1 / 26

Backward Thinking

Backward Thinking. Confessions of a Numerical Analyst Keith Evan Schubert. Simple Problem. Consider the problem ax=b The resulting x value is. Simple Problem 2. Consider the problem ax=b The resulting x value is. What’s Up?.

fpelayo
Download Presentation

Backward Thinking

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Backward Thinking Confessions of a Numerical Analyst Keith Evan Schubert

  2. Simple Problem • Consider the problem ax=b • The resulting x value is

  3. Simple Problem 2 • Consider the problem ax=b • The resulting x value is

  4. What’s Up? • The condition number (sensitivity to perturbations) is about 400. • A condition number of 1 is perfect. • Perturbation is 0.01, so 0.01*400=4. • Components of x can vary by this much!

  5. What Can We Do? • Rather than solve it the standard way • X=a\b • X=(ATA)-1atb • Consider the following: • X=(ATA+i)-1atb •  =.01 • Then:

  6. Lucky Guess?

  7. Does It Always Work? • No • Consider  X0 • Consider si2(si is singular value of A) X± • Picking the wrong value can get junk

  8. Skyline • Consider a 1 dimensional picture • Use height instead of color • Result looks like the silhouette of a city’s skyline • Have smog which blurs and softens • Don’t know exactly how much blur • Want to get sharp edges

  9. Getting Garbage

  10. Getting Improvement

  11. Why Backward? • Forward errors • Explicitly account for each error source • (X+d1)(y+d2)=xy+(yd1+xd2+d1d2) • Backward errors • Check that my algorithm acting on data will give me a solution that is “near” to the actual system acting on a nearby set of data • I.E. My algorithm with good data should do about as well as a perfect calculation on ok data

  12. Picture Please! Inherent errors in A b Perfect Calculations b* Errors due to algorithm best My Algorithm Actual Data (x) Nearby Data (x*)

  13. Least Squares • Usually we don’t have an invertible matrix • Need to find an estimated solution • Criterion: minimize ||ax-b|| • Normal equation • ATA x = ATb • Solution • X = (ATA)-1atb

  14. Backward Error • Criterion: minimize ||Ax-b||/(||A|| ||x||+||b||) • Normal Equations • Solution:

  15. Non Convex

  16. Finding The Root

  17. Look At Critical Region

  18. Informal Algorithm • Get (A,b) • svd(A)  [u1 u2],,v • U1b  b1 • Use rootfinder (bisection, Newton, etc.) to get  in [-sn2,0] • vT(2- I)-1  b1  x

  19. What You Get

  20. Least Squares

  21. Total Least Squares

  22. Tikhonov

  23. Backward Error

  24. Original

  25. Comparison

  26. Final Thoughts • BE is always optimistic in that it presumes that the real system is “better” • Even with this it is “robust” • There is a perturbed version of this algorithm which can be either optimistic or pessimistic • That version is not fully proven

More Related