</math> in <math>\Omega</math>. Given a reference function <math>u_d</math>, the optimization problem is given by:

:<math>
\begin{array}[t]{rl}
\min\limits_{u,f} & \int\limits_\Omega (u - u_d)^2 \, \mathrm{d}x \\
\text{s.t.} & \begin{array}[t]{rll}
- \Delta u(x) &= f(x) \qquad & \forall x \in \Omega \\[1.5ex]
\frac{\partial u}{\partial \nu}(x) &= -u(x) \qquad & \forall x \in \partial \Omega \\
f(x) &\in [C_1, C_2] \qquad & \forall x \in \Omega
\end{array}
\end{array}
</math>

== Weak formulation ==

Some PDE discretization techniques (such as [[wikipedia:Finite Element Method|finite element methods]]) require the use of [[wikipedia:Weak solution|weak formulations]] of the original problem. The weak formulation of the Poisson problem with Robin-type boundary conditions as described above is obtained using [[wikipedia:Green's identities|Green's identities]]:

:<math>
\int\limits_\Omega \left\langle \nabla u, \nabla v \right\rangle \,\mathrm{d}x + \int\limits_{\partial \Omega} u \cdot v \,\mathrm{d}s(x) = \int\limits_\Omega f \cdot v \,\mathrm{d}x \qquad \forall v \in V
</math>

Here, <math>V</math> is a suitable space of test functions. The optimization problem then takes the following form:

:<math>
\begin{array}[t]{rl}
\min\limits_{u,f} & \int\limits_\Omega (u - u_d)^2 \, \mathrm{d}x \\
\text{s.t.} & \begin{array}[t]{rll}
\int\limits_\Omega \left\langle \nabla u, \nabla v \right\rangle \,\mathrm{d}x + \int\limits_{\partial \Omega} u \cdot v \,\mathrm{d}s(x) &= \int\limits_\Omega f \cdot v \,\mathrm{d}x \qquad &\forall v \in V \\[1.5ex]
f(x) &\in [C_1, C_2] \qquad & \forall x \in \Omega
\end{array}
\end{array}
</math>

= Parameters =

For testing purposes, we consider <math>\Omega = B(0, 1) = \left\lbrace x \in \mathbb{R}^n \colon \|x\| < 1 \right\rbrace</math> with <math>n = 2</math>. The reference function is given by:

:<math>
u_d(x) = \|x\|^2
</math>

Additionally, let

:<math>
(C_1, C_2) := (-10, 10).
</math>

= Reference solution =

The reference solution was generated using finite element discretizations. The mesh was generated using FEniCS' mesh generation component with a resolution setting of <math>100</math>. The function space was constructed using first-degree Lagrangian elements. The optimization problem was solved using CasADi's IPOPT interface. The exact code used to solve the problem alongside solution data can be found under [[Controlled Heating (FEniCS/Casadi)]]. The optimal objective function value is <math>0.0145336</math>.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
Image:controlled_heating_solution_3d.png| False color 3D surface plot of the solution.
Image:controlled_heating_solution_color.png| False color 2D plot of the solution.
Image:controlled_heating_control_3d.png| False color 3D surface plot of the control function.
Image:controlled_heating_control_color.png| False color 2D plot of the control function.
Image:controlled_heating_error_3d.png| False color 3D surface plot of the error function.
Image:controlled_heating_error_color.png| False color 2D plot of the error function.
</gallery>

= Source Code =

Model descriptions are available in

* [[:Category: Casadi | Casadi code]] using FEniCS at [[Controlled Heating (FEniCS/Casadi)]]

[[Category:PDE model]]

Controlled Heating (Casadi/FEniCS) - Solution Data

2016-01-12T21:54:22Z

MirkoHahn: Improved table layout

The following is a tabular version of the solution data generated by the code at [[Controlled Heating (FEniCS/Casadi)]]

