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  1. QUESTION What is ‘foundations of computational mathematics’?






  7. Who’s Algorithm is Best? • Test examples? • Heuristics? • Fight it out? • Clearly define problem (focm)


  9. IMAGE PROCESSING Model “Real Images” Metric “Human Visual System” Stochastic Mathematical Metric Deterministic Smoothness Classes K Lp Norms Lp Norms

  10. Kolmogorov Entropy • Given  > 0, N(K) smallest number of  balls that cover K

  11. Kolmogorov Entropy • Given  > 0, N(K) smallest number of  balls that cover K

  12. Kolmogorov Entropy • Given  > 0, N(K) smallest number of  balls that cover K • H(K):= log (N(K)) Best encoding with distortion  of K

  13. ENTROPY NUMBERS dn(K) := inf { : H(K)  n} • This is best distortion for K with bit budget n • Typically dn(K) n-s

  14. SUMMARY • Find right metric • Find right classes • Determine Kolmogorov entropy • Build encoders that give these entropy bounds

  15. COMPACT SETS IN Lp FOR d=2 Sobolev embedding line 1/q= /2+1/p Smoothness (1/q, )  Lq Lp 1/q (1/p,0) Lq Space 2

  16. COMPACT SETS IN L2 FOR d=2 Smoothness (1,1)-BV (1/q, )  Lq L2 1/q (1/2,0) Lq Space 2

  17. ENTROPY OF K Entropy of Besov Balls B (Lq ) in Lp is nd Is there a practical encoder achieving this simultaneously for all Besov balls? ANSWER: YES Cohen-Dahmen-Daubechies-DeVore wavelet tree based encoder

  18. f = S S cI yI j I  D j [T0 |B0|S0|T1|U1 |B1|S1|T2|U2 |B2|S2|. . . ] Lead tree & bits Level 1 tree, update & new bits, signs Level 2 tree, update & new bits, signs COHEN-DAUBECHIES-DAHMEN-DEVORE • Partition growth into subtrees • Decompose image D j :=T j \ T j-1

  19. WHAT DOES THIS BUY YOU? • Explains performance of best encoders: Shapiro, Said-Pearlman • Classifies images according to their compressibility (DeVore-Lucier) • Handles metrics other than L2 • Tells where to improve performance: Better metric, Better classes (e.g. not rearrangement invariant)

  20. DTED DATA SURFACE Grand Canyon

  21. Z-Values Grid POSTINGS Postings

  22. FIDELITY • L2 metric not appropriate

  23. FIDELITY • L2 metric not appropriate • L better

  24. OFFSET If surface is offset by a lateral error of , the L norm may be huge L error

  25. Hausdorff error OFFSET But Hausdorff error is not large. L error

  26. CAN WE FIND dn(K)? • K bounded functions : dN(K)  n-1 for N=nd+1 • K continuous functions: dN(K)  n-1, for N= nd log n • K bounded variation in d=1: dn(K)  n-1 • K class of characteristic functions of convex sets dn(K)  n-1

  27. Example: functions in BV, d=1 Assume f monotone; encode first (jk) and last (jk) square in column. Then k |jk-jk| M n. Can encode all such jk with C M n bits. jk jk k

  28. ANTICIPATED IMPACTDTED • Clearly define the problem • Expose new metrics to data compression community • Result in better and more efficient encoders

  29. NUMERICAL PDEs u solution to PDE uh or u n is a numerical approximation uh typically piecewise polynomial (FEM) un linear combination of n wavelets different from image processing because u is unknown

  30. MAIN INGREDIENTS • Metric to measure error • Number of degrees of freedom / computations • Linear (SFEM) or nonlinear (adaptive) method of approximation using piecewise polynomials or wavelets • Inversion of an operator Right question: Compare error with best error that could be obtained using full knowledge of u


  32. CLASSICAL ELLIPTIC THEOREM Variational formulation gives energy norm Ht THEOREM: If u in Ht+s then SFEM gives ||u-uh ||Ht < hs |u|Ht+s Can replace Ht+s by Bs+t (L2 ) Approx. order hs equivalent to u in Bs+t (L2 ) . . ) h 8 8

