algorithmic treatment of nonlinear datamishra/systmod_presentations/...p. grohs jun-15-2012, liege...

Algorithmic Treatment of Nonlinear DataP. GrohsETH Zurich, Seminar for Applied Mathematics

Jun-15-2012, Liege University

Outline

MotivationWeighted Averages in ManifoldsWaveletsGeodesic Finite Elements

P. Grohs Jun-15-2012, Liege University p. 2

Nonlinear Data

Source: X. Pennec


Nonlinear Data

Source: I. UrRahman


Nonlinear Data


Nonlinear Data

Source: A. Ivancevic


Nonlinear Data

Source: G. Sundaramoorthi


And many more!

Spherical data (SAR inferogram)Grassmann manifold (array signal processing)Stiefel manifold (low rank approximations)Quaternions (motion design). . .


And many more!Spherical data (SAR inferogram)

Grassmann manifold (array signal processing)Stiefel manifold (low rank approximations)Quaternions (motion design). . .


And many more!Spherical data (SAR inferogram)Grassmann manifold (array signal processing)

Stiefel manifold (low rank approximations)Quaternions (motion design). . .


And many more!Spherical data (SAR inferogram)Grassmann manifold (array signal processing)Stiefel manifold (low rank approximations)

Quaternions (motion design). . .


And many more!Spherical data (SAR inferogram)Grassmann manifold (array signal processing)Stiefel manifold (low rank approximations)Quaternions (motion design)

. . .


And many more!Spherical data (SAR inferogram)Grassmann manifold (array signal processing)Stiefel manifold (low rank approximations)Quaternions (motion design). . .


Goals/Challenges

Interpolation and Approximation (also for inverseproblems)CompressionNoise RemovalFeature Extraction


Goals/ChallengesInterpolation and Approximation (also for inverseproblems)

CompressionNoise RemovalFeature Extraction


Goals/ChallengesInterpolation and Approximation (also for inverseproblems)Compression

Noise RemovalFeature Extraction


Goals/ChallengesInterpolation and Approximation (also for inverseproblems)CompressionNoise Removal

Feature Extraction


Goals/ChallengesInterpolation and Approximation (also for inverseproblems)CompressionNoise RemovalFeature Extraction


Desiderata

Objectivity (respecting natural invariances)Universality (valid for general manifolds)Theoretical Guarantees (same theoretical results asfor linear case)Tractability (same overall complexity as linear case)

In short, generalize linear tools to manifold-valued case.


DesiderataObjectivity (respecting natural invariances)

Universality (valid for general manifolds)Theoretical Guarantees (same theoretical results asfor linear case)Tractability (same overall complexity as linear case)



DesiderataObjectivity (respecting natural invariances)Universality (valid for general manifolds)

Theoretical Guarantees (same theoretical results asfor linear case)Tractability (same overall complexity as linear case)



DesiderataObjectivity (respecting natural invariances)Universality (valid for general manifolds)Theoretical Guarantees (same theoretical results asfor linear case)

Tractability (same overall complexity as linear case)In short, generalize linear tools to manifold-valued case.


DesiderataObjectivity (respecting natural invariances)Universality (valid for general manifolds)Theoretical Guarantees (same theoretical results asfor linear case)Tractability (same overall complexity as linear case)



The Central Problem

f1 : Ω→ M f2 : Ω→ M


The Central Problem

f1 : Ω→ M f2 : Ω→ M

f1 + f2 =??


The Central Problem

f1 : Ω→ M f2 : Ω→ M

f1 + f2 =??linear functionalanalysis!

classicalapproximationtheory!anything based onvector spaces!


The Central Problem

f1 : Ω→ M f2 : Ω→ M

f1 + f2 =??linear functionalanalysis!classicalapproximationtheory!

anything based onvector spaces!


The Central Problem

f1 : Ω→ M f2 : Ω→ M

f1 + f2 =??linear functionalanalysis!classicalapproximationtheory!anything based onvector spaces!


Weighted Two-Point Averages

f1 : Ω→ M f2 : Ω→ M


Weighted Two-Point Averages

f1 : Ω→ M f2 : Ω→ M

tf1(x) + (1− t)f2(x) = γf2(x)f1(x) (t)

(γqp (·) is the (locally) unique geodesic connecting the points p ∈ M and q ∈ M)


Observation: Actually, many constructions can be formulatedentirely using weighted averages∑

i

wipi ,∑

i

wi = 1,wi ∈ R!