<pre>
# x y u f e
0.980785 0.19509 0.756845 9.99192 0.243155
0.92388 0.382683 0.756886 9.99302 0.243114
0.83147 0.55557 0.756711 9.98685 0.243289
0.707107 0.707107 0.756764 9.98888 0.243236
0.55557 0.83147 0.756912 9.99096 0.243088
0.382683 0.92388 0.756901 9.99209 0.243099
0.19509 0.980785 0.756927 9.9924 0.243073
6.12323e-17 1 0.756793 9.98999 0.243207
-0.19509 0.980785 0.756857 9.99214 0.243143
-0.382683 0.92388 0.756898 9.99211 0.243102
-0.55557 0.83147 0.756778 9.98712 0.243222
-0.707107 0.707107 0.756965 9.99269 0.243035
-0.83147 0.55557 0.756855 9.99179 0.243145
-0.92388 0.382683 0.756841 9.99211 0.243159
-0.980785 0.19509 0.756848 9.99273 0.243152
-1 1.22465e-16 0.75689 9.99254 0.24311
-0.980785 -0.19509 0.756889 9.98991 0.243111
-0.92388 -0.382683 0.756941 9.99253 0.243059
-0.83147 -0.55557 0.756764 9.98554 0.243236
-0.707107 -0.707107 0.756888 9.99367 0.243112
-0.55557 -0.83147 0.756919 9.99087 0.243081
-0.382683 -0.92388 0.75694 9.99269 0.24306
-0.19509 -0.980785 0.756896 9.99196 0.243104
-1.83697e-16 -1 0.756796 9.98754 0.243204
0.19509 -0.980785 0.756948 9.99201 0.243052
0.382683 -0.92388 0.756886 9.99106 0.243114
0.55557 -0.83147 0.756931 9.99268 0.243069
0.707107 -0.707107 0.756925 9.99298 0.243075
0.83147 -0.55557 0.756912 9.98827 0.243088
0.92388 -0.382683 0.756796 9.98786 0.243204
0.980785 -0.19509 0.756869 9.989 0.243131
1 0 0.756976 9.99291 0.243024
2.77556e-17 0 -6.25001e-05 -4.50376 6.25001e-05
-0.5 0.0492457 0.250264 -1.49361 0.00216095
-0.236839 -0.443094 0.250129 -1.52124 0.00229589
-0.145845 0.480785 0.250193 -1.19344 0.0022317
0.263161 -0.425506 0.248464 -0.762101 0.0018453
0.388375 0.318731 0.250064 -1.84646 0.00236129
0.551688 -0.0913973 0.305251 -8.91502 0.00746203
-0.570885 -0.305145 0.431957 -9.91018 -0.0129334
-0.500385 0.410655 0.431986 -9.79375 -0.012963
0.187907 0.619447 0.432089 -9.76917 -0.0130656
0.0211771 -0.66218 0.459375 -9.8436 -0.0204441
0.671421 0.19386 0.531364 -9.83766 -0.0429759
0.568581 -0.397621 0.521273 -9.91333 -0.0398868
0.290396 0.0421361 0.0860976 -4.0558 7.35968e-06
-0.319998 -0.171043 0.131594 -4.67665 6.046e-05
0.105328 0.347218 0.131644 -4.44657 1.02592e-05
-0.280481 0.230185 0.131695 -3.3357 -4.02993e-05
-0.732454 -0.0885431 0.615336 9.70953 -0.071007
-0.0560529 0.735654 0.615257 9.7814 -0.0709281
-0.480551 -0.559821 0.615157 9.75243 -0.0708282
0.471973 0.575101 0.62782 9.82353 -0.0743208
-0.362101 0.642815 0.615372 9.75036 -0.0710434
-0.71194 0.215965 0.627656 9.78072 -0.0741573
0.280174 -0.684945 0.619952 9.81955 -0.0723044
-0.23754 -0.705395 0.628148 9.84237 -0.0741404
0.00805534 -0.289149 0.0836196 -3.22912 5.25803e-05
0.73969 -0.234604 0.683088 9.94616 -0.0809075
0.225735 -0.188712 0.0865704 -3.72925 -1.67002e-06
0.495372 -0.612545 0.70044 9.96192 -0.0798354
0.772827 -0.00965354 0.678557 9.95094 -0.0812027
0.590876 0.394514 0.556664 -9.64157 -0.0518887
0.486612 0.103674 0.245958 -0.911672 0.00158129
-0.0633382 0.208798 0.0475888 -4.27442 1.95261e-05
-0.217143 0.0213867 0.0474684 -5.46371 0.000139992
0.404031 -0.281887 0.241459 -1.17691 0.0012426
-0.081288 -0.825331 0.748775 9.98672 -0.0609971
-0.127264 -0.150358 0.0387711 -4.24722 3.25719e-05
-0.240554 0.793002 0.748108 9.98094 -0.06139
0.0734623 -0.473595 0.229496 -1.04531 0.000192834
0.139163 -0.804545 0.736059 9.97778 -0.0693995
0.124575 0.151795 0.0384553 -5.18816 0.000105413
-0.326839 0.464548 0.315357 -9.0663 0.00727149
-0.411188 -0.391859 0.315483 -9.20797 0.00714536
0.0136677 0.56784 0.315248 -9.59643 0.00738071
-0.655049 -0.478357 0.730066 9.97908 -0.0721513
-0.75168 -0.324351 0.738445 9.9781 -0.0682197
-0.554262 -0.124185 0.315419 -9.00137 0.00720947
0.31246 0.485301 0.326624 -9.09247 0.00652387
-0.54914 0.596959 0.730154 9.9792 -0.0722398
0.12494 0.801439 0.730168 9.97578 -0.0722536
0.350638 0.729857 0.728571 9.98038 -0.0729332
-0.697031 0.41205 0.728374 9.97463 -0.0727365
-0.529419 0.229919 0.326585 -9.52655 0.00656239
-0.836804 0.082418 0.75883 9.98664 -0.0517969
-0.402594 -0.7532 0.768524 9.99073 -0.0391328
0.379965 -0.11076 0.156761 -2.82206 -0.000119512
0.652078 -0.545993 0.766097 9.98954 -0.0427832
-0.12643 -0.574541 0.340856 -9.6867 0.00522499
0.749536 -0.405394 0.767195 9.98654 -0.0410471
0.664251 0.545136 0.77168 9.9892 -0.0332771
0.755801 0.344224 0.749868 9.9847 -0.0601417
-0.187363 0.638664 0.465037 -9.87491 -0.0220403
-0.0773876 -0.423268 0.185397 -4.34905 -0.000251641
-0.210162 -0.286204 0.125995 -5.05136 8.54533e-05
0.239687 0.261792 0.125745 -5.23442 0.000240303
0.411586 -0.770023 0.779021 9.99081 -0.0166831
0.528305 0.252685 0.337443 -9.36265 0.00551282
0.826304 0.144013 0.757014 9.98524 -0.053496
0.629251 0.045827 0.404132 -9.75431 -0.00607519
0.55302 -0.244902 0.363778 -9.69106 0.00202993
0.41816 -0.436642 0.363473 -9.60227 0.00204097
-0.663286 0.059742 0.465847 -9.85792 -0.0223294
0.196352 -0.560165 0.348019 -9.61734 0.00431906
0.0724796 -0.142668 0.0255369 -4.7523 7.04514e-05
0.147374 0.00606884 0.0217047 -4.787 5.12928e-05
0.357048 0.173204 0.157464 -4.59386 1.90437e-05
0.141934 -0.345494 0.13958 -2.26469 -6.88225e-05
-0.68868 0.565185 0.785504 9.99157 0.0082109
0.262654 0.844046 0.783498 9.99226 -0.00209685
-0.421698 -0.0704574 0.183143 -3.85949 -0.000349848
-0.452563 -0.22477 0.252677 -2.14015 0.00265777
0.168632 0.476655 0.252873 -2.03437 0.00276352
-0.0479211 0.361982 0.133286 -4.12658 4.20513e-05
-0.380849 0.332098 0.252686 -2.15002 0.00264909
-0.404873 0.157467 0.189046 -4.39454 -0.000327995
-0.226471 0.362635 0.183136 -4.72806 -0.000343372
0.338789 -0.555026 0.436896 -9.88863 -0.014064
-0.323944 -0.574013 0.453276 -9.82967 -0.0188459
-0.381373 0.78269 0.777921 9.99136 -0.0198719
-0.561138 -0.683749 0.78362 9.99421 -0.0012311
-0.117739 0.862662 0.777888 9.99213 -0.0198396
-0.880268 -0.0866988 0.783541 9.99165 -0.00115236
-0.219446 -0.843573 0.778424 9.99138 -0.0186522
0.889933 -0.0876507 0.78631 9.9917 0.0133528
0.27926 -0.826794 0.778853 9.98983 -0.0172789
-0.663172 -0.205424 0.522092 -9.86441 -0.0400954
-0.807777 -0.20109 0.75166 9.98543 -0.0587195
0.44111 0.44335 0.39519 -9.84535 -0.00405308
-0.847281 0.217361 0.779761 9.99077 -0.0146298
0.582646 0.653383 0.780052 9.98901 -0.0136658
0.0745801 0.682126 0.505704 -9.72299 -0.0348458
-0.434999 0.535775 0.513584 -9.8667 -0.0373057
-0.535072 -0.4296 0.505259 -9.85919 -0.0344006
0.952332 0.288887 0.759302 9.99253 0.231091
-0.110049 -0.70105 0.555067 -9.69706 -0.0514853
0.154026 -0.679141 0.526899 -9.72737 -0.0419414
-0.494029 0.715814 0.777534 9.99194 -0.02108
0.00678047 0.844655 0.76185 9.98731 -0.0483616
0.350303 0.604651 0.531323 -9.85842 -0.0430074
-0.599258 0.333839 0.505024 -9.856 -0.0344655
-0.802818 0.338368 0.778069 9.98946 -0.0190593
-0.774361 -0.445901 0.786245 9.99105 0.0122171
0.952332 -0.288887 0.759397 9.99125 0.230996
0.858805 -0.206106 0.783102 9.99258 -0.00307634
0.478866 0.728706 0.778558 9.98975 -0.0182321
-0.597011 0.4849 0.672253 9.92414 -0.0807018
-0.179055 0.146946 0.0537598 -3.2458 -0.000105999
0.0348397 -0.877537 0.781257 9.99176 -0.00997248
0.265225 -0.0752935 0.0760448 -4.45478 -3.15915e-05
0.255432 -0.305373 0.158581 -5.1398 -8.22633e-05
0.408634 0.00871913 0.167221 -3.95355 -0.000162714
-0.334103 -0.294455 0.198816 -2.3139 -0.000487338
0.0509876 0.453993 0.209073 -2.56043 -0.000363435
0.232067 0.729488 0.666442 9.93376 -0.0804339
0.684837 -0.111608 0.521333 -9.69139 -0.0398752
0.662356 -0.324741 0.615217 9.68373 -0.071045
0.283002 0.371291 0.218133 -2.07716 -0.000185919
-0.769288 -0.631339 0.759316 9.99116 0.231076
-0.649402 -0.599941 0.78349 9.9939 -0.00183789
0.0975452 0.990393 0.759458 9.99354 0.230935
0.604673 -0.652139 0.785168 9.99307 0.00574566
-0.347799 0.038671 0.122454 -3.64311 6.40497e-06
-0.104142 0.0556651 0.0138508 -5.10539 9.33295e-05
0.884611 0.0375834 0.783986 9.99355 -3.70514e-05
0.82202 -0.315537 0.782115 9.98964 -0.00683351
0.514656 -0.499183 0.571416 2.90255 -0.0573619
0.39088 -0.657978 0.665807 9.90194 -0.0800855
0.516664 -0.724429 0.785187 9.9917 0.00655245
0.245493 0.147451 0.082124 -4.28781 -0.000115057
0.877675 0.469127 0.7594 9.9917 0.230993
0.753225 0.464365 0.783581 9.99192 -0.000596975
-0.166564 0.262748 0.0966931 -4.12225 8.70912e-05
-0.0269128 -0.523218 0.269646 -4.38259 0.00483554
-0.242365 0.539801 0.345497 -9.47007 0.00462873
0.72944 0.0915842 0.609993 9.7237 -0.0695232
-0.877675 0.469127 0.759278 9.98861 0.231115
0.459519 -0.186828 0.244526 -0.543815 0.00153561
-0.100677 -0.259799 0.0776909 -4.5674 -5.96211e-05
0.0496625 0.243076 0.0614653 -5.00611 8.71874e-05
0.0115744 0.117517 0.0138843 -4.14672 5.98208e-05
-0.230673 -0.0950845 0.0621791 -5.2334 7.21425e-05
0.469127 0.877675 0.759429 9.99202 0.230964
-0.615882 0.160193 0.413179 -9.86112 -0.00820689
0.628589 0.29316 0.520919 -9.84016 -0.0398524
