QUESTION

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QUESTION

• What is foundations of computational
• mathematics?

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FOCM

• DATA COMPRESSION
• ADAPTIVE PDE SOLVERS

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COMPRESSION - ENCODING
DATA SET Image Signal Surface BIT STREAM 1100111100...
Function f B(f)(B1,,Bn)
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COMPRESSION - ENCODING
1100111100...
f B(f)(B1,,Bn)
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DECODER
BIT STREAM B FUNCTION gB
B Image Signal Surface
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DECODER
BIT STREAM B FUNCTION gB
B
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Whos Algorithm is Best?

• Test examples?
• Heuristics?
• Fight it out?
• Clearly define problem (focm)

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MUST DECIDE

• METRIC TO MEASURE ERROR
• MODEL FOR OBJECTS TO BE COMPRESSED

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IMAGE PROCESSING
Model Real Images
Metric Human Visual System

Stochastic
Mathematical Metric
Deterministic
Smoothness Classes K
Lp Norms
Lp Norms
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Kolmogorov Entropy

• Given ? gt 0, N? (K) smallest number of ? balls
  that cover K

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Kolmogorov Entropy

• Given ? gt 0, N? (K) smallest number of ? balls
  that cover K

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Kolmogorov Entropy

• Given ? gt 0, N? (K) smallest number of ? balls
  that cover K
• H?(K) log (N?(K))
• Best encoding with distortion ? of K

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Encoders and Kolmogorov Entropy
Codebook
Approximants Code x1 0000
x2 0001 x3 0010 x4 0011
... xN(?)
bmb2b1b0
?-balls with centers xj
Max of bits ? log2N(?)
Compact set K
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ENTROPY NUMBERS

• dn(K) inf ? H?(K) ? n
• This is best distortion for K with bit budget n
• Typically dn(K) ? n-s

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SUMMARY

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

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COMPACT SETS IN Lp FOR d2
Sobolev embedding line 1/q ?/21/p

Smoothness
(1/q, ?)

?
Lq
Lp
1/q
(1/p,0)
Lq Space
2
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COMPACT SETS IN L2 FOR d2

Smoothness

(1,1)-BV
(1/q, ?)

?
Lq
L2
1/q
(1/2,0)
Lq Space
2
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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

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COHEN-DAUBECHIES-DAHMEN-DEVORE

• Partition growth into subtrees
• Decompose image

D j T j \ T j-1
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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)

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DTED DATA SURFACE
Grand Canyon
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POSTINGS
Postings
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FIDELITY

• L2 metric not appropriate

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FIDELITY

• L2 metric not appropriate
• L? better

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OFFSET

• If surface is offset by a lateral error of ?,
  the L? norm may be huge

L? error
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OFFSET

• But Hausdorff error is not large.

L? error
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CAN WE FIND dn(K)?

• K bounded functions dN(K) ? n-1 for
  Nnd1
• K continuous functions dN(K) ? n-1, for N nd
  log n
• K bounded variation in d1 dn(K) ? n-1
• K class of characteristic functions of convex
  sets
• dn(K) ? n-1

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Example functions in BV, d1

• 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
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ANTICIPATED IMPACTDTED

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

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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

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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

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EXAMPLE OF ELLIPTIC EQUATION

• POISSON PROBLEM

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CLASSICAL ELLIPTIC THEOREM

• Variational formulation gives energy norm Ht
• THEOREM If u in Hts then SFEM
  gives
• u-uh Ht lt hs uHts
• Can replace Hts by Bst (L2 )
• Approx. order hs equivalent to u in Bst (L2 )

.
.
)
h
8
8
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HYPERBOLIC

• Conservation Law
• ut divx(f(u))0, u(x,0)u0(x)
• THEOREM If u0 in BV then
• u(,,t)-uh(.,t)L1 lt h1/2 u0 BV
• u0 in BV implies u in BV this is equivalent
  to approximation of order h in L1

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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

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FORM OF NONLINEAR APPROXIMATION

• Good Theorem For a range of s gt0, 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

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AFEMs

• Initial partition P0 and Galerkin soln. u0
• General iterative step Pj Pj1 and uj
  uj1
• 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 Pj1

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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

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)
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BINEV-DAHMEN-DEVORE

• New AFEM Algorithm
• 1. Add coarsening step
• 2. Fundamental analysis of completion
• 3. Utilize principles of nonlinear
  approximation

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BINEV-DAHMEN-DEVORE

• THEOREM (BDD) Poisson problem, for a
  certain range of s gt0. 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).

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)
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ADAPTIVE WAVELET METHODS

• General elliptic problem
• Auf
• ???????????????????????????????????
• ????????????????????????????????????
• Problem in wavelet coordinates
• A u f
• A l2 l2
• Av v

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FORM OF WAVELET METHODS

• Choose a set ?? of wavelet indices
• Find Gakerkin solution u? from span????????
• Check residual and update ? ??

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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???? ? ?j1
• 3. Stop when residual error small enough

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ADAPTIVE WAVELETSCOHEN-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).

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CDD SECOND VIEW

• u n1 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

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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).

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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

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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