Quadratic B-Splines

p−1

p0

p1

p2


Quadratic B-Splines

q−2

q−1

q0

q1

q2

q2i = 34 pi + 1

4 pi+1

q2i+1 = 14 pi + 3

4 pi+1


Quadratic B-Splines

Sp(i) =∑

j∈Z a(i − 2j)p(j) a = [. . . ,0, 14 ,

34 ,

34 ,

14 ,0, . . . ]


Quadratic B-Splines∑

j∈Z a(i − 2j) = 1


Quadratic B-Splines

limit function with control polygon


Example:motion design/interpolation

limit functionwith control polygon??


Example: finite elements

0 1

1

f (x) =∑

i ϕi (x)fiwith shape functions ϕi , satisfying∑

i ϕi (x) ≡ 1.

x-dependent weighted average of points fi !


Weighted Averages in Manifolds


Goal of this Section

Come up with geometric notion of weighted mean

p∗ =N⊕

i=1

wipi ∈ M

of points pi ∈ M with weights wi ∈ R, such that∑N

i=1 wi = 1.

Should also be efficiently computable!


Linear Averages Revisited

In Rd we have

p∗ =N∑

i=1

wipi ⇔N∑

i=1

wi−−→p∗pi = 0

Necessary: Computationof difference vectorbetween two points.

p∗

p1

p2

p3

p4

p∗ = 14 (p1 + p2 + p3 + p4)


Computation of Difference Vectors

(a) Retraction pair basedon exponential map

(b) Retraction pair basedon closest point projection

(c) Retraction pair basedon vertical projection

Figure: Different retractions for the circle.


Retraction Pairs

Definition (G (2012))

Two functions (f ,g),

f : TM → M, g : M ×M → TM,

such that

f (p,0) = p, f (p,g(p,q)) = q for all p, q ∈ M,

and∂

∂ξf (p,0) = Id

are called a retraction pair for the manifold M.

Standard example: f = expM and g = logM .


Constructing operations from retraction pairs

Point-vector addition p ⊕ ξ for ξ ∈ TpMdefined by f (p, ξ).

Point-point difference q p ∈ TpM such that

p ⊕ (q p) = q

defined by g(p,q). Weighted average

p∗ =⊕i∈I

wipi ,∑i∈I

wi = 1, and pi ∈ M.

defined implicitly by ∑i∈I

wig(p∗,pi ) = 0.


Constructing operations from retraction pairs Point-vector addition p ⊕ ξ for ξ ∈ TpM

defined by f (p, ξ). Point-point difference q p ∈ TpM such that

p ⊕ (q p) = q


p∗ =⊕i∈I

wipi ,∑i∈I



wig(p∗,pi ) = 0.



defined by f (p, ξ).

Point-point difference q p ∈ TpM such that

p ⊕ (q p) = q


p∗ =⊕i∈I

wipi ,∑i∈I



wig(p∗,pi ) = 0.




p ⊕ (q p) = q


p∗ =⊕i∈I

wipi ,∑i∈I



wig(p∗,pi ) = 0.




p ⊕ (q p) = q

defined by g(p,q).

Weighted average

p∗ =⊕i∈I

wipi ,∑i∈I



wig(p∗,pi ) = 0.




p ⊕ (q p) = q


p∗ =⊕i∈I

wipi ,∑i∈I



wig(p∗,pi ) = 0.


Well-Definedness

Theorem (G 2012)

The notion of average is locally well-defined for any retraction pair(f ,g).

M Riemannian manifold, f = expM , g = logM ; Karcher mean(Karcher ∼ 1970, Cartan ∼ 1930).

Computation is efficient as long as the compuation of f and g isefficient (simple algorithm: fixed-point iterationpn+1∗ ← f (pn

∗,∑

i wig(pn∗,pi )))!