0.539127 0.489879 0.595639 9.42943 -0.0649989
0.0130212 -0.770809 0.674994 9.94469 -0.0806775
-0.952332 -0.288887 0.759415 9.99191 0.230977
0.127659 -0.237077 0.072429 -3.51179 7.35121e-05
-0.586363 -0.0189373 0.338719 -9.56484 0.00546169
-0.631339 0.769288 0.75935 9.99141 0.231042
-0.0780909 0.628108 0.407378 -9.81972 -0.0067603
0.838362 0.260778 0.781083 9.99123 -0.0102277
0.494299 0.352404 0.367089 -9.72653 0.00143116
-0.0259606 -0.183698 0.0342943 -5.24391 0.000124487
-0.409183 -0.643453 0.661428 9.90878 -0.0799648
-0.135259 -0.0451516 0.0204303 -4.32698 -9.66428e-05
0.790324 -0.124793 0.716891 9.96906 -0.0767056
0.511633 0.00363101 0.258382 -2.37984 0.00339964
0.583895 0.138771 0.357073 -9.71233 0.00311741
-0.304112 -0.784386 0.759129 9.98625 -0.0513841
0.164181 -0.0956187 0.036167 -3.73752 -6.84969e-05
-0.852575 -0.30984 0.788603 9.99462 0.0342825
-0.28314 0.123531 0.0953992 -4.43294 2.90427e-05
0.0905347 -0.579869 0.338982 -9.5435 0.00546249
-0.395866 -0.49649 0.410746 -9.81568 -0.00753473
0.33075 -0.204937 0.15125 -5.0735 0.000144922
0.345389 -0.365263 0.250415 -1.84008 0.00229579
0.476803 -0.353265 0.347765 -9.57571 0.00437235
-0.805465 -0.0157802 0.723813 9.97798 -0.0747901
-0.0482323 0.461484 0.215576 -2.23485 -0.000281889
-0.225109 -0.545335 0.343012 -9.64676 0.00505203
0.3599 0.827985 0.788078 9.99403 0.0270094
-0.755473 0.492032 0.787799 9.99321 0.0250359
-0.695817 0.313348 0.662182 9.91715 -0.0798332
0.990393 -0.0975452 0.759546 9.9942 0.230847
0.769288 -0.631339 0.759389 9.99162 0.231003
0.0975452 -0.990393 0.759376 9.99003 0.231017
-0.990393 0.0975452 0.759428 9.99334 0.230965
0.288887 -0.952332 0.759556 9.99458 0.230836
-0.0975452 -0.990393 0.759329 9.99177 0.231063
0.631339 -0.769288 0.759296 9.98811 0.231097
-0.990393 -0.0975452 0.759442 9.99404 0.230951
-0.631339 -0.769288 0.75947 9.9942 0.230923
-0.288887 -0.952332 0.759403 9.99142 0.23099
-0.769288 0.631339 0.759422 9.99286 0.230971
0.631339 0.769288 0.759466 9.99321 0.230927
-0.288887 0.952332 0.759385 9.99323 0.231008
0.288887 0.952332 0.759398 9.9896 0.230994
-0.0975452 0.990393 0.759246 9.98717 0.231147
0.990393 0.0975452 0.759414 9.99169 0.230979
0.769288 0.631339 0.759329 9.99091 0.231064
-0.877675 -0.469127 0.75931 9.98906 0.231082
-0.952332 0.288887 0.759388 9.99258 0.231004
-0.469127 0.877675 0.759257 9.98775 0.231135
-0.469127 -0.877675 0.759408 9.99208 0.230985
0.469127 -0.877675 0.75931 9.99102 0.231083
0.877675 -0.469127 0.759491 9.99245 0.230901
0.166081 -0.440751 0.221921 -2.38245 -7.65499e-05
0.0493148 -0.378338 0.145436 -5.36972 0.000135814
-0.273435 0.695335 0.633572 9.86261 -0.0753148
-0.217822 0.886894 0.788999 9.99386 0.0450275
-0.146471 0.770223 0.695007 9.95704 -0.0803086
-0.315801 0.853899 0.788958 9.99513 0.0399157
-0.141642 -0.90053 0.788948 9.99439 0.0420686
-0.322958 -0.0577357 0.107417 -5.4943 0.000217959
-0.734849 -0.53618 0.788831 9.99416 0.0386608
-0.655287 -0.371188 0.645058 9.88634 -0.0778769
-0.61452 0.671542 0.788934 9.99472 0.0396697
-0.568228 -0.520006 0.673956 9.94733 -0.0806668
0.137866 0.899776 0.788925 9.99292 0.0396788
-0.225874 -0.19123 0.0876088 -4.17818 -2.10046e-05
0.145382 0.245252 0.0812461 -5.15099 3.81914e-05
-0.137235 -0.348401 0.140128 -4.75143 8.84318e-05
0.138074 -0.901885 0.789018 9.99468 0.0434427
-0.459297 0.627505 0.685733 9.95465 -0.0810168
0.433415 0.2329 0.24094 -1.25988 0.00115089
0.644349 -0.203395 0.48451 -9.85978 -0.0279546
-0.046066 -0.087173 0.00972763 -3.66338 -6.41993e-06
0.218771 -0.755476 0.698844 9.96167 -0.0802386
0.427157 -0.528473 0.492052 -9.90883 -0.0303052
0.264914 0.566051 0.394445 -9.85172 -0.00385091
0.39531 0.524705 0.449298 -9.86389 -0.0177126
-0.496941 0.142476 0.263331 -3.68098 0.00391783
-0.508676 0.319287 0.357667 -9.67328 0.00302872
-0.918402 0.041209 0.789091 9.99418 0.0560691
-0.903835 0.145091 0.789144 9.99467 0.0488263
-0.78025 0.154687 0.710776 9.96598 -0.0780579
-0.330351 -0.693791 0.671294 9.9309 -0.0808157
-0.408312 0.420534 0.338113 -9.39755 0.00545442
-0.640691 -0.0930267 0.431984 -9.86673 -0.0128454
-0.49171 -0.349742 0.361702 -9.79495 0.00239515
-0.332581 0.556033 0.433119 -9.78714 -0.0133355
0.100387 0.594996 0.361752 -9.68738 0.00234543
-0.572184 -0.213782 0.372651 -9.78051 0.00044589
-0.31547 -0.388405 0.248396 -1.05724 0.00198436
0.657171 -0.445447 0.708971 9.96799 -0.0786749
-0.434124 0.250459 0.249174 -1.61805 0.00201938
0.745293 0.251263 0.698532 9.96325 -0.0799361
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-0.792515 -0.283591 0.759537 9.98004 -0.0510328
0.120512 0.949315 0.78137 9.99002 0.134351
0.611797 0.72008 0.785324 9.99309 0.107487
0.640125 -0.629268 0.787115 9.98864 0.0186229
0.638807 -0.60861 0.782787 9.98855 -0.00430651
0.647205 -0.61453 0.785939 9.98864 0.0105815
-0.410162 -0.704745 0.734861 9.95922 -0.0699623
-0.293285 -0.060874 0.0898688 -2.25071 -0.00014695
0.0465013 -0.562607 0.311337 -8.44983 0.00735223
0.372197 -0.610531 0.56639 -6.50026 -0.0551112
-0.80963 -0.455111 0.7885 9.99364 0.0741271
-0.0548653 0.902839 0.78817 9.98971 0.0299587
-0.0838826 -0.800127 0.722108 9.9549 -0.0748692
0.0895831 -0.843946 0.764742 9.98016 -0.0444717
0.402759 -0.467554 0.38221 -9.27849 -0.00138944
-0.201522 -0.403326 0.203818 -2.3558 -0.000534803
-0.597908 0.344465 0.513302 -9.83479 -0.0371515
-0.587311 0.361468 0.512494 -9.72552 -0.036901
-0.583234 0.344014 0.487393 -9.76138 -0.0288848
-0.759264 0.109682 0.668812 9.89957 -0.0803006
-0.299528 -0.846084 0.787012 9.9855 0.0185626
0.7431 -0.335695 0.734782 9.9615 -0.0698934
-0.374848 0.062845 0.144466 -4.40606 -5.228e-06
-0.396345 0.594127 0.565121 -9.10714 -0.0550447
0.512215 -0.705608 0.778496 9.98649 -0.0182492
-0.044644 -0.322773 0.106301 -1.59985 -0.000125831
-0.0448919 -0.306806 0.0962753 -2.33069 -0.000130096
-0.00862618 0.5368 0.281948 -5.83542 0.00628096
-0.201353 0.677242 0.547988 -9.49093 -0.0487877
-0.830025 -0.26171 0.777801 9.98686 -0.0203679
0.735841 0.247716 0.683595 9.92145 -0.0807698
0.731983 0.232677 0.670304 9.88813 -0.0803663
0.718116 0.245486 0.654911 9.8742 -0.078957
0.501982 0.214188 0.29083 -7.13561 0.00703228
-0.484399 0.713022 0.77334 9.98433 -0.030298
-0.471826 0.720971 0.77304 9.97769 -0.0306215
0.880641 0.30251 0.788072 9.99318 0.078969
0.226507 -0.674236 0.558963 -9.1397 -0.0530627
-0.578414 0.298388 0.438125 -9.71607 -0.0145271
0.0496608 0.0648212 0.00674004 -2.17446 -7.20622e-05
0.06699 0.0754899 0.0102978 -1.52315 -0.000111446
0.460766 -0.581904 0.624154 9.65985 -0.0732367
0.686981 -0.586964 0.788088 9.99135 0.0283815
0.0971055 0.0723527 0.0147423 -2.49894 -7.79191e-05
-0.756628 -0.543994 0.788063 9.99147 0.0803527
-0.76533 -0.536652 0.787624 9.99125 0.086101
0.0613195 -0.333422 0.115023 -2.75902 -9.20364e-05
0.483933 0.592978 0.666233 9.85663 -0.0804183
0.40248 -0.291948 0.245549 -0.357389 0.00167482
0.407295 -0.304867 0.255764 -1.65739 0.00306926
-0.458902 -0.0108359 0.211109 -1.77591 -0.000400129
-0.45992 -0.0279414 0.212743 -0.9366 -0.000435514
0.570107 0.0353292 0.319246 -8.41372 0.00702368
-0.843593 0.327004 0.78818 9.99122 0.0304018
-0.520522 0.383962 0.431005 -9.72649 -0.0126345
-0.113949 0.966333 0.77385 9.99367 0.172934
-0.100382 -0.825035 0.750367 9.97666 -0.0596077
0.853314 0.390899 0.786915 9.99234 0.0940327
0.379958 0.625992 0.603846 9.4932 -0.0676122
-0.443393 -0.481039 0.444034 -9.77395 -0.0160382
-0.452606 -0.470922 0.44218 -9.76218 -0.0155597
-0.46369 -0.478205 0.465948 -9.7573 -0.0222592
0.187698 -0.791677 0.73289 9.95494 -0.0709062
0.4315 -0.351876 0.302605 -7.82559 0.00740365
-0.383573 -0.771335 0.773037 9.98718 -0.0309516
-0.346474 -0.516347 0.389638 -9.55829 -0.00297965
0.0196503 0.233367 0.0550371 -1.95061 -0.000190711
0.0339948 0.212754 0.0465266 -2.18895 -0.000106755
0.459993 0.290639 0.289322 -6.31151 0.00674247
-0.577826 -0.465155 0.623007 9.7674 -0.0727542
-0.760632 0.392594 0.769622 9.98321 -0.0369296
-0.973515 -0.188219 0.762963 9.99238 0.220195
-0.974716 -0.176025 0.763578 9.99211 0.217478
0.735083 0.632025 0.775721 9.99361 0.164083
0.860089 0.328355 0.788948 9.99112 0.0586229
-0.862892 -0.148198 0.780042 9.9868 -0.0134967
0.450069 0.025593 0.203677 -1.71806 -0.000460365
-0.137567 -0.00352233 0.0189149 -1.62584 2.23058e-05
0.311551 -0.61901 0.519357 -9.53088 -0.0391199
-0.750449 -0.352243 0.748626 9.97513 -0.0613773
-0.75793 -0.359663 0.757323 9.98036 -0.0535086
-0.0620439 -0.983771 0.766228 9.99308 0.205426
0.00414652 -0.803847 0.72137 9.9576 -0.0751833
0.262355 -0.792839 0.754066 9.98011 -0.0566423
-0.466727 0.187538 0.250848 -1.15506 0.00215551
-0.485535 0.183967 0.265386 -2.81822 0.00420288
-0.0479956 -0.09725 0.0117788 -1.43402 -1.76398e-05
0.900267 -0.188528 0.789095 9.99082 0.056928
0.777325 0.604834 0.76664 9.99364 0.203419
</pre>

