Solving Minimal ProblemsNumerics

KALLE ÅSTRÖM, CENTRE FOR MATHEMATICAL SCIENCES,

LUND UNIVERSITY, SWEDEN

Solving System of Polynomial EquationsStep by Step once more

1. System of equations

2. Expand by multiplying each equation with a number of monomials

3. Choose a set of Basis monomials

4. Choose an action variable, e g

5. Calculate monomials to reduce R and the rest E

6. Re-order monomials

7. Gaussian elimination

Solving System of Polynomial EquationsStep by Step once more

8. Gaussian elimination

9. Form Action matrix

10. Solve eigenvalue problem

11. Extract solutions from eigenvalues/eigenvectors

12. What is the main cause of bad numerics?

Solving System of Polynomial EquationsStep by Step once more

8. From Gaussian elimination

9. Form Action matrix

10. Solve eigenvalue problem

11. Extract solutions from eigenvalues/eigenvectors

12. What is the main cause of bad numerics?

Ways to improve numerics

1. Redo the problem formulation

– Simpler equations, fewer unknowns

2. Choice of polynomials to multiply with (expansion)

3. Choice of basis monomials

4. Numerical linear algebra (SVD, QR)

5. Exploit symmetry

QR with column pivoting- as a method for choosing basis monomials

1. Choose a set P of monomials (slightly larger than B, before)

2. Reduce E and R part as before

3. Use QR with column pivoting to choose basis monomials B among P that minimize the condition number

QR with column pivoting(Byröd, Josephson, Åström, IJCV 2009)

SVD method - as a method for choosing basis polynomials

1. Choose a set P of monomials (slightly larger than B, before)

2. Reduce E and R part as before

3. Use QR with column pivoting to choose basis monomials B among P that minimize the condition number

4. Theoretically better, but more complicated to code and longer time to compute.

Optimizing (i) expansion and (ii) set of permissible monomials

1. Generate a small benchmark set of examples

2. Change the set of permissible monomials.

3. Evaluate on the benchmark set.

4. Perform local optimization.

5. Difficult optimization problem, many local minima.

6. Potential for improvement

Optimizing (i) expansion and (ii) set of permissible monomials

1. Generate a small benchmark set of examples

2. Change the set of permissible monomials.

3. Evaluate on the benchmark set.

4. Perform local optimization.

5. Difficult optimization problem, many local minima.

6. Potential for improvement