Examples of Retraction Pairs


Sphere

M = x ∈ Rd : ‖x‖2 = 1TpM = ξ ∈ Rd : ξ>p = 0

expM(p, ξ) = cos(‖ξ‖2)p + sin(‖ξ‖2)ξ

‖ξ‖2

logM(p,q) = arccos(p>q)q − (p>q)p‖q − (p>q)p‖2


The Lie Group of Euclidean Motions

SE(d) = A =

(R x0 1

): x ∈ Rd ,R ∈ SO(3)

TASE(d) = ξ =

(ξ1 ξ20 0

): ξ1 ∈ Rd , ξ2 ∈ sl3

expSE(d)(A, ξ) = A exp(ξ)

logSE(d)(A,B) = log(A−1B)


The Symmetric Space of SPD-Matrices

SPD(d) = A ∈ Rd×d : A 0

TASPD(d) = ξ ∈ Rd×d : ξ> = ξ

expM(A, ξ) = A1/2 exp(A−1/2ξA−1/2)A1/2

logM(A,B) = A1/2 log(A−1/2BA−1/2)A1/2


The Stiefel Manifold

St(n,p) = A ∈ Rn×p : A>A = Ip

TASt(n,p) = ξ ∈ Rn×p : ξ>A+A>ξ = 0p

expM(A, ξ) = some complicated expression

logM(A,B) = no explicit expression known!

Important: All operations computable inorder pn operations (not order n2).

Sources: TOP: [Lubich, Nonnenmacher

2010], Bottom: [Sundaramoorthi etal

2011]


Closest Point Projection

Idea: Define f (A, ξ) by closest-point projection onto St(n,p) fromRn×p:

f (A, ξ) := π(A + ξ).

Algorithmic realization using polar decomposition of A + ξ:

π(A + ξ) = U,

whereA + ξ = UP

is the polar decomposition of A + ξ.

Efficient computation possible.


Computation of the Inverse

Theorem (G 2012)

Assume that −A>B is a Hadamard matrix (only eigenvalues in thenegative half-plane). Then

g(A,B) = 2B∫ ∞

0exp(−tA>B) exp(−tB>A)dt − A.

Fast computation is possible (for instance by alternating projections)!


Summary

We are able to define computationally accessible and flexible notionsof geometric means in virtually all (structured) manifolds of practicalinterest.


A first Application: Subdivision


Subdivision in SE(3)


Interactive Motion Design


Modeling in the Presence of Obstacles (Pottmann,Wallner 2006)


Smoothness EquivalenceSmoothness equivalence property: nonlinear limit function is assmooth as corresponding linear one.

Theorem (G, SIAM J. Math. Anal. (2010))

The smoothness equivalence property is true for the geometricanalog.

The aim is to establish a (d , f )-proximity condition of the form

‖∆d (T − S)p‖∞ . Ωf (p),

Ωf (p) :=∑j∈Γf

f∏k=1

‖∆k p‖jk∞,

Γf :=

j ∈ Nf

0 :f∑

i=1

iji = f , ‖j‖1 > 1

,

∆p(i) := p(i + 1)− p(i).A perturbation result shows that Cr -smoothness requires(r − 1, r + 1).


Smoothness Equivalence

DefinitionS with mask a has polynomial generation degree (PGD) (d , f ) if for allpolynomials p1, . . . ,pk with deg(

∏pj ) < f it holds that∏

j (Spj ) (i)− S(∏

j pj

)(i) = q(i) with some polynomial q of degree

< d .

TheoremAssume S has PGD (d , f ) and

‖∆j p‖∞ . Ω′j (p) j = 1, . . . ,d , (1)

then the geometric analog T satisfies a proximity condition of order(d , f ) with S. The geometric analog of a (stable) linear Cr scheme isCr provided that (1) holds.


Proof Sketch:Step 1 Taylor expansion of ∆d (T − S)p in p.Step 2 Reduction to existence of ’rewriting rule’ of terms∑

i∈Zk

AiΨ (pi1 , . . . ,pik )

as as linear combinations of terms of the form

Ψ(∆j1pi1 , . . . ,∆

jk pik), ‖j‖1 = f ,

Ψ, Ψ k -multilinear forms.Step 3 Translation of rewriting rule to algebraic properties of the

generating function A(x) of (Ai)i∈Zk (vanishing derivatives at(1, . . . ,1)).

Step 4 Show that these algebraic properties are equivalent to PGD oforder (d , f ).

Step 5 Show that stable S ∈ Cr satisfies PGD of order (r − 1, r + 1).


Main Result

TheoremThe smoothness equivalence true is true for the geometric analog.

Proof.Using the previous theorem we need to show that

‖∆j p‖∞ . Ω′j (p) j = 1, . . . , r − 1,


Main Result

TheoremThe smoothness equivalence property is true for the Riemanniananalog.