Source Inversion (FEniCS/Casadi)

2016-01-12T21:42:26Z

MirkoHahn: /* Solution */

This page contains a discretized version of the mixed integer PDE constrained optimization problem [[Source Inversion]] in [http://www.casadi.org CasADi 2.4.3] format. The weak version of the problem was discretized using finite element methods (via [http://www.fenicsproject.org FEniCS]) using first-degree Lagrangian elements on an equidistant mesh with <math>128</math> cells.

=== CasADi ===

<source lang="Python">
from dolfin import *

import math
import numpy
import scipy, scipy.sparse
import casadi

# Simple Branch and Bound solver
def solve_bnb(solver, root, integers):
# Initialize the node queue
Q = [(numpy.inf, root)]

# Global solution data
x_opt = None
f_opt = numpy.inf

# Continuous solution data
x0 = None
f0 = numpy.inf

# Main loop
seqnum = 0
while Q:
seqnum += 1

# Extract node with lowest lower bound and check if fathoming is necessary
# (should not happen)
node = Q.pop()
if f_opt < numpy.inf and node[0] >= f_opt:
print "Node %4d (%4d left): %9g >= %9g - PRE-FATHOM" % (seqnum, len(Q), node[0], f_opt)
continue
node = node[1]

# Solve the node
result = solver(node)

# Local solution data
x = result['x']
f = result['f'].get()[0]

# Store continuous solution data if none has been stored yet
if x0 is None:
x0 = casadi.DMatrix(x)
f0 = f

# Check if branch can be fathomed
if f >= f_opt:
print "Node %4d (%4d left): %9g >= %9g - POST-FATHOM" % (seqnum, len(Q), f, f_opt)
continue

# Check for violations of integrality (fixed tolerance 1e-5)
viol = [abs(casadi.floor(x[i] + 0.5) - x[i]) for i in integers]
idx = [(integers[i], viol[i]) for i in range(len(integers)) if viol[i] > 1e-5]
if not idx:
# Register new global solution
x_opt = x
f_opt = f

# Cull the branch and bound tree
pre = len(Q)
Q = [n for n in Q if n[0] < f_opt]
post = len(Q)

print "Node %4d (%4d left): f_opt = %9g - *** SOLUTION *** (%d culled)" % (seqnum, len(Q), f, pre - post)
else:
# Branch on first violation
idx = idx[0][0]

# Generate two new nodes (could reuse node structure)
ln = {
'x0': casadi.DMatrix(x),
'lbx': node['lbx'],
'ubx': casadi.DMatrix(node['ubx']),
'lbg': node['lbg'],
'ubg': node['ubg'],
'lam_x0': casadi.DMatrix(result['lam_x']),
'lam_g0': casadi.DMatrix(result['lam_g'])
}
un = {
'x0': casadi.DMatrix(x),
'lbx': casadi.DMatrix(node['lbx']),
'ubx': node['ubx'],
'lbg': node['lbg'],
'ubg': node['ubg'],
'lam_x0': casadi.DMatrix(result['lam_x']),
'lam_g0': casadi.DMatrix(result['lam_g'])
}

ln['ubx'][idx] = casadi.floor(x[idx])
un['lbx'][idx] = casadi.ceil(x[idx])

lower_node = (f, ln)
upper_node = (f, un)

# Insert new nodes in queue (inefficient for large queues)
Q.extend([lower_node, upper_node])
Q.sort(cmp=lambda x,y: cmp(y[0], x[0]))
print "Node %4d (%4d left): %9g - BRANCH ON %d" % (seqnum, len(Q), f, idx)

return {
'x0': x0,
'f0': f0,
'x': x_opt,
'f': f_opt
}

# Converts FEniCS matrix to scipy.sparse.coo_matrix
def matrix_to_coo(A, shape=None):
nnz = A.nnz()
col = numpy.empty(nnz, dtype='uint64')
row = numpy.empty(nnz, dtype='uint64')
val = numpy.empty(nnz)

if shape is None:
shape = (A.size(0), A.size(1))

it = 0
for i in range(A.size(0)):
col_local, val_local = A.getrow(i)
nnz_local = col_local.shape[0]

col[it:it+nnz_local] = col_local[:]
row[it:it+nnz_local] = i
val[it:it+nnz_local] = val_local[:]
it += nnz_local

return scipy.sparse.coo_matrix((val, (row, col)), shape=shape)

# Returns FEniCS expression for elementary source term
def source_term(x0):
return Expression("100.0 * exp(-pow(x[0] - x0, 2) / 0.02)", x0=x0)

# Grid resolutions
nx = 128
nu = 8

# Generate mesh and function space
mesh = UnitIntervalMesh(nx)
V = FunctionSpace(mesh, 'CG', 1)

# Define the bilinear form and assemble it
u = TrialFunction(V)
v = TestFunction(V)
a = inner(grad(u), grad(v)) * dx + u * v * ds
A = assemble(a)

# Create the linear form for the exact source term and perform assembly
f = source_term(3.0 / math.pi) + source_term(0.5)
L = f * v * dx
b = assemble(L)

# Solve for the reference solution
u_ref = Function(V)
solve(A, u_ref.vector(), b)

# Define the control grid and the elementary source terms
ctrl_grid = numpy.linspace(0.0, 1.0, num=nu+1)[:-1]
ctrl_grid += 0.5 * ctrl_grid[1]
src_expr = [source_term(x0) for x0 in ctrl_grid]

# Assemble the linear forms corresponding to the elementary source terms and
# construct the second half of the coefficient matrix
b = [assemble(-f * v * dx) for f in src_expr]
B = scipy.column_stack([v.array() for v in b])

# Construct the full coefficient matrix
C = scipy.sparse.vstack([
scipy.sparse.hstack([matrix_to_coo(A), B]),
scipy.sparse.coo_matrix(([1.0]*nu, ([0]*nu, [V.dim() + i for i in range(nu)])), shape=(1, V.dim() + nu))
])

# Transfer the optimization constraints to CasADi
C = casadi.DMatrix(C)
lb = casadi.DMatrix.zeros(C.shape[0], 1)
ub = casadi.DMatrix.zeros(C.shape[0], 1)

lb[-1] = 0.0
ub[-1] = 5.0

lbx = numpy.array([-numpy.inf]*V.dim() + [0.0]*nu)
ubx = numpy.array([ numpy.inf]*V.dim() + [1.0]*nu)

# Define the objective functional and assemble it
u = Function(V)
J = (u_ref - u)**2 * dx
dJdu = derivative(J, u)
dJdu2 = derivative(dJdu, u)

H = casadi.DMatrix(matrix_to_coo(assemble(dJdu2), shape=(V.dim() + nu, V.dim() + nu)))
g = casadi.DMatrix(numpy.concatenate((assemble(dJdu).array(), [0.0]*nu)))
c = assemble(J)

# Build the CasADi NLP
x = casadi.MX.sym('X', V.dim() + nu)
nlp = casadi.MXFunction('nlp',
casadi.nlpIn(x=x),
casadi.nlpOut(
f=0.5*casadi.quad_form(x, H) + casadi.inner_prod(g, x) + c,
g=casadi.mul(C, x)
))

# Build NLP solver
solver = casadi.NlpSolver('solver', 'ipopt', nlp, {
'print_level': 0,
'print_time': False
})
result = solve_bnb(solver, {
'lbx': lbx,
'ubx': ubx,
'lbg': lb,
'ubg': ub
}, range(V.dim(), V.dim() + nu))

# Evaluate result
print ""
print "----------------------------"
print " f = %g" % result['f']

u.vector()[:] = result['x'][:V.dim()].toArray()[:,0]
</source>

== Results ==

Node solutions are calculated using IPOPT (3.12, default settings, using linear solver MUMPS on Ubuntu 15.10 for x86-64 with Linux 4.2.0-19-generic, Intel(R) Core(TM) i7-4790K CPU @ 4.00GHz, 16 GB RAM). The objective function value is <math>0.858837</math>. The integer solution is found after processing 25 nodes.

