Variation of Karcher Simplex Volume

12. Variation of Karcher Simplex Volume

Remark. <a> Note thatJ˙ andX˙ denote derivatives with respect to dierent direc

tions. In this section, the parameterrtakes the rôle oftin section 6.

<b> In the notation of Grohs et al. (2013),X˙ =∇1log(a, p).

Proof. ad primum: Without restriction, we can assume that p(t) and a(r) describe geodesics, as no second derivatives of them enter. Analogous to 1.23, consider a varia

tion of geodesics

c(r, s, t) := exp_{p(t) σ}^s(exp_p(t))⁻¹a(r) . As in 1.23, one can see that

X_p(t)|c(r,s,t)=_σ^s∂_sc(r, s, t), J_r=∂_r, J_t=∂_t

because∂r and∂tare Jacobi elds with the desired values ats= 0ands=σ. ad sec.: We have ¹_σ∇_a(r)˙ X˙_p(t) =DrDt∂sc(r, s, σ) = DtDr∂sc(r, s, σ) +R(∂r, ∂t)∂s, and becausec(r, s, σ) =a(r)is independent oft, we have∂t= 0there. And the former term isDtDs∂rc(r, s, σ) =DtJ˙r(σ), q. e. d.

12.4 Lemma. Situation as before. Abbreviate`:=d(a(r), p(t)),V := ˙a(r),W := ˙p(t) and T :=∂<sub>s</sub>. Then if C<sub>0</sub>`²< ^π₄²,

|DtJr| ≤90C0,1(r)^s(σ−s)_σ |V| |W| |T|(r)

≤90C_0,1(r)s|V| |W| |T|(r), |D_tJ˙_r| ≤50C_0,1(r)|V| |W| |T|(r).

Proof. The claim is similar to 6.6, and so is the proof: We again have to nd a dieren

tial equation forU :=DtJrto apply 6.5. By the usual laws of covariant dierentiation and∂r=Jr,∂t=Jt, we have

D_tJ¨_r=D_tD_sD_s∂_r=D_sD_tD_s∂_r+R(J_t, T) ˙J_r

=D_sD_sD_t∂_r+D_sR(J_t, T)J_r+R(J_t, T) ˙J_r

=D²_ssU+ (D_sR)(J_t, T)J_r+R( ˙J_t, T)J_r+ 2R(J_t, T) ˙J_r. On the other hand, using the Jacobi equation,

−DtJ¨r=DtR(Jr, T)T= (DtR)(Jr, T)T+R(U, T)T+R(Jr,J˙t)T+R(Jr, T) ˙Jt. So withA:=−R(·, T)T, we haveU¨ =AU +B, where||A|| ≤C₀|T|². Let us assume that we consider some geodesic with|T|= 1(as usual, the correct power of|T|follows from a scaling argument). Then, because 6.3 holds forJ_r as well as forJ_t,

|B|=|(D_sR)(J_t, T)J_r+ (D_tR)(J_r, T)T

+R( ˙J_t, T)J_r+ 2R(J_t, T) ˙J_r+R(J_r,J˙_t)T+R(J_r, T) ˙J_t|

≤2C1|V| |W|+ 15C01

σ|V| |W|.

The claim onDtJ˙r follows fromDtJ˙r=DtDs∂r=DsDt∂r+R(∂t, ∂r)∂s, q. e. d.

12. Variation of Karcher Simplex Volume 12.5 Lemma. Notation as before, andA_λ as in 5.7. For varying verticesp_i(t)∈M, let x_t

be the corresponding Karcher triangulations, andV =dx v. Then

Aλx˙t|λ=−λⁱX˙i|_xt(λ), Aλ∇Vx˙ =−vⁱX˙i−λⁱ∇²_V,x˙Xi−λⁱ∇VX˙i−Avx.˙ Proof. ad primum: Along t 7→ xt(λ), consider the vector eld U(t) := λⁱXi(t)|_xt(λ). As has been stated in 5.2, this vector eld vanishes for allr, soU˙ = 0. For those who believe, the shorthand proof is

0 =D_t|_t=0(λⁱX_i(t)|_xt) =λⁱD_t|_t=0(X_i(t)|_x0) +λⁱD_t|_t=0(X_i(0)|_xt)