  33. HYPERBOLIC Conservation Law: ut + divx(f(u))=0, u(x,0)=u0(x) THEOREM: If u0 in BV then ||u(,,t)-uh(.,t)||L1 < h1/2 |u0| BV u0 in BV implies u in BV; this is equivalent to approximation of order h in L1 . . )

  34. ADAPTIVE METHODS Wavelet Methods (WAM) : approximates u by a linear combination of n wavelets AFEM: approximates u by piecewise polynomial on partition generated by adaptive subdivision

  35. FORM OF NONLINEAR APPROXIMATION Good Theorem: For a range of s >0, if u can be approximated with accuracy O(n-s) using full knowledge of u then numerical algorithm produces same accuracy using only information about u gained during the computation. Here n is the number of degrees of freedom Best Theorem: In addition bound the number of computations by Cn

  36. AFEMs • Initial partition P0 and Galerkin soln. u0 • General iterative step Pj Pj+1 and uj uj+1 i. Examine residual (a posteriori error estimators) to determine cells to be subdivided (marked cells) ii. Subdivide marked cells - results in hanging nodes. iii. Remove hanging nodes by further subdivision (completion) resulting in Pj+1

  37. FIRST FUNDAMENTALTHEOREMS Doerfler, Morin-Nochetto-Siebert: Introduce strategy for marking cells: a posterio estimators plus bulk chasing Rule for subdivision: newest vertex bisection • THEOREM (D,MNS): For Poisson problem algorithm convergence . . ) . . )

  38. BINEV-DAHMEN-DEVORE New AFEM Algorithm: 1. Add coarsening step 2. Fundamental analysis of completion 3. Utilize principles of nonlinear approximation

  39. BINEV-DAHMEN-DEVORE THEOREM (BDD): Poisson problem, for a certain range of s >0. If u can be approximated with order O(n-s ) in energy norm using full knowledge of u, then BDD adaptive algorithm does the same. Moreover, the number of computations is of order O(n). . . )

  40. ADAPTIVE WAVELET METHODS General elliptic problem Au=f   Problem in wavelet coordinates A u= f A: l2 l2 ||Av|| ~ ||v||

  41. FORM OF WAVELET METHODS • Choose a set  of wavelet indices • Find Gakerkin solution u from span{} • Check residual and update 

  42. COHEN-DAHMEN-DEVOREFIRST VIEW For finite index set  A u  = f u  Galerkin sol. Generate sets j , j = 0,1,2, … Form of algorithm: 1. Bulk chase on residual several iterations • j j~ • 2. Coarsen: j~  j+1 • 3. Stop when residual error small enough

  43. ADAPTIVE WAVELETS:COHEN-DAHMEN-DEVORE • THEOREM (CDD): For SPD problems. If u can be approximated with O(n-s ) using full knowledge of u (best n term approximation), then CDD algorithm does same. Moreover, the number of computations is O(n).

  44. CDD: SECOND VIEW u n+1 = u n - (A u n -f ) This infinite dimensional iterative process converges Find fast and efficient methods to compute Au n , f when u n is finitely supported. Compression of matrix vector multiplication Au n

  45. SECOND VIEW GENERALIZES • Wide range of semi-elliptic, and nonlinear THEOREM (CDD): For wide range of linear and nonlinear elliptic problems. If u can be approximated with O(n-s ) using full knowledge of u (best n term approximation), then CDD algorithm does same. Moreover, the number of computations is O(n).

  46. WHAT WE LEARNED • Proper coarsening controls size of problem • Remain with infinite dimensional problem as long as possible • Adaptivity is a natural stabilizer, e.g. LBB conditions for saddle point problems are not necessary

  47. WHAT focm CAN DO FOR YOU • Clearly frame the computational problem • Give benchmark of optimal performance • Discretization/Analysis/Solution interplay • Identify computational issues not apparent in computational heuristics • Guide the development of optimal algorithms