== Solution ==

The following is a Gnuplot compatible tabular version of the control data determined by the solver:

<pre>
# x w w0
0.0625 0 2.91549e-08
0.1875 0 2.75108e-08
0.3125 0 6.746e-08
0.4375 1 0.46917
0.5625 0 0.543817
0.6875 0 1.33205e-07
0.8125 0 1.51069e-07
0.9375 1 0.897707
</pre>

The following is a Gnuplot compatible tabular version of the solution functions:

<pre>
# x u u0 u_d
0 19.845 18.7521 18.7182
0.0078125 20.0001 18.8986 18.8645
0.015625 20.1551 19.0451 19.0107
0.0234375 20.3101 19.1916 19.1569
0.03125 20.4652 19.3381 19.3032
0.0390625 20.6202 19.4846 19.4494
0.046875 20.7752 19.6311 19.5956
0.0546875 20.9303 19.7776 19.7419
0.0625 21.0853 19.9241 19.8881
0.0703125 21.2403 20.0706 20.0343
0.078125 21.3953 20.2171 20.1806
0.0859375 21.5503 20.3635 20.3268
0.09375 21.7053 20.51 20.473
0.101562 21.8603 20.6565 20.6193
0.109375 22.0153 20.8029 20.7655
0.117188 22.1702 20.9494 20.9117
0.125 22.3251 21.0958 21.0579
0.132812 22.4799 21.2422 21.2042
0.140625 22.6347 21.3886 21.3504
0.148438 22.7894 21.5349 21.4966
0.15625 22.944 21.6812 21.6428
0.164062 23.0985 21.8275 21.7889
0.171875 23.2528 21.9737 21.9351
0.179688 23.407 22.1197 22.0812
0.1875 23.561 22.2657 22.2273
0.195312 23.7146 22.4116 22.3733
0.203125 23.868 22.5573 22.5193
0.210938 24.0209 22.7028 22.6652
0.21875 24.1734 22.848 22.811
0.226562 24.3254 22.993 22.9567
0.234375 24.4766 23.1377 23.1022
0.242188 24.6271 23.282 23.2476
0.25 24.7767 23.4259 23.3927
0.257812 24.9252 23.5692 23.5376
0.265625 25.0725 23.7119 23.6822
0.273438 25.2184 23.854 23.8263
0.28125 25.3627 23.9952 23.97
0.289062 25.5052 24.1355 24.1131
0.296875 25.6457 24.2748 24.2556
0.304688 25.7839 24.413 24.3972
0.3125 25.9196 24.5498 24.538
0.320312 26.0525 24.6852 24.6777
0.328125 26.1823 24.8189 24.8162
0.335938 26.3088 24.9509 24.9533
0.34375 26.4316 25.0809 25.0889
0.351562 26.5505 25.2087 25.2226
0.359375 26.6651 25.3342 25.3542
0.367188 26.7753 25.4572 25.4836
0.375 26.8807 25.5774 25.6105
0.382812 26.9811 25.6947 25.7346
0.390625 27.0762 25.8089 25.8556
0.398438 27.1659 25.9198 25.9733
0.40625 27.2499 26.0271 26.0873
0.414062 27.3281 26.1308 26.1973
0.421875 27.4004 26.2305 26.3032
0.429688 27.4666 26.3262 26.4046
0.4375 27.5268 26.4177 26.5012
0.445312 27.5809 26.5047 26.5927
0.453125 27.6289 26.5873 26.6791
0.460938 27.6708 26.6652 26.7599
0.46875 27.7069 26.7384 26.8351
0.476562 27.7371 26.8066 26.9045
0.484375 27.7617 26.87 26.968
0.492188 27.7808 26.9283 27.0255
0.5 27.7947 26.9816 27.0768
0.507812 27.8035 27.0298 27.1221
0.515625 27.8076 27.0729 27.1613
0.523438 27.8072 27.1109 27.1945
0.53125 27.8026 27.1438 27.2217
0.539062 27.794 27.1718 27.2431
0.546875 27.7818 27.1948 27.2589
0.554688 27.7662 27.213 27.2692
0.5625 27.7476 27.2264 27.2743
0.570312 27.7261 27.2352 27.2743
0.578125 27.7021 27.2395 27.2696
0.585938 27.6759 27.2394 27.2603
0.59375 27.6476 27.2351 27.2469
0.601562 27.6174 27.2269 27.2295
0.609375 27.5857 27.2147 27.2084
0.617188 27.5525 27.199 27.184
0.625 27.5181 27.1797 27.1565
0.632812 27.4826 27.1573 27.1261
0.640625 27.4461 27.1317 27.0932
0.648438 27.4087 27.1032 27.058
0.65625 27.3706 27.0721 27.0207
0.664062 27.3318 27.0385 26.9815
0.671875 27.2924 27.0025 26.9406
0.679688 27.2525 26.9644 26.8983
0.6875 27.2119 26.9242 26.8545
0.695312 27.1709 26.8821 26.8096
0.703125 27.1293 26.8383 26.7635
0.710938 27.0871 26.7928 26.7164
0.71875 27.0443 26.7457 26.6683
0.726562 27.0008 26.6971 26.6193
0.734375 26.9566 26.6469 26.5693
0.742188 26.9115 26.5953 26.5184
0.75 26.8654 26.5422 26.4666
0.757812 26.8183 26.4875 26.4137
0.765625 26.7699 26.4312 26.3598
0.773438 26.72 26.3732 26.3046
0.78125 26.6686 26.3135 26.2481
0.789062 26.6153 26.2517 26.1902
0.796875 26.56 26.188 26.1306
0.804688 26.5024 26.1219 26.0691
0.8125 26.4423 26.0534 26.0057
0.820312 26.3794 25.9822 25.9399
0.828125 26.3134 25.9082 25.8717
0.835938 26.244 25.8311 25.8007
0.84375 26.171 25.7506 25.7267
0.851562 26.094 25.6665 25.6494
0.859375 26.0129 25.5785 25.5685
0.867188 25.9272 25.4865 25.4837
0.875 25.8368 25.3902 25.3947
0.882812 25.7413 25.2893 25.3013
0.890625 25.6406 25.1837 25.2032
0.898438 25.5345 25.0732 25.1002
0.90625 25.4227 24.9576 24.9919
0.914062 25.305 24.8368 24.8782
0.921875 25.1815 24.7107 24.7589
0.929688 25.0519 24.5791 24.6338
0.9375 24.9162 24.4421 24.5029
0.945312 24.7745 24.2996 24.3659
0.953125 24.6267 24.1516 24.2228
0.960938 24.4728 23.9982 24.0737
0.96875 24.313 23.8395 23.9184
0.976562 24.1474 23.6756 23.7572
0.984375 23.9762 23.5067 23.5899
0.992188 23.7995 23.3328 23.4169
1 23.6175 23.1542 23.2381
</pre>

[[Category:Casadi]]

Source Inversion (FEniCS/Casadi)

2016-01-12T21:41:50Z

MirkoHahn: /* Solution */

Source Inversion (FEniCS/Casadi)

2016-01-12T21:41:28Z

MirkoHahn: /* Solution */

This page contains a discretized version of the mixed integer PDE constrained optimization problem [[Source Inversion]] in [http://www.casadi.org CasADi 2.4.3] format. The weak version of the problem was discretized using finite element methods (via [http://www.fenicsproject.org FEniCS]) using first-degree Lagrangian elements on an equidistant mesh with <math>128</math> cells.

