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Applications of the Hodge Decomposition to Biological Structure and Function Modeling Applications of the Hodge Decomposition to Biological Structure and Function Modeling Andrew Gillette joint work with Chandrajit Bajaj and John Luecke Department of Mathematics, Institute of Computational Engineering and Sciences University of Texas at Austin, Austin, Texas 78712, USA http://www.math.utexas.edu/users/agillette university-logo Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 1/32 University Introduction Molecular dynamics are governed by electrostatic forces of attraction and repulsion. These forces are described as the solutions of a PDE over the molecular surfaces. Molecular surfaces may have complicated topological features affecting the solution. The Hodge Decomposition relates topological properties of the surface to solution spaces of PDEs over the surface. ∼ (space of forms) = (solutions to ∆u = f 6≡ 0) ⊕ (non-trivial deRham classes)university-logo Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 2/32 University Outline 1 The Hodge Decomposition for smooth differential forms 2 The Hodge Decomposition for discrete differential forms 3 Applications of the Hodge Decomposition to biological modeling university-logo Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 3/32 University Outline 1 The Hodge Decomposition for smooth differential forms 2 The Hodge Decomposition for discrete differential forms 3 Applications of the Hodge Decomposition to biological modeling university-logo Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 4/32 University Differential Forms Let Ω denote a smooth n-manifold and Tx (Ω) the tangent space of Ω at x. A k-form ω is a mapping from Ω to the space of alternating k-tensors on the tangent space of Ω at the input point: k k ∗ ω : Ω → Λ [Tx (Ω) ], ω(x) : Tx (Ω) → R, Mi=1 where ω(x) is an alternating k-tensor. The space of k-forms is denoted Λk (Ω). Alternate Characterization of k-forms A k-form represents an intrinsically k-dimensional phenomena and can be integrated over a k-dimensional region. q u E B electric electric fields are magnetic fields are charge density is potential is valued based on a are dual to electric fields valued over a volumeuniversity-logo point−valued linear current flow and valued on planes Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 5/32 University The Exterior Derivative Operator The exterior derivative operator denoted by d is d : Λk (Ω) → Λk+1(Ω), defined as follows. Let I := {i1,..., ik } denote an increasing sequence of k indices (ij < ij+1) and let dxI = dx1 ∧···∧ dxk . Given ω = I aIdxI define P ∂aI dω := daI ∧ dxI where daI := dxi ∂xi XI Xi∈I The exterior derivative generalizes the familiar operators grad, curl, and div for Ω= R3: d d d 0 −−−−−→ Λ0(Ω) −−−−−→ Λ1(Ω) −−−−−→ Λ2(Ω) −−−−−→ Λ3(Ω) −−−−−→ 0 grad curl div Example: Given f ∈ Λ0(R3), we have f : R3 → R. Accordingly: ∂f ∂f ∂f ∂f ∂f ∂f grad f := , , = dx + dy + dz =: df „ ∂x ∂y ∂z « ∂x ∂y ∂z university-logo Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 6/32 University Stokes’ Theorem Stokes’ Theorem Given a compact, oriented n-dimensional manifold Ω with boundary ∂Ω and a smooth (n − 1) form ω on Ω, the following equality holds: ω = dω Z∂Ω ZΩ Alternate Characterization of d Let ω be a k-form on a compact oriented n-manifold Ω (0 ≤ k < n). Then dω is the unique (k + 1)-form such that on any (k + 1)-dimensional submanifold Π ⊂ Ω the following holds dω = ω ZΠ Z∂Π university-logo Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 