=λⁱ( ˙X_i+∇x˙0X_i)|x0, (12.5a) a rather uncommon use of the chain rule. For all others, this argument is justied by a calculation in coordinates: LetλⁱXi=U^j∂j andx(t) = ˙˙ x^j(t)∂j be the needed coor

dinate representation. AsU^k(t) = 0for allkand allt, we haveU˙^k(t) = 0, and this is by usual Euclidean chain rule∂_tU^k(t, x¹(t), . . . , xⁿ(t)) +∂<sub>`</sub>U<sup>k</sup>(t, x<sup>1</sup>(t), . . . , x<sup>n</sup>(t)) ˙x<sup>`</sup>(t). And if allU^k vanish, we can addΓ^k<sub>`j</sub>U<sup>j</sup>x˙<sup>`</sup> without harm, which gives

0 = (∂tU^k+∂`U<sup>k</sup>x˙<sup>`</sup>+ Γ^k<sub>`j</sub>U<sup>j</sup>x˙<sup>`</sup>)∂k =DtU+∇x˙U.

ad sec.: Dierentiating 12.5a once more leads to

0 =λⁱ∇V∇x˙Xi+∇V(λⁱX˙i) =λⁱ∇²_V,x˙Xi+λⁱ∇_∇Vx˙Xi+vⁱX˙i+λⁱ∇VX˙i+vⁱ∇x˙Xi, q. e. d.

12.6 Proposition. Situation as in 12.2, and let the variation ofpibe given byp˙i(t) =dx wi

for some vectorwi∈T re. Then for u:=λⁱwi, we have

|x˙−dx u|.C_0,1⁰ h|x|,˙ |∇dx vx˙−dx∇^g_v^eu|.C_0,1⁰ h|u| |x|.˙

Proof. ad primum: At the vertexe_i of the standard simplex,d_eix wand the variation

˙

p_i agree. At another point λ∈∆,

dλx wi=P de_ix wi+O(C_0,1⁰ h²|wi|) =Pp˙i+O(C_0,1⁰ h²|wi|) by 6.24, which meansdx u=λⁱPp˙i+O(C_0,1⁰ h|u|)by denition ofu, and

˙

x=−λⁱX˙_i+O(C₀⁰h²|x|)˙ by 12.5

= λⁱPp˙i+O(C₀⁰h²|x|)˙ by 12.3 and 6.3.

ad sec.: The derivative ofuis, by Euclidean calculus, just∇^g_v^eu=vⁱw_i, so the latter term isdx∇^g_v^eu=vⁱdx w_i. The covariant derivative∇²X_iin 12.5 is estimated by 6.14, and the∇VX˙ term by 12.4, so we have

|Aλ∇dx vx˙ −vⁱdx wi| ≤vⁱ|X˙i+dx wi|+λⁱ|∇²_V,x˙Xi|+λⁱ|∇VX˙i|+|Avx|˙ .C0h²vⁱ|p˙i|+C_0,1⁰ h|v| |x|,˙

q. e. d.

Discrete Vector Fields

12.7 Denition. For a piecewise barycentric mappingx:rK→M, let PX¯ x:=T M|x(K⁰)=G

pi,i∈K⁰

Tp_iM

be the disjoint union of all vertex tangent spaces. For U¯ = (U_i) ∈ PX¯ _x, dene a piecewise interpolation: It induces a variation ofxby deningx_t[ ¯U]to be the piecewise barycentric mapping with respect to verticesexp_p

itU_i (where we keept so small that x(λ) and xt(λ) stay in a convex neighbourhood of each other). We call U¯ 7→ U :=

˙

xt[ ¯U]|t=0 theP¹-interpolation ofU¯ and

P¹Xx:={U: ¯U ∈PX¯ x}

the set of piecewise smooth, globally continuous test vector elds.

12.8 Observation. As a nite sum of vector spaces with scalar products g and g^e, PX¯ _x carries the natural inner products

`²ghhhV ,¯ W¯iii=X

i

ghVi, Wii, `²g^ehhhV ,¯ W¯iii=X

i

g^ehve,i, we,ii,

whereasP¹Xxhas the scalar productsL²g andL²g^ethat are induced fromL²X(T M): L²ghhhV, Wiii=

ˆ

x(rK)

ghV, Wi, L²g^ehhhV, Wiii= X

e∈Kⁿ

ˆ

re

g^eh¯v,wi,¯

where V = dxv¯ and W = dxw¯. As both are isomorphic nite-dimensional vector spaces, all these norms are equivalent. The equivalence constants between`<sup>2</sup>gand`²g^e and betweenL²g andL²g^e are the ones from 6.17a and 7.3a, whereas the equivalence constant betweenL²g^eand`²g^edepends on the maximal and minimal simplex volume.