=== CasADi ===

<source lang="Python">
from dolfin import *

import math
import numpy
import scipy, scipy.sparse
import casadi

# Simple Branch and Bound solver
def solve_bnb(solver, root, integers):
# Initialize the node queue
Q = [(numpy.inf, root)]

# Global solution data
x_opt = None
f_opt = numpy.inf

# Continuous solution data
x0 = None
f0 = numpy.inf

# Main loop
seqnum = 0
while Q:
seqnum += 1

# Extract node with lowest lower bound and check if fathoming is necessary
# (should not happen)
node = Q.pop()
if f_opt < numpy.inf and node[0] >= f_opt:
print "Node %4d (%4d left): %9g >= %9g - PRE-FATHOM" % (seqnum, len(Q), node[0], f_opt)
continue
node = node[1]

# Solve the node
result = solver(node)

# Local solution data
x = result['x']
f = result['f'].get()[0]

# Store continuous solution data if none has been stored yet
if x0 is None:
x0 = casadi.DMatrix(x)
f0 = f

# Check if branch can be fathomed
if f >= f_opt:
print "Node %4d (%4d left): %9g >= %9g - POST-FATHOM" % (seqnum, len(Q), f, f_opt)
continue

# Check for violations of integrality (fixed tolerance 1e-5)
viol = [abs(casadi.floor(x[i] + 0.5) - x[i]) for i in integers]
idx = [(integers[i], viol[i]) for i in range(len(integers)) if viol[i] > 1e-5]
if not idx:
# Register new global solution
x_opt = x
f_opt = f

# Cull the branch and bound tree
pre = len(Q)
Q = [n for n in Q if n[0] < f_opt]
post = len(Q)

print "Node %4d (%4d left): f_opt = %9g - *** SOLUTION *** (%d culled)" % (seqnum, len(Q), f, pre - post)
else:
# Branch on first violation
idx = idx[0][0]

# Generate two new nodes (could reuse node structure)
ln = {
'x0': casadi.DMatrix(x),
'lbx': node['lbx'],
'ubx': casadi.DMatrix(node['ubx']),
'lbg': node['lbg'],
'ubg': node['ubg'],
'lam_x0': casadi.DMatrix(result['lam_x']),
'lam_g0': casadi.DMatrix(result['lam_g'])
}
un = {
'x0': casadi.DMatrix(x),
'lbx': casadi.DMatrix(node['lbx']),
'ubx': node['ubx'],
'lbg': node['lbg'],
'ubg': node['ubg'],
'lam_x0': casadi.DMatrix(result['lam_x']),
'lam_g0': casadi.DMatrix(result['lam_g'])
}

ln['ubx'][idx] = casadi.floor(x[idx])
un['lbx'][idx] = casadi.ceil(x[idx])

lower_node = (f, ln)
upper_node = (f, un)

# Insert new nodes in queue (inefficient for large queues)
Q.extend([lower_node, upper_node])
Q.sort(cmp=lambda x,y: cmp(y[0], x[0]))
print "Node %4d (%4d left): %9g - BRANCH ON %d" % (seqnum, len(Q), f, idx)

return {
'x0': x0,
'f0': f0,
'x': x_opt,
'f': f_opt
}

# Converts FEniCS matrix to scipy.sparse.coo_matrix
def matrix_to_coo(A, shape=None):
nnz = A.nnz()
col = numpy.empty(nnz, dtype='uint64')
row = numpy.empty(nnz, dtype='uint64')
val = numpy.empty(nnz)

if shape is None:
shape = (A.size(0), A.size(1))

it = 0
for i in range(A.size(0)):
col_local, val_local = A.getrow(i)
nnz_local = col_local.shape[0]

col[it:it+nnz_local] = col_local[:]
row[it:it+nnz_local] = i
val[it:it+nnz_local] = val_local[:]
it += nnz_local

return scipy.sparse.coo_matrix((val, (row, col)), shape=shape)

# Returns FEniCS expression for elementary source term
def source_term(x0):
return Expression("100.0 * exp(-pow(x[0] - x0, 2) / 0.02)", x0=x0)

# Grid resolutions
nx = 128
nu = 8

# Generate mesh and function space
mesh = UnitIntervalMesh(nx)
V = FunctionSpace(mesh, 'CG', 1)

# Define the bilinear form and assemble it
u = TrialFunction(V)
v = TestFunction(V)
a = inner(grad(u), grad(v)) * dx + u * v * ds
A = assemble(a)

# Create the linear form for the exact source term and perform assembly
f = source_term(3.0 / math.pi) + source_term(0.5)
L = f * v * dx
b = assemble(L)

# Solve for the reference solution
u_ref = Function(V)
solve(A, u_ref.vector(), b)

# Define the control grid and the elementary source terms
ctrl_grid = numpy.linspace(0.0, 1.0, num=nu+1)[:-1]
ctrl_grid += 0.5 * ctrl_grid[1]
src_expr = [source_term(x0) for x0 in ctrl_grid]

# Assemble the linear forms corresponding to the elementary source terms and
# construct the second half of the coefficient matrix
b = [assemble(-f * v * dx) for f in src_expr]
B = scipy.column_stack([v.array() for v in b])

# Construct the full coefficient matrix
C = scipy.sparse.vstack([
scipy.sparse.hstack([matrix_to_coo(A), B]),
scipy.sparse.coo_matrix(([1.0]*nu, ([0]*nu, [V.dim() + i for i in range(nu)])), shape=(1, V.dim() + nu))
])

# Transfer the optimization constraints to CasADi
C = casadi.DMatrix(C)
lb = casadi.DMatrix.zeros(C.shape[0], 1)
ub = casadi.DMatrix.zeros(C.shape[0], 1)

lb[-1] = 0.0
ub[-1] = 5.0

lbx = numpy.array([-numpy.inf]*V.dim() + [0.0]*nu)
ubx = numpy.array([ numpy.inf]*V.dim() + [1.0]*nu)

# Define the objective functional and assemble it
u = Function(V)
J = (u_ref - u)**2 * dx
dJdu = derivative(J, u)
dJdu2 = derivative(dJdu, u)

H = casadi.DMatrix(matrix_to_coo(assemble(dJdu2), shape=(V.dim() + nu, V.dim() + nu)))
g = casadi.DMatrix(numpy.concatenate((assemble(dJdu).array(), [0.0]*nu)))
c = assemble(J)

# Build the CasADi NLP
x = casadi.MX.sym('X', V.dim() + nu)
nlp = casadi.MXFunction('nlp',
casadi.nlpIn(x=x),
casadi.nlpOut(
f=0.5*casadi.quad_form(x, H) + casadi.inner_prod(g, x) + c,
g=casadi.mul(C, x)
))

# Build NLP solver
solver = casadi.NlpSolver('solver', 'ipopt', nlp, {
'print_level': 0,
'print_time': False
})
result = solve_bnb(solver, {
'lbx': lbx,
'ubx': ubx,
'lbg': lb,
'ubg': ub
}, range(V.dim(), V.dim() + nu))

# Evaluate result
print ""
print "----------------------------"
print " f = %g" % result['f']

u.vector()[:] = result['x'][:V.dim()].toArray()[:,0]
</source>

== Results ==

Node solutions are calculated using IPOPT (3.12, default settings, using linear solver MUMPS on Ubuntu 15.10 for x86-64 with Linux 4.2.0-19-generic, Intel(R) Core(TM) i7-4790K CPU @ 4.00GHz, 16 GB RAM). The objective function value is <math>0.858837</math>. The integer solution is found after processing 25 nodes.

== Solution ==

The following is a Gnuplot compatible tabular version of the control data determined by the solver:

<pre>
# x w w0
0.0625 0 2.91549e-08
0.1875 0 2.75108e-08
0.3125 0 6.746e-08
0.4375 1 0.46917
0.5625 0 0.543817
0.6875 0 1.33205e-07
0.8125 0 1.51069e-07
0.9375 1 0.897707
</pre>

The following is a Gnuplot compatible tabular version of the solution found using FEM approximations:

