1. Boundary Value Problems and Least Squares Minimization
Daniel Baur ETH Zurich, Institut für Chemie- und Bioingenieurwissenschaften ETH Hönggerberg / HCI F128 – Zürich
Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE

2. Boundary Value Problems (BVP) for ODEs
Problem definition:
Find a solution for a system of ODEs
Subject to the boundary conditions (BCs):
The total number of BCs has to be equal to the number of equations!
Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE

3. Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE
Shooting method
A first approach is to transform the BVP into an initial value problem (IVP), by guessing the missing initial conditions and using the BC to refine the guess, until convergence is reached
This way, the same algorithms as for IVPs can be used, but the convergence can be very problematic
Too high: reduce initial velocity!
Target
Too low: increase initial velocity!
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4. Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE
Relaxation methods
Relaxation methods try to find a solution that «relaxes» from the disturbance that are the initial conditions
Example: Dimensionless tubular reactor
The solution «relaxes» into the steady state solution
Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE

5. Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE
Visualization
Take a look at the discretized internal profile y
yin y0 y1 y2 y3 y4 y5 yN
yN+1 x
L/Δx L
Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE

6. Relaxation methods (continued)
Therefore, we can iterate until convergence:
For a first order reaction, we have a linear system
With the «boundary conditions»
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7. Matrix formulation of the linear system
If we cast the linear system into matrix form, we get
Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE

8. Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE
Iterative methods
Instead of simply iterating, we can use the properties of the matrix to speed up the solution
The main idea is to reformulate the problem in the following way
The goal is to choose S in a way that is easy to solve (e.g. diagonal, tridiagonal, triangular), but in some sense similar to A to keep ρ[S-1*(S-A)] small
The iteration formula therefore reads
Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE

9. Jacobi’s method The Jacobi method chooses S to be a diagonal matrix
The procedure is therefore given by
This method is guaranteed to converge if A is diagonally dominant, i.e.
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Collocation method
A more sophisticated approach is the collocation method; it is based on approximating the unknown function with a sum of polynomials multiplied with unknown coefficients
The coefficients are determined by forcing the approximated solution to satisfy the ODE at a number of points equal to the number of coefficients
Matlab has a built-in function bvp4c which implements such a method
Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE

11. Collocation method (continued)
We require that at every point
Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE

12. Transformation into a first order ODE
bvp4c solves first order ODEs, so if we remember the «trick» and transform our ODE, we get
And the boundary conditions
Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE

13. Least squares minimization
In data fitting, we generally try to solve a problem of the form
where f is some model with parameters β, X are the experimental conditions and y are the results x y
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14. Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE
Some nomenclature
In prospect of the statistics lecture, please note the differences in nomenclature!
p parameters (in linear regression p = m+1)
m variables (plus constant term)
n measurements / data points
Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE

15. Linear least squares minimization
If the model f(β,X) is linear (this means that the parameters β only appear in linear terms!), we saw that the unique minimizer to the problem reads
However, since XTX is much harder to solve (worse conditioned) than X, we can try to find another decomposition where Q is an orthogonal matrix (Q-1 = QT) and R is an upper triangular matrix
Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE

16. Linear least squares minimization (continued)
Substituing our originial problem
This yields the same minimizer, but is easier to solve
Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE

17. Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE
Assignment 1
Solve the steady state of the dimensionless tubular reactor using Jacobi’s method.
Use Pe = 10; N = 150; Da = 1; ystart = zeros(N,1); n = 1;
Define the core of the matrix A; This is easiest done using diag():
After this, change the first and the last row of A to reflect the boundary conditions and define b (see slide 6).
Use A = sparse(A); and b = sparse(b); to tell Matlab that the matrices contain mostly zeros. Check with spy(A) if this is true. This will vastly speed up all calculations.
Use Jacobi’s iteration formula and iterate until the change in y is smaller than 1e-8 or 1e5 iterations are exceeded.
Plot the results using plot(linspace(0,1,N), y)
Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE

18. Assignment 1 (continued)
Solve the same steady state of the dimensionless tubular reactor using bvp4c
bvp4c uses a call of the form sol = bvp4c(ode_fun, bc_fun, solinit);
ode_fun is a function taking as inputs a scalar t and a vector y, returning as an output dy / dt
bc_fun is a function taking as inputs vectors where the boundary conditions are evaluated, returning as output the residual at the boundary
solinit is an initial guess for the solution: solinit =
sol is a struct containing the solution and other parameters
Plot the solution using plot(sol.x, sol.y(1,:)) and compare with the solution from the first part.
More on the usage of bvp4c on the next slide
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19. Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE
Usage of bvp4c
In our case use the following
Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE

20. Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE
Exercise 2
Find online the example files from the lecture for the linear least squares problem.
We fit a linear model of the form
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21. Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE
Assignment 2
Run example01.m to see the results of the fit.
Read through the code and ask questions if you have any.
Is the model really linear, even though it contains x2?
In the definition of the data matrix X on line 7, there is a vector of ones. Why is that? Check with the definition of a linear model in matrix form
Note: no programming will be required for this assignment.
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22. Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE
Exercise 3
Find online the example files from the lecture for the non-linear least squares problem.
We fit a model of the form
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23. Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE
Assignment 3
Run example02.m to see the results of the fitting.
Read through the code and ask questions if you have any.
The functions provide a Hessian and a Jacobian specific to the problem. What could be done to make a solver more general and easier to use (remember back when we implemented the Newton method!)?
In what cases can the Newton method fail?
In what cases can the Gauss-Newton method fail?
Note: no programming will be required for this assignment.
Daniel Baur / Numerical Methods for Chemical Engineers / BVP and PDE