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Mathematics of Inverse Problems

Gain insights on optimization fundamentals, convex analysis, variational principles, and more at IMA Summer School. Learn key ideas and real-world applications. Enhance your mathematical problem-solving skills for future careers.

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Mathematics of Inverse Problems

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  1. Mathematics of Inverse Problems IMA Summer School June 15-July 3 University of Delaware, Newark Delaware Week 3: Inverse Problems as Optimization Problems Jonathan Borwein, FRSCwww.carma.newcastle.edu.au/~jb616 Canada Research Chairin Collaborative Technology Laureate ProfessorUniversity of Newcastle, NSW Revised 04-07-09 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAA

  2. Our Goals for the Week • Convex Analysis • Duality and Optimality Conditions • Fixed Points and Monotone Mappings • Variational Principles • Stability and Regularity • Models and Algorithm Design • Some Concrete Examples • Some Experimentation A brief introduction to some key ideas from optimization that should be useful later in your careers

  3. Mathematics is not a spectator subject

  4. Outline of Week’s Lectures Day 1:A big-pictureOverviewof the Week Day 2 Convex Duality and Applications Day 3 Variational Principles & Applications Day 4 Monotone & Non-expansive Maps Day 5 Algebraic Reconstruction Methods and Interactive Geometry Days will spill over...

  5. Primary Source The primary source isChapters 3-5, 7-8 of

  6. Contents

  7. Assuring a minimum The basis of all optimization is: • When this holds we are in business. • If not we have to work harder to establish the minimum exists • e.g., the isoperimetric problem (of Queen Dido).

  8. The Fermat (location) problem with a twist

  9. Day 1: An Overview of the WeekandHow to Maximize Surprise

  10. Day 2: Convex Duality and Applications

  11. ENIAC 20Mb file =100,000 ENIACS

  12. A set is convex iff its indicator function is • A function is convex iff the epigraph is Convex functions

  13. A subtle convex function

  14. Day 2: topics CANO2 CFC3 Topics for today: subgradients • Symbolic convex analysis (in Maple) • formalizing our model for the week (Potter and Arun)

  15. Subgradients CFC3 Max formula = subgradient and Fenchel duality Tangent cone to ellipse includes vertical line

  16. SCA CFC3

  17. NMR (back)

  18. characterizes p= PC(x) y x PC is “firmly nonexpansive”: for some nonexpansive T Application to Reflections C RC(y) RC(x)

  19. Dualizing Potter and Arun (back)

  20. Day 3: Variational Principles and Applications

  21. 2009 Record Pi Computation

  22. Day 3: topics • Topics for today • tangency of convex sets and CANO2 +... • an application of metric regularity • two smooth variational principles • computing projections with KKT multipliers

  23. Tangency Regular limiting normals (back)

  24. Ekeland’s principle in Euclidean Space

  25. Pareto optimality Ekeland’s principle is Pareto optimality for an ice-cream (second-order) cone • the osculating function is nonsmooth at the important point:

  26. SVP in action This can be fixed as I discovered in 1986

  27. 2 SVPs (usually a norm) (back) (back)

  28. Error bounds and the distance to the intersection of two convex sets Holds Fails (back)

  29. Asplund spaces An important corollary is (back)

  30. Computing Projections using Multipliers

  31. The Raleigh Quotient (back)

  32. Day 4: Monotonicity & Applications

  33. “Die Mathematikersindeine Art Franzosen; redet man mitihnen, so übersetzensiees in ihreSprache, und dannistesalsobaldganzetwasanderes. [Mathematicians are a kind of Frenchman: whatever you say to them they translate into their own language, and right away it is something entirely different.]” (Johann Wolfgang von Goethe, 1748-1932) Maximen und Reflexionen, no. 1279, p.160 Penguin Classic ed. Goethe about Us

  34. ofMoebius transformationsand Chemical potentials Rogness and Arnoldat IMA Helaman Ferguson Sculpture Continuing from before

  35. Day 4: topics Topics for today • Cuscos and Fenchel Duality as decoupling • Multifunction Section 5.1.4 from TOVA • Sum theorem for maximal monotones • Monotonicity of the Laplacian • Potter and Arun revisited • iterates of firmly non-expansivemappings • implementing our model for the week • (Potter and Arun)

  36. Minimal and non-minimal Cuscos

  37. Fenchel Duality as Decoupling (back)

  38. The Sum Theorem (back)

  39. Laplacians as MaximalMonotone Operators (back)

  40. Nonexpansive Maps P(x) x F=Fix(P) (back)

  41. (Firmly) Nonexpansive Maps andso iteration provide solutions to Potter and Arun’s formulation (back)

  42. Day 5: Closing the Circle: Interactive Algorithmic Analysis

  43. A really new work: based on Potter and Arun’s model

  44. The Old Math And the New?

  45. Topics: Algebraic Phase Reconstruction and Discovery Alternating Projections and Reflections Parallelization Related ODES and Linearizations Proofs and a final Variation on a Theme Periodicity with reflections on half line and circle

  46. x B A (back) ‘2=N’: Inverse Problems as Feasibility Problems

  47. Algebraic Phase Reconstruction Projectors and Reflectors: PA(x) is the metric projection or nearest point and RA(x) reflects in the tangent: x is red x A 2008 Finding exoplanet Fomalhaut in Piscis with projectors A PA(x) RA(x) "All physicists and a good many quite respectable mathematicians are contemptuous about proof." - G. H. Hardy (1877-1947) 2007 Solving Sudoku with reflectors

  48. Consider the simplest case of a line A of height h and the unit circle B. With the iteration becomes In a wide variety of problems (protein folding, 3SAT, Sudoku) the set B is non-convex but “divide and concur” works better than theory can explain. It is: For h=0 we will prove convergence to one of the two points in A Å B iff we do not start on the vertical axis (where we have chaos). For h>1 (infeasible) it is easy to see the iterates go to infinity (vertically). For h=1we converge to an infeasible point. For h in (0,1) the pictures are lovely but full proofs escape us. Two representative pictures follow: An ideal problem to introduce early graduates to research, with many open accessible extensions in 2 or 3 dimensions APR: Why does it work?

  49. Interactive APRin Cinderella Recall the simplest case of a line A of height hand unit circle B. With the iteration becomes The pictures are lovely but full proofs escape us. A Cinderella picture of two steps from (4.2,-0.51) follows:

  50. CAS+IGP: the Grief is in the GUI Robust data from Maple Numerical errors in using double precision

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