FOCM 2002 QUESTION What is ‘foundations of computational mathematics’?

FOCM 2002 FOCM DATA COMPRESSION ADAPTIVE PDE SOLVERS

FOCM 2002 COMPRESSION - ENCODING DATA SET Image Signal Surface BIT STREAM Function f B(f)=(B 1,…,B n )

FOCM 2002 COMPRESSION - ENCODING f B(f)=(B 1,…,B n )

FOCM 2002 DECODER BIT STREAM B FUNCTION g B B Image Signal Surface

FOCM 2002 DECODER BIT STREAM B FUNCTION g B B

FOCM 2002 Who’s Algorithm is Best?  Test examples?  Heuristics?  Fight it out?  Clearly define problem (focm)

FOCM 2002 MUST DECIDE  METRIC TO MEASURE ERROR  MODEL FOR OBJECTS TO BE COMPRESSED

FOCM 2002 Model “Real Images” Metric “Human Visual System” Stochastic Mathematical Metric Deterministic Lp Norms Smoothness Classes K L p Norms IMAGE PROCESSING

FOCM 2002  Given  > 0, N  (K) smallest number of  balls that cover K Kolmogorov Entropy

FOCM 2002 Kolmogorov Entropy  Given  > 0, N  (K) smallest number of  balls that cover K

FOCM 2002 Kolmogorov Entropy  Given  > 0, N  (K) smallest number of  balls that cover K  H  (K):= log (N  (K)) Best encoding with distortion  of K

FOCM 2002 Encoders and Kolmogorov Entropy  -balls with centers x j Compact set K ApproximantsCode x x x x …… x N(  ) b m …b 2 b 1 b 0 Codebook Max # of bits  log 2 N(  )

FOCM 2002 ENTROPY NUMBERS d n (K) := inf {  : H  (K)  n}  This is best distortion for K with bit budget n  Typically d n (K)  n -s

FOCM 2002 SUMMARY  Find right metric  Find right classes  Determine Kolmogorov entropy  Build encoders that give these entropy bounds

FOCM 2002 LpLp (1/p,0) LqLq 1/q  Smoothness L q Space (1/q, ) (1/q,  ) 2 COMPACT SETS IN L p FOR d=2 Sobolev embedding line 1/q=  /2+1/p

FOCM 2002 L2L2 (1,1) - BV (1/2,0) LqLq 1/q  Smoothness L q Space (1/q, ) (1/q,  ) 2 COMPACT SETS IN L 2 FOR d=2

FOCM 2002 ENTROPY OF K Entropy of Besov Balls B  (L q ) in L p 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

FOCM 2002 COHEN-DAUBECHIES- DAHMEN-DEVORE  Partition growth into subtrees  Decompose image   j :=T j \ T j-1 f =  f =  c I  I I  j j [ T 0 | B 0 | S 0 | T 1 | U 1 | B 1 | S 1 | T 2 | U 2 | B 2 | S 2 |... ] Lead tree & bits Level 1 tree, update & new bits, signs Level 2 tree, update & new bits, signs

FOCM 2002 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 L 2  Tells where to improve performance: Better metric, Better classes (e.g. not rearrangement invariant)

FOCM 2002 DTED DATA SURFACE Grand Canyon

FOCM 2002 POSTINGS Postings Grid Z-Values

FOCM 2002 FIDELITY  L 2 metric not appropriate

FOCM 2002 FIDELITY  L 2 metric not appropriate  L  better

FOCM 2002 OFFSET If surface is offset by a lateral error of , the L  norm may be huge  L  error

FOCM 2002 OFFSET But Hausdorff error is not large. Hausdorff error  L  error

FOCM 2002 CAN WE FIND d n (K)?  K bounded functions : d N (K)  n -1 for N=n d+1  K continuous functions: d N (K)  n -1, for N= n d log n  K bounded variation in d=1: d n (K)  n -1  K class of characteristic functions of convex sets d n (K)  n -1

FOCM 2002 Example: functions in BV, d=1 Assume f monotone; encode first (j k ) and last (j k ) square in column. Then  k |j k -j k |  M n. Can encode all such j k with C M n bits. k jkjk jkjk

FOCM 2002 ANTICIPATED IMPACT DTED  Clearly define the problem  Expose new metrics to data compression community  Result in better and more efficient encoders

FOCM 2002 NUMERICAL PDEs u solution to PDE u h or u n is a numerical approximation u h typically piecewise polynomial (FEM) u n linear combination of n wavelets different from image processing because u is unknown

FOCM 2002 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

FOCM 2002 EXAMPLE OF ELLIPTIC EQUATION POISSON PROBLEM

FOCM 2002 CLASSICAL ELLIPTIC THEOREM Variational formulation gives energy norm H t THEOREM: If u in H t+s then SFEM gives ||u-u h || H t < h s |u| H t+s Can replace H t+s by B s+t (L 2 ) Approx. order h s equivalent to u in B s+t (L 2 ) 8 8 h.. )

FOCM 2002 HYPERBOLIC Conservation Law: u t + div x (f(u))=0, u(x,0)=u 0 (x) THEOREM: If u 0 in BV then ||u(,,t)-u h (.,t)|| L1 < h 1/2 |u 0 | BV u 0 in BV implies u in BV; this is equivalent to approximation of order h in L 1 )..

FOCM 2002 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

FOCM 2002 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

FOCM 2002 AFEMs  Initial partition P 0 and Galerkin soln. u 0  General iterative step P j P j+1 and u j u j+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 P j+1

FOCM 2002 FIRST FUNDAMENTAL THEOREMS 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 ).... )

FOCM 2002 BINEV-DAHMEN-DEVORE New AFEM Algorithm: 1. Add coarsening step 2. Fundamental analysis of completion 3. Utilize principles of nonlinear approximation

FOCM 2002 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)... )

FOCM 2002 ADAPTIVE WAVELET METHODS General elliptic problem Au=f        Problem in wavelet coordinates A u= f A: l 2 l 2 ||Av|| ~ ||v||

FOCM 2002 FORM OF WAVELET METHODS  Choose a set  of wavelet indices  Find Gakerkin solution u  from span{   }  Check residual and update   

FOCM 2002 COHEN-DAHMEN-DEVORE FIRST 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

FOCM 2002 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).

FOCM 2002 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

FOCM 2002 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).

FOCM 2002 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

FOCM 2002 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