<pre>
# x u u0 u_d
0 19.845 18.7521 18.7182
0.0078125 20.0001 18.8986 18.8645
0.015625 20.1551 19.0451 19.0107
0.0234375 20.3101 19.1916 19.1569
0.03125 20.4652 19.3381 19.3032
0.0390625 20.6202 19.4846 19.4494
0.046875 20.7752 19.6311 19.5956
0.0546875 20.9303 19.7776 19.7419
0.0625 21.0853 19.9241 19.8881
0.0703125 21.2403 20.0706 20.0343
0.078125 21.3953 20.2171 20.1806
0.0859375 21.5503 20.3635 20.3268
0.09375 21.7053 20.51 20.473
0.101562 21.8603 20.6565 20.6193
0.109375 22.0153 20.8029 20.7655
0.117188 22.1702 20.9494 20.9117
0.125 22.3251 21.0958 21.0579
0.132812 22.4799 21.2422 21.2042
0.140625 22.6347 21.3886 21.3504
0.148438 22.7894 21.5349 21.4966
0.15625 22.944 21.6812 21.6428
0.164062 23.0985 21.8275 21.7889
0.171875 23.2528 21.9737 21.9351
0.179688 23.407 22.1197 22.0812
0.1875 23.561 22.2657 22.2273
0.195312 23.7146 22.4116 22.3733
0.203125 23.868 22.5573 22.5193
0.210938 24.0209 22.7028 22.6652
0.21875 24.1734 22.848 22.811
0.226562 24.3254 22.993 22.9567
0.234375 24.4766 23.1377 23.1022
0.242188 24.6271 23.282 23.2476
0.25 24.7767 23.4259 23.3927
0.257812 24.9252 23.5692 23.5376
0.265625 25.0725 23.7119 23.6822
0.273438 25.2184 23.854 23.8263
0.28125 25.3627 23.9952 23.97
0.289062 25.5052 24.1355 24.1131
0.296875 25.6457 24.2748 24.2556
0.304688 25.7839 24.413 24.3972
0.3125 25.9196 24.5498 24.538
0.320312 26.0525 24.6852 24.6777
0.328125 26.1823 24.8189 24.8162
0.335938 26.3088 24.9509 24.9533
0.34375 26.4316 25.0809 25.0889
0.351562 26.5505 25.2087 25.2226
0.359375 26.6651 25.3342 25.3542
0.367188 26.7753 25.4572 25.4836
0.375 26.8807 25.5774 25.6105
0.382812 26.9811 25.6947 25.7346
0.390625 27.0762 25.8089 25.8556
0.398438 27.1659 25.9198 25.9733
0.40625 27.2499 26.0271 26.0873
0.414062 27.3281 26.1308 26.1973
0.421875 27.4004 26.2305 26.3032
0.429688 27.4666 26.3262 26.4046
0.4375 27.5268 26.4177 26.5012
0.445312 27.5809 26.5047 26.5927
0.453125 27.6289 26.5873 26.6791
0.460938 27.6708 26.6652 26.7599
0.46875 27.7069 26.7384 26.8351
0.476562 27.7371 26.8066 26.9045
0.484375 27.7617 26.87 26.968
0.492188 27.7808 26.9283 27.0255
0.5 27.7947 26.9816 27.0768
0.507812 27.8035 27.0298 27.1221
0.515625 27.8076 27.0729 27.1613
0.523438 27.8072 27.1109 27.1945
0.53125 27.8026 27.1438 27.2217
0.539062 27.794 27.1718 27.2431
0.546875 27.7818 27.1948 27.2589
0.554688 27.7662 27.213 27.2692
0.5625 27.7476 27.2264 27.2743
0.570312 27.7261 27.2352 27.2743
0.578125 27.7021 27.2395 27.2696
0.585938 27.6759 27.2394 27.2603
0.59375 27.6476 27.2351 27.2469
0.601562 27.6174 27.2269 27.2295
0.609375 27.5857 27.2147 27.2084
0.617188 27.5525 27.199 27.184
0.625 27.5181 27.1797 27.1565
0.632812 27.4826 27.1573 27.1261
0.640625 27.4461 27.1317 27.0932
0.648438 27.4087 27.1032 27.058
0.65625 27.3706 27.0721 27.0207
0.664062 27.3318 27.0385 26.9815
0.671875 27.2924 27.0025 26.9406
0.679688 27.2525 26.9644 26.8983
0.6875 27.2119 26.9242 26.8545
0.695312 27.1709 26.8821 26.8096
0.703125 27.1293 26.8383 26.7635
0.710938 27.0871 26.7928 26.7164
0.71875 27.0443 26.7457 26.6683
0.726562 27.0008 26.6971 26.6193
0.734375 26.9566 26.6469 26.5693
0.742188 26.9115 26.5953 26.5184
0.75 26.8654 26.5422 26.4666
0.757812 26.8183 26.4875 26.4137
0.765625 26.7699 26.4312 26.3598
0.773438 26.72 26.3732 26.3046
0.78125 26.6686 26.3135 26.2481
0.789062 26.6153 26.2517 26.1902
0.796875 26.56 26.188 26.1306
0.804688 26.5024 26.1219 26.0691
0.8125 26.4423 26.0534 26.0057
0.820312 26.3794 25.9822 25.9399
0.828125 26.3134 25.9082 25.8717
0.835938 26.244 25.8311 25.8007
0.84375 26.171 25.7506 25.7267
0.851562 26.094 25.6665 25.6494
0.859375 26.0129 25.5785 25.5685
0.867188 25.9272 25.4865 25.4837
0.875 25.8368 25.3902 25.3947
0.882812 25.7413 25.2893 25.3013
0.890625 25.6406 25.1837 25.2032
0.898438 25.5345 25.0732 25.1002
0.90625 25.4227 24.9576 24.9919
0.914062 25.305 24.8368 24.8782
0.921875 25.1815 24.7107 24.7589
0.929688 25.0519 24.5791 24.6338
0.9375 24.9162 24.4421 24.5029
0.945312 24.7745 24.2996 24.3659
0.953125 24.6267 24.1516 24.2228
0.960938 24.4728 23.9982 24.0737
0.96875 24.313 23.8395 23.9184
0.976562 24.1474 23.6756 23.7572
0.984375 23.9762 23.5067 23.5899
0.992188 23.7995 23.3328 23.4169
1 23.6175 23.1542 23.2381
</pre>

[[Category:Casadi]]

Source Inversion (FEniCS/Casadi)

2016-01-12T21:37:35Z

MirkoHahn:

This page contains a discretized version of the mixed integer PDE constrained optimization problem [[Source Inversion]] in [http://www.casadi.org CasADi 2.4.3] format. The weak version of the problem was discretized using finite element methods (via [http://www.fenicsproject.org FEniCS]) using first-degree Lagrangian elements on an equidistant mesh with <math>128</math> cells.

=== CasADi ===

<source lang="Python">
from dolfin import *

import math
import numpy
import scipy, scipy.sparse
import casadi

# Simple Branch and Bound solver
def solve_bnb(solver, root, integers):
# Initialize the node queue
Q = [(numpy.inf, root)]

# Global solution data
x_opt = None
f_opt = numpy.inf

# Continuous solution data
x0 = None
f0 = numpy.inf

# Main loop
seqnum = 0
while Q:
seqnum += 1

# Extract node with lowest lower bound and check if fathoming is necessary
# (should not happen)
node = Q.pop()
if f_opt < numpy.inf and node[0] >= f_opt:
print "Node %4d (%4d left): %9g >= %9g - PRE-FATHOM" % (seqnum, len(Q), node[0], f_opt)
continue
node = node[1]

# Solve the node
result = solver(node)

# Local solution data
x = result['x']
f = result['f'].get()[0]

# Store continuous solution data if none has been stored yet
if x0 is None:
x0 = casadi.DMatrix(x)
f0 = f

# Check if branch can be fathomed
if f >= f_opt:
print "Node %4d (%4d left): %9g >= %9g - POST-FATHOM" % (seqnum, len(Q), f, f_opt)
continue

# Check for violations of integrality (fixed tolerance 1e-5)
viol = [abs(casadi.floor(x[i] + 0.5) - x[i]) for i in integers]
idx = [(integers[i], viol[i]) for i in range(len(integers)) if viol[i] > 1e-5]
if not idx:
# Register new global solution
x_opt = x
f_opt = f

# Cull the branch and bound tree
pre = len(Q)
Q = [n for n in Q if n[0] < f_opt]
post = len(Q)

print "Node %4d (%4d left): f_opt = %9g - *** SOLUTION *** (%d culled)" % (seqnum, len(Q), f, pre - post)
else:
# Branch on first violation
idx = idx[0][0]

# Generate two new nodes (could reuse node structure)
ln = {
'x0': casadi.DMatrix(x),
'lbx': node['lbx'],
'ubx': casadi.DMatrix(node['ubx']),
'lbg': node['lbg'],
'ubg': node['ubg'],
'lam_x0': casadi.DMatrix(result['lam_x']),
'lam_g0': casadi.DMatrix(result['lam_g'])
}
un = {
'x0': casadi.DMatrix(x),
'lbx': casadi.DMatrix(node['lbx']),
'ubx': node['ubx'],
'lbg': node['lbg'],
'ubg': node['ubg'],
'lam_x0': casadi.DMatrix(result['lam_x']),
'lam_g0': casadi.DMatrix(result['lam_g'])
}

ln['ubx'][idx] = casadi.floor(x[idx])
un['lbx'][idx] = casadi.ceil(x[idx])

lower_node = (f, ln)
upper_node = (f, un)

# Insert new nodes in queue (inefficient for large queues)
Q.extend([lower_node, upper_node])
Q.sort(cmp=lambda x,y: cmp(y[0], x[0]))
print "Node %4d (%4d left): %9g - BRANCH ON %d" % (seqnum, len(Q), f, idx)

return {
'x0': x0,
'f0': f0,
'x': x_opt,
'f': f_opt
}

# Converts FEniCS matrix to scipy.sparse.coo_matrix
def matrix_to_coo(A, shape=None):
nnz = A.nnz()
col = numpy.empty(nnz, dtype='uint64')
row = numpy.empty(nnz, dtype='uint64')
val = numpy.empty(nnz)

if shape is None:
shape = (A.size(0), A.size(1))

it = 0
for i in range(A.size(0)):
col_local, val_local = A.getrow(i)
nnz_local = col_local.shape[0]

col[it:it+nnz_local] = col_local[:]
row[it:it+nnz_local] = i
val[it:it+nnz_local] = val_local[:]
it += nnz_local

return scipy.sparse.coo_matrix((val, (row, col)), shape=shape)

# Returns FEniCS expression for elementary source term
def source_term(x0):
return Expression("100.0 * exp(-pow(x[0] - x0, 2) / 0.02)", x0=x0)