12.9 Denition. Situation as in 12.2, and U¯ ∈ PX¯ _x. Inside every simplex e ∈ K^m, the vectorUi,i∈e, can be represented asUi=drix_ew^e_i (without any summation). Dene a piecewise linear, globally discontinuous vector eld u|re := λⁱw_i^e and a piecewise smooth vector eldu¯ by requiringdxu¯=U everywhere.

12.10 Conclusion. By denition,dx∇^x_v^∗gu¯=∇dx vx˙t[ ¯U]. From 12.6, we hence know that

|u−u|¯ .C_0,1⁰ h|u|, |∇^x_v^∗gu¯− ∇^g_v^eu|.C_0,1⁰ h|v||u|,

where all norms are | · |x^∗g norms. The same estimates hold for the jump [dx u]f = dxeλⁱw^e_i−dxe⁰λⁱw^e_i⁰ ofuacross a facet f=e∩e⁰.

12. Variation of Karcher Simplex Volume Area Differentials

12.11 Observation. Situation as in 12.2. If the verticespi of a Karcher triangulation vary

smoothly with velocity(Ui)∈PX¯ x, the area change ofS =x(rK)¯ is also smooth, hence has a dierential

dvolK¯ : ¯PXx→R. (12.11a) The volume is additive ins∈K¯ⁿ, and the variations of dierent vertices are linearly independent, so it suces to compute the dierentialdvolⁱ_s,g :TrirK→R of|x(rs)|g

with respect to the variation of x(ri), i ∈ s. Correspondingly, let dvolⁱ_s,ge be the analogous dierential of|s|g^e.

Remark. We do not think that a notational distinction between this area dierential and the volume form from 9.6 is neccessary. For readers who disagree, we remark that in 12.11a, the d denotes a dierential and is hence written in italics, whereas it is upright as part of the volume formdvol.

12.12 Proposition. Situation as in 12.2. Then|dvolⁱ_s,g−dvolⁱ_s,ge|.C_0,1⁰ h|s|_ge.

Proof. By 12.1,dvolⁱ_s,g(w) =−´

div^SZ, where Z= ˙xis induced by the vertex varia

tionp˙i=dx w. Ifv˜jform ag-orthogonal basis ofT rs, this is´

h∇dx˜v_jx, dx˙ ˜vji. By the comparison of volume elements forgandg^e in 3.20,

dvol^j_s,g(w) =− ˆ

x(rs),g

gh∇dxv˜_jx, dx˙ ˜vji=−(1 +O(C₀⁰h²)) ˆ

x(rs),g^e

gh∇dx˜v_jx, dx˙ v˜ji,

and, noting that there is a g^e-orthonormal basisvj ofT rs with |vj−˜vj| .C₀⁰h² by 3.6, the integrand is

gh∇^g_dx˜v

jx, dx˙ ˜v_ji=x^∗gh∇^g_˜vj^eu,v˜_ji+O(C_0,1⁰ h|u|) by 12.6

=x^∗gh∇^g_vj^eu, vji+O(C_0,1⁰ h|u|) +O(C₀⁰h²||∇u||)

=g^eh∇^g_v^e

ju, v_ji+O(C_0,1⁰ h|u|) +O(C₀⁰h²||∇u||),

and the last right-hand-side term isdiv^(rs,ge)u, q. e. d.

12.13 Remark. <a> It is common knowledge that dvolⁱ_∆ = dλⁱ|∆|, proven by inserting

divu= dλⁱ(w) for u =λⁱw into 12.1. Classically, one says for triangles (Polthier 2002, eqn. 4.3) that the gradient of the area functional with respect to vertex variations is the ^π₂ rotation of the opposite edge vector, which is exactly (dλⁱ)^].