Proof.Using the previous theorem we need to show that

‖∆jTp‖∞ . Ω′j (p) j = 1, . . . , r − 1,

This is shown using a bootstrapping argument.


Approximation Order

DefinitionS has approximation order r if for all f ∈ Cr∥∥∥f − S∞f

∣∣hZ

( ·h

)∥∥∥∞

. hr+1.

Theorem (G, J. Approx. Theory (2010))

Approximation order properties of the linear scheme S carry over tothe nonlinear scheme T .

These results imply that almost anything that can be done withB-splines can be done in general manifolds.


LiteratureH. Karcher, “Mollifier Smoothing and Riemannian Center of Mass”,Comm. Pure and Appl. Anal., (1977).

J. Wallner and N. Dyn, “Convergence and C1-analysis of subdivisionschemes on manifolds by proximity.”, CAGD (2005).

P. Grohs, “A general proximity analysis of nonlinear subdivisionschemes”, SIAM J. Math. Anal. (2010).

O. Ebner, “Nonlinear Markov semigroups and refinement schemes onmetric spaces” arxiv:1112.6003 (2012).

P. Grohs, “Geometric multiscale decompositions of dynamic low-rankmatrices.”, SAM Report 2012-04, (2012).


Wavelet Transforms


Interpolatory Wavelets (Decomposition)given dense samples Pn(f ) := (f (i/2n))n∈Zof continuous function f


Interpolatory Wavelets (Decomposition)

downsampling



prediction by interpolatory subdivision schemep(·) 7→ Sp(·) =

∑j∈Z a(· − 2j)p(j)



store error



next scale


Geometric Multiscale Decomposition

Let Pnf (i) := f ( i2n ) and T a manifold-valued analog of the subdivision

scheme S (replace weighted averages by nonlinear averages).

DefinitionLet f : R→ M continuous. The geometric multiscale decompositionis defined as (

P0(f ),d1T (f ), . . .

),

wheredn

T (f ) := Pn(f ) TPn−1(f ).

We have

Pn(f ) = (. . .T (T (Pp0(f )⊕ d1T (f ))⊕ d2

T (f ))⊕ . . . )⊕ dnT (f ).


Norm Equivalences

Theorem (G-Wallner, Appl. Comp. Harmon. Anal. (2009))

Given S interpolatory with polynomial reproduction ≥ d andsmoothness ≥ r . Denote T its geometric analog. Let f : Zk → Mcontinuous with decomposition (P0

T (f ),d1T (f ), . . . ). Then

(i) If f ∈ Bα∞,∞ with α < d, then

‖dnT (f )‖∞ . 2−αn.

(ii) If‖dn

T (f )‖∞ . 2−αn.

with α < r , then f ∈ Bα∞,∞.


Stability

Is the nonlinear reconstruction procedure stable?

Yes, a typical result from (G, Constr. Approx. (2010)) for Riemannianmanifolds:Define for v, w ∈ TM the distance

dist(v, w) := ‖v − Ptπ(v)π(w)

(w)‖2.

Theorem (G, Constr. Approx. (2010))Given f ∈ Bα∞,∞ for some α > 0 with decomposition (P0(f ), d1

T (f ), . . . ). Let (Q0, e1, . . . ) be another sequence of coefficientswith ∥∥∥dist

(dn

T (f )(·), en(·))∥∥∥∞

. 2−αn.

Then there exists g ∈ Bα∞,∞ with

(P0(g), d1T (g), . . . ) = (Q0

, e1, . . . )

and‖f g(·)‖∞ . ‖P0(f ) Q0(·)‖∞ +

∑n≥1

∥∥∥dist(

dnT (f )(·), en(·)

)∥∥∥∞.


Stability

Is the nonlinear reconstruction procedure stable?Yes, a typical result from (G, Constr. Approx. (2010)) for Riemannianmanifolds:Define for v, w ∈ TM the distance

dist(v, w) := ‖v − Ptπ(v)π(w)

(w)‖2.

Theorem (G, Constr. Approx. (2010))Given f ∈ Bα∞,∞ for some α > 0 with decomposition (P0(f ), d1

T (f ), . . . ). Let (Q0, e1, . . . ) be another sequence of coefficientswith ∥∥∥dist

(dn

T (f )(·), en(·))∥∥∥∞

. 2−αn.