# Grid resolutions
nx = 128
nu = 8

# Generate mesh and function space
mesh = UnitIntervalMesh(nx)
V = FunctionSpace(mesh, 'CG', 1)

# Define the bilinear form and assemble it
u = TrialFunction(V)
v = TestFunction(V)
a = inner(grad(u), grad(v)) * dx + u * v * ds
A = assemble(a)

# Create the linear form for the exact source term and perform assembly
f = source_term(3.0 / math.pi) + source_term(0.5)
L = f * v * dx
b = assemble(L)

# Solve for the reference solution
u_ref = Function(V)
solve(A, u_ref.vector(), b)

# Define the control grid and the elementary source terms
ctrl_grid = numpy.linspace(0.0, 1.0, num=nu+1)[:-1]
ctrl_grid += 0.5 * ctrl_grid[1]
src_expr = [source_term(x0) for x0 in ctrl_grid]

# Assemble the linear forms corresponding to the elementary source terms and
# construct the second half of the coefficient matrix
b = [assemble(-f * v * dx) for f in src_expr]
B = scipy.column_stack([v.array() for v in b])

# Construct the full coefficient matrix
C = scipy.sparse.vstack([
scipy.sparse.hstack([matrix_to_coo(A), B]),
scipy.sparse.coo_matrix(([1.0]*nu, ([0]*nu, [V.dim() + i for i in range(nu)])), shape=(1, V.dim() + nu))
])

# Transfer the optimization constraints to CasADi
C = casadi.DMatrix(C)
lb = casadi.DMatrix.zeros(C.shape[0], 1)
ub = casadi.DMatrix.zeros(C.shape[0], 1)

lb[-1] = 0.0
ub[-1] = 5.0

lbx = numpy.array([-numpy.inf]*V.dim() + [0.0]*nu)
ubx = numpy.array([ numpy.inf]*V.dim() + [1.0]*nu)

# Define the objective functional and assemble it
u = Function(V)
J = (u_ref - u)**2 * dx
dJdu = derivative(J, u)
dJdu2 = derivative(dJdu, u)

H = casadi.DMatrix(matrix_to_coo(assemble(dJdu2), shape=(V.dim() + nu, V.dim() + nu)))
g = casadi.DMatrix(numpy.concatenate((assemble(dJdu).array(), [0.0]*nu)))
c = assemble(J)

# Build the CasADi NLP
x = casadi.MX.sym('X', V.dim() + nu)
nlp = casadi.MXFunction('nlp',
casadi.nlpIn(x=x),
casadi.nlpOut(
f=0.5*casadi.quad_form(x, H) + casadi.inner_prod(g, x) + c,
g=casadi.mul(C, x)
))

# Build NLP solver
solver = casadi.NlpSolver('solver', 'ipopt', nlp, {
'print_level': 0,
'print_time': False
})
result = solve_bnb(solver, {
'lbx': lbx,
'ubx': ubx,
'lbg': lb,
'ubg': ub
}, range(V.dim(), V.dim() + nu))

# Evaluate result
print ""
print "----------------------------"
print " f = %g" % result['f']

u.vector()[:] = result['x'][:V.dim()].toArray()[:,0]
</source>

== Results ==

Node solutions are calculated using IPOPT (3.12, default settings, using linear solver MUMPS on Ubuntu 15.10 for x86-64 with Linux 4.2.0-19-generic, Intel(R) Core(TM) i7-4790K CPU @ 4.00GHz, 16 GB RAM). The objective function value is <math>0.858837</math>. The integer solution is found after processing 25 nodes.

== Solution ==

The following is a Gnuplot compatible tabular version of the control data determined by the solver:

<pre>
# x w w0
0.0625 0 2.91549e-08
0.1875 0 2.75108e-08
0.3125 0 6.746e-08
0.4375 1 0.46917
0.5625 0 0.543817
0.6875 0 1.33205e-07
0.8125 0 1.51069e-07
0.9375 1 0.897707
</pre>

The following is a Gnuplot compatible tabular version of the solution found using FEM approximations:

<pre>
# x u u0
0 19.845 18.7521
0.0078125 20.0001 18.8986
0.015625 20.1551 19.0451
0.0234375 20.3101 19.1916
0.03125 20.4652 19.3381
0.0390625 20.6202 19.4846
0.046875 20.7752 19.6311
0.0546875 20.9303 19.7776
0.0625 21.0853 19.9241
0.0703125 21.2403 20.0706
0.078125 21.3953 20.2171
0.0859375 21.5503 20.3635
0.09375 21.7053 20.51
0.101562 21.8603 20.6565
0.109375 22.0153 20.8029
0.117188 22.1702 20.9494
0.125 22.3251 21.0958
0.132812 22.4799 21.2422
0.140625 22.6347 21.3886
0.148438 22.7894 21.5349
0.15625 22.944 21.6812
0.164062 23.0985 21.8275
0.171875 23.2528 21.9737
0.179688 23.407 22.1197
0.1875 23.561 22.2657
0.195312 23.7146 22.4116
0.203125 23.868 22.5573
0.210938 24.0209 22.7028
0.21875 24.1734 22.848
0.226562 24.3254 22.993
0.234375 24.4766 23.1377
0.242188 24.6271 23.282
0.25 24.7767 23.4259
0.257812 24.9252 23.5692
0.265625 25.0725 23.7119
0.273438 25.2184 23.854
0.28125 25.3627 23.9952
0.289062 25.5052 24.1355
0.296875 25.6457 24.2748
0.304688 25.7839 24.413
0.3125 25.9196 24.5498
0.320312 26.0525 24.6852
0.328125 26.1823 24.8189
0.335938 26.3088 24.9509
0.34375 26.4316 25.0809
0.351562 26.5505 25.2087
0.359375 26.6651 25.3342
0.367188 26.7753 25.4572
0.375 26.8807 25.5774
0.382812 26.9811 25.6947
0.390625 27.0762 25.8089
0.398438 27.1659 25.9198
0.40625 27.2499 26.0271
0.414062 27.3281 26.1308
0.421875 27.4004 26.2305
0.429688 27.4666 26.3262
0.4375 27.5268 26.4177
0.445312 27.5809 26.5047
0.453125 27.6289 26.5873
0.460938 27.6708 26.6652
0.46875 27.7069 26.7384
0.476562 27.7371 26.8066
0.484375 27.7617 26.87
0.492188 27.7808 26.9283
0.5 27.7947 26.9816
0.507812 27.8035 27.0298
0.515625 27.8076 27.0729
0.523438 27.8072 27.1109
0.53125 27.8026 27.1438
0.539062 27.794 27.1718
0.546875 27.7818 27.1948
0.554688 27.7662 27.213
0.5625 27.7476 27.2264
0.570312 27.7261 27.2352
0.578125 27.7021 27.2395
0.585938 27.6759 27.2394
0.59375 27.6476 27.2351
0.601562 27.6174 27.2269
0.609375 27.5857 27.2147
0.617188 27.5525 27.199
0.625 27.5181 27.1797
0.632812 27.4826 27.1573
0.640625 27.4461 27.1317
0.648438 27.4087 27.1032
0.65625 27.3706 27.0721
0.664062 27.3318 27.0385
0.671875 27.2924 27.0025
0.679688 27.2525 26.9644
0.6875 27.2119 26.9242
0.695312 27.1709 26.8821
0.703125 27.1293 26.8383
0.710938 27.0871 26.7928
0.71875 27.0443 26.7457
0.726562 27.0008 26.6971
0.734375 26.9566 26.6469
0.742188 26.9115 26.5953
0.75 26.8654 26.5422
0.757812 26.8183 26.4875
0.765625 26.7699 26.4312
0.773438 26.72 26.3732
0.78125 26.6686 26.3135
0.789062 26.6153 26.2517
0.796875 26.56 26.188
0.804688 26.5024 26.1219
0.8125 26.4423 26.0534
0.820312 26.3794 25.9822
0.828125 26.3134 25.9082
0.835938 26.244 25.8311
0.84375 26.171 25.7506
0.851562 26.094 25.6665
0.859375 26.0129 25.5785
0.867188 25.9272 25.4865
0.875 25.8368 25.3902
0.882812 25.7413 25.2893
0.890625 25.6406 25.1837
0.898438 25.5345 25.0732
0.90625 25.4227 24.9576
0.914062 25.305 24.8368
0.921875 25.1815 24.7107
0.929688 25.0519 24.5791
0.9375 24.9162 24.4421
0.945312 24.7745 24.2996
0.953125 24.6267 24.1516
0.960938 24.4728 23.9982
0.96875 24.313 23.8395
0.976562 24.1474 23.6756
0.984375 23.9762 23.5067
0.992188 23.7995 23.3328
1 23.6175 23.1542
</pre>

[[Category:Casadi]]

Controlled Heating (Casadi/FEniCS) - Solution Data

2016-01-12T21:23:19Z

MirkoHahn: Created page with "The following is a tabular version of the solution data generated by the code at Controlled Heating (FEniCS/Casadi) <pre> # x y u f e 0.980785 0.19509 0.756845 9.99192 0...."