One has to take care to transfer this to the subsimplex situation, becausediv^(rs,ge)u

= 0ifwis perpendicular tors, so one needs a formdλⁱ_sthat acts likedλⁱ onT rsand vanishes onT rs^⊥. For example, ifrs= conv(e0, . . . , en)andv0= gradλ⁰∈T∆,

dλⁱ_s(w) =dλⁱ

w−hw0, wi

|v₀|² v0

=wⁱ−_|dλ^w0⁰|²hdλⁱ, dλ⁰i.

It is easier to transfer the gradientv_sⁱ ofλⁱinrstoT re, as its vector components stay the same:div^(s,ge)u=g^ehv_sⁱ, wi. By 3.2a, thisv_sⁱ is characterised as the vector inT rs that is perpendicular to the facets\ {i}oppositeiwith lengthsh⁻¹_i , the reciprocal of rs's height aboves\ {i}from 3.2a.

<b> For computational purposes, 12.12 is insatisfactory, as onlydvols,g^e(u)is numeri

cally accessible, notdvols,g^e(¯u). But this is only an easy combination with 12.10, which will be spelled out for the area gradients in the following paragraphs.

Area Gradients

12.14 Observation. The area dierentialsdvols,ganddvols,g^e can be expressed as gradient with respect to dierent norms on PX¯ x. The gradients with respect to `<sup>2</sup>g and `²g^e correspond to the mean curvature vector of Polthier (2002), whereas the gradients with respect to orL²gandL²g^egive the construction from Dziuk (1991).

12.15 Corollary. IfH_{`</sub>2g∈Tp<sub>i</sub>M is the gradient of|x(rs)|<sub>g</sub>with respect to a variation ofpi, andH<sub>`}2g^e=|s|_gegradλⁱ is the corresponding gradient of|s|_ge, then|dx H`<sup>2</sup>g<sup>e</sup>−H<sub>`</sub>2g| .C_0,1⁰ h|H_`2g^e|.

12.16 Denition. The discrete mean curvature vectorH_L2g ∈P¹Xx is the solution of L²ghhhH_L2g, Viii= dvolK,g¯ (V)for allV ∈P¹X. The approximate mean curvature vector HL²g^e∈P¹Xxis the solution ofL²g^ehhhHL²g^e, Viii=dvolK,g¯ ^e(V)for allV ∈P¹X. 12.17 Observation. The dierentials of the right-hand sides can be represented asL² prod

ucts of linear maps: If vectors Zi =dx wi ∈Tp_iM induce a variationx˙ = dxz¯ of x, and if we denez=λⁱwi, as well asdλ:ei7→gradλⁱ, then

L²ghhhH_L2g, Ziii=hhhdx,∇^x∗gziii¯ _L2g(T rK⊗x^∗T M), (12.17a) L²g^ehhhHL²g^e, Ziii=hhhdλ,∇^geziiiL²g^e(T rK⊗x^∗T M). (12.17b) To see 12.17a, we take an orthonormal basisE_i=dx y_i indiv^SZ=h∇EiZ, E_ii. Then we have an integrand of the formhα(yi), β(y_i)ifor two linear mapsα, β. A computation in coordinates easily shows that this ishα, βi.

For 12.17b, the computation is even simpler:z=λⁱv_i (without summation) for the gradientvi ofλⁱ has derivative∇z=vi⊗dλⁱ, which maps ek to vk, sodivz=v_iⁱ = h∇e_iz,gradλⁱi.

12.18 Proposition. Situation as before. Then HL²g−HL²g^e L².C_0,1⁰ h(1 +h HL²g^e L²). Proof. Similar to 10.13: The functionals on the right-hand side of 12.17b only dier by a factor of 1 +O(C_0,1⁰ h|S|), and the bilinear forms on the left fulll |L²ghU, Vi − L²g^ehU, Vi|.C₀⁰h² U V , so

H_L2g−H_L2g^e 2

L²g=L²ghHL²g−H_L2g^e, H_L2g−H_L2g^ei

≤ |L²ghH_L2g, H_L2g−H_L2g^ei −L²g^ehH_L2g^e, H_L2g−H_L2g^ei|

+|(L²g−L²g^e)hHL²g^e, HL²g−HL²g^ei|

.|(dvolK,g¯ −dvolK,g¯ ^e)(H_L2g−H_L2g^e)|

+C₀⁰h² H_L2g^e H_L2g−H_L2g^e

Im Dokument numerical approximation in riemannian manifolds by karcher means (Seite 97-103)