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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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Who’s 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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Model

“Real Images”

Metric

“Human Visual System”

Stochastic Mathematical MetricDeterministic

Lp NormsSmoothness Classes K Lp Norms

IMAGE PROCESSING

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Given > 0, N (K) smallest number of balls that cover K

Kolmogorov Entropy

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Kolmogorov Entropy

Given > 0, N (K) smallest number of balls that cover K

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Kolmogorov Entropy

Given > 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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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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Lp

•

(1/p,0)(1/p,0)Lq 1/q1/q

•

SmoothnessSmoothness

Lq Space

(1/q, (1/q, ))

22

COMPACT SETS IN Lp FOR d=2

Sobolev embedding line 1/q= /2+1/p

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L2

(1,1)(1,1)--BVBV

• •

(1/2,0)(1/2,0)Lq 1/q1/q

•

SmoothnessSmoothness

Lq Space

(1/q, (1/q, ))

22

COMPACT SETS IN L2 FOR d=2

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ENTROPY OF K

Entropy of Besov Balls B (Lq ) in Lp is nd

Is there a practical encoder achieving thissimultaneously 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 j :=T j \ T j-1

f = f = cI I I j j

[T0 | B0 | S0 | T1 | U1 | B1 | S1 | T2 | U2 | B2 | S2 | . . . ]Lead tree & bits Level 1

tree, update & new bits, signs Level 2tree, update & new bits, signs

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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

Grid Z-Values

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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.

Hausdorff error

L error

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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

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Example: functions in BV, d=1

Assume f monotone; encode first (jk) and last

(jk) square in column. Then k |jk-jk| M n.

Can encode all such jk with C M n bits.

k

jk

jk

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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 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 )

8

8h

..)

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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

< 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 >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 freedomBest 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 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

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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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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 >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).

..)

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ADAPTIVE WAVELET METHODS

General elliptic problem

Problem in wavelet coordinates A u= f A:2 2

||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~

j+1

·3. Stop when residual error small enough

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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).

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CDD: SECOND VIEW

u n+1 = u n - (A u n -f )

This infinite dimensional iterative process converges

Find fast and efficient methods to computeAu 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