Then there exists g ∈ Bα∞,∞ with

(P0(g), d1T (g), . . . ) = (Q0

, e1, . . . )

and‖f g(·)‖∞ . ‖P0(f ) Q0(·)‖∞ +

∑n≥1

∥∥∥dist(

dnT (f )(·), en(·)

)∥∥∥∞.


Best N-term Approximation

Assume M a general manifold with operations ⊕, and weightedaverage induced by retraction pair.

Theorem (G 2012)

Let f : [−1,1]→ M, f∣∣[−1,0)

, f∣∣[0,1]∈ Cs. Then, using N wavelet

coefficients, one can approximate f up to an error of order N−s in theL2 norm.

This is as good as if there were no singularity!


Proof Sketch

Step 1: Collect coefficients not contributing to the singularity up toscale log2(N) which needs O(N) coefficients. By a result of[G-Wallner ACHA (2009)] these coefficients decay of order 2js, wherej is the scale.

Step 2: For each scale j there are only a fixed number L ofcoefficients which feel the singularity. Collect also these up to scale2s log2(N). Coefficient count: 2Ls log2(N) = O(N).

Step 3: Away from an interval I of size ∼ 2−2s log2(N) we have onlythresholded coefficients which decay of order 2−sj . By a result of [GConstr. Approx. (2010)], the reconstruction precedure is stable, whichimplies that on [0,1] \ I there is a uniform (and thus L2-) error of orderN−s.

Step 4: On I we have a uniformly bounded error. Since I is of size2−2s log2(N) ∼ N−2s, the L2 error is of the same order.


Proof SketchStep 1: Collect coefficients not contributing to the singularity up toscale log2(N) which needs O(N) coefficients. By a result of[G-Wallner ACHA (2009)] these coefficients decay of order 2js, wherej is the scale.

Step 2: For each scale j there are only a fixed number L ofcoefficients which feel the singularity. Collect also these up to scale2s log2(N). Coefficient count: 2Ls log2(N) = O(N).

Step 3: Away from an interval I of size ∼ 2−2s log2(N) we have onlythresholded coefficients which decay of order 2−sj . By a result of [GConstr. Approx. (2010)], the reconstruction precedure is stable, whichimplies that on [0,1] \ I there is a uniform (and thus L2-) error of orderN−s.

Step 4: On I we have a uniformly bounded error. Since I is of size2−2s log2(N) ∼ N−2s, the L2 error is of the same order.


Remarks

This result is optimal and as good as the corresponding result forEuclidean data.

Wavelet trafo is nonadaptive, compression goes simply bythresholding small coefficients; the singularity is foundautomatically.


Examples


Flight Data US Air Compression

original data 10 : 1 compression

Data source: www-stat.stanford.edu/$\sim$symmlab


www-stat.stanford.edu/$\sim $symmlab

Dynamical Low-Rank MatricesThe matrices of fixed rank form a smooth manifold:

Mn,p :=

A ∈ Rn×n : rank(A) = p.

Piecewise smooth curves γ : t 7→ A(t) ∈Mn,p can be decomposedinto smooth curves on the Stiefel manifold:

Theorem (Smooth Singular Value Decomposition)

Assume that γ as above is a (piecewise) Cs smooth curve. Thenthere exist (piecewise) Cs smooth curves

γU : t 7→ U(t) ∈ St(n,p), γV : t 7→ V(t) ∈ St(n,p), γS : t 7→ S(t) ∈ Rp×p

such thatA(t) = U(t)S(t)V(t)>.


Occurences

Video compression Latent semantic indexing Low-rank (tensor) approximation of high-dimensional evolution

equations


A Smooth Example

Smooth matrix curve

A(t) = U(t)Σ(t)V(t)> = sin (tE) + noise ∈ R10×10.

Use rank 2 approximation.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 17

6

5

4

3

2

7

6

5

4

3

original data

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 17

6

5

4

3

2

7

6

5

4

3

thresholded coefficients


0 20 40 60 80 100 120 140

0.16

0.18

0.2

0.22

0.24

0.26

coordinate plot

0 20 40 60 80 100 120 1402

4

6

8

10

12

14

16

18

20

22

two largest singular values


A singular Example

Singular matrix curve

A(t) = U(t)Σ(t)V(t)> = sin (tE) +(1− t2)1/2

cos (tF) ∈ R10×10.

Use rank 3 approximation.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 17

6

5

4

3

2

7

6

5

4

3

original data

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 17

6

5

4

3

2

7

6

5

4

3

thresholded coefficients


Real Data

A hyperspectral image is an image where intensities corresponding tomany different wavelengths are recorded separately. The result is aparametrized family A(t) of matrices, t being the wavelength. A(t) isa smooth curve!

Applications: Agriculture, Mineraology, Surveillance, ...


Real Data

A hyperspectral image is an image where intensities corresponding tomany different wavelengths are recorded separately. The result is aparametrized family A(t) of matrices, t being the wavelength. A(t) isa smooth curve!Applications: Agriculture, Mineraology, Surveillance, ...


Hyperspectral Image Compression

Given hyperspectal image A ∈ R144×144×128.1. Decompose A into small 12× 12× 128 blocks2. Approximate each block at each wavelength with a rank 2

approximation3. Perform wavelet thresholding on each low-rank approximation

Results in more than 90% saving with a relative error of order 0.01.


Further Work

Feature detection, inpainting, ... Wavelet analysis in curve shape space


LiteratureI. UrRahman etal., “Multiscale Representation of manifold-valueddata”, SIAM Mult. Mod. Sim., (2006).

P. Grohs and J. Wallner, “Interpolatory wavelets for manifold-valueddata”, ACHA (2009).

P. Grohs, “Stability of manifold-valued subdivision schemes andmultiscale transforms”, Constr. Approx. (2010).

O. Koch and Ch. Lubich, “Dynamical low-rank approximation” SIMAX(2007).

P. Grohs, “Geometric multiscale decompositions of dynamic low-rankmatrices.”, SAM Report 2012-04, (2012).


Geodesic Finite Elements1

1ongoing joint work with Prof. Oliver Sander and Hanne Hardening (FU Berlin)


Problem

Seek minimizer Φ : Ω→ M of manifold-valued variational problemargminJ(Φ), J : H1(Ω,M)→ R.

Applications:Liquid crystals: S2, PR2, SO(3)Cosserat shells: S2, SO(3)Image processing: S2

Problem: DiscretizationFinite elements cannot be used, because space M is nonlinearKey problem: enforce nonlinear constraints given by M


Recall...

0 1

1

f (x) =∑

i ϕi (x)fiwith shape functions ϕi , satisfying∑

i ϕi (x) ≡ 1.

x-dependent weighted average of points fi !


Geodesic Finite Elements

Let ϕpi be the p-th order scalar Lagrangian shape functions. ∆

reference simplex with coordinates ξ.

DefinitionGFE function

Υp(v , ξ) =⊕

i

ϕpi (ξ)vi .

GFE space associated to a simplicial decomposition: restriction toeach element must be GFE function. DOF’s: Control points vi ∈ M.


Results (in preparation)

Yields H1-conforming discretization. Nonlinear Bramble-Hilbert lemma (optimal approximation order

of GFE spaces). Nonlinear Céa lemma (coerciveness⇒ minimization of J on

GFE space yields quasioptimal approximation)


Harmonic Mappings

Harmonic energy

J(Φ) =

∫Ω

‖dΦ∣∣Φ(x)‖2

Φ(x)dx

with Dirichlet BC’s.

Used as simple model in liquid-crystal theory, coercivity results e.g.by [Haskins-Speight (2003)].


Ω = [0,1]2, M = S2

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

0.01 0.1 1

err

or

mesh size

L2 errorh1 errorO(h^2)O(h^3)

Source: Oliver Sander


Remarks

Discrete problems solved using Riemannian trust region schemedeveloped by Absil etal.

Method assumes existence of smooth solution which is notalways the case (but in many cases it is!)

Future work will also address time-dependent problems withpossible applications in image processing.


LiteratureP.-A. Absil, R. Mahoney and R. Sepulchre “Optimization algorithmson matrix manifolds”, Princeton University Press, (2008).

O. Sander., “Geodesic finite elements for Cosserat rods”, Int. J. Num.Meth. Eng., (2011).

O. Sander, “Geodesic finite elements on simplicial grids”, Int. J. Num.Meth. Eng. (2012) to appear.

P. Grohs, “Finite elements of arbitrary order and quasiinterpolation forRiemannian data”, SAM report 2011-56 (2011).

P. Grohs, H. Hardening and O. Sander, “A convergence theory forGFEs” in preparation.


The End

Questions?


algorithmic treatment of nonlinear datamishra/systmod_presentations/...p. grohs jun-15-2012, liege...

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