7/32 University Inner Products We would like to construct an operator going the other way in the sequence: δ δ δ δ 0 ←−−−−− Λ0(Ω) ←−−−−− Λ1(Ω) ←−−−−− Λ2(Ω) ←−−−−− Λ3(Ω) ←−−−−− 0 To do so, we must first fix an inner product on our space. An inner product (·, ·)V on a vector space V is linear in each variable, symmetric, and non-degenerate, i.e. given α ∈ V : (α, β)= 0 ∀β ⇐⇒ α = 0. An inner product (·, ·)V on V extends to an inner product (·, ·) on Λ(V ) Let {e1,..., en} be an orthonormal basis of V . We say that elements of the form ei1 ∧ . ∧ eik have grading k. Define the inner product of elements with different gradings to be zero. Define (ei1 , ej1 )V · · · (ei1 , ejk )V e e e e det 0 . .. 1 1 0 1 ( i1 ∧ . ∧ ik , j1 ∧ . ∧ jk ) := . ∈ {− , , } B (e , e ) · · · (e , e ) C @ ik j1 V ik jk V A Extend the above definitions bilinearly for arbitrary elements of Λ(V ). university-logo Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 8/32 University The Hodge Star Operator The Hodge Star operator denoted by ⋆ is ⋆ : Λk (Ω) → Λn−k (Ω). Let {e1,..., en} be an orthonormal basis for V with positive orientation. For 0 < k < n, let σ ∈ Sn satisfy σ(1) < · · · <σ(k) and σ(k + 1) < · · · <σ(n). Define ⋆ eσ(1) ∧ . ∧ eσ(k) := sign(σ) eσ(k+1) ∧ . ∧ eσ(n) . ` ´ ` ´ For k = 0 and k = n, ⋆ is defined by ⋆(1)= ±e1 ∧···∧ en and ⋆(e1 ∧···∧ en)= ±1. Example: ⋆dxdz = ⋆dx1 ∧ dx3 = sign (1 3 2)dx2 = −dx2 = −dy Alternate Characterization of ⋆ The Hodge Star is the unique operator satisfying the relationship α ∧ ⋆β =(α, β)µ ∀α, β ∈ Λk (V ), where µ := e1 ∧···∧ en is the volume element of Λ(V ). university-logo Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 9/32 University The Codifferential Operator The (exterior) codifferential operator denoted by δ is δ : Λk (Ω) → Λk−1(Ω) defined as the formal adjoint to d with respect to the inner product (·, ·) on Λ(Ω). That is, given η ∈ Λk (Ω), δη is the unique element of Λk−1(Ω) satisfying (dω, η)=(ω,δη) for all ω ∈ Λk−1(Ω). Alternate Characterization of δ Given an inner product on Λ(V ), where V has dimension n, the action of δ on a k-form is given by δ =(−1)nk+1 ⋆ d⋆ Example: δ(xyzdxdz)=(−1)3(2)+1 ⋆ d(−xyzdy)= ⋆d(xyzdy) = ⋆(yzdxdy − xydydz)= yzdz − xydx university-logo Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 10/32 University The Laplacian Operator The Laplacian operator denoted by ∆ is ∆ : Λk (Ω) → Λk (Ω) given by ∆ := dδ + δd Note that ∆ reduces to the typical Laplacian operator when k = 0. By the definitions, d0 = ∇ and δ1 = div. Hence n ∂2 ∆0 = d 1δ0 + δ1d0 = δ1d0 = div∇ = ∇·∇ = . − ∂x 2 Xi=1 i Therefore, understanding solutions to the differential form problem ∆ω = α for a given form α will shed light on the Laplace and Poisson PDE problems. university-logo Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 11/32 University The Hodge Decomposition The space of harmonic k-forms is Hk := {ω ∈ Λk (Ω) : ∆ω = 0} The Hodge Decomposition Theorem For each integer k with 0 ≤ k ≤ n, Hk is finite dimensional and we have the following orthogonal direct sum decompositions of Λk (Ω) of smooth k-forms on Ω: Λk (Ω) = ∆(Λk ) ⊕ Hk = dδ(Λk ) ⊕ δd(Λk ) ⊕ Hk = d(Λk−1) ⊕ δ(Λk+1) ⊕ Hk . Consequently, the equation ∆ω = α has a solution ω ∈ Λk (Ω) if and only if the k-form α is orthogonal to the space of harmonic k-forms. university-logo Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 12/32 University Relation to Cohomology Recall that the deRham complex on Ω is d d d − 0 −−−−−→ Λ0(Ω) −−−−−→0 Λ1(Ω) −−−−−→·1 · · −−−−−→n 1 Λn(Ω) −−−−−→ 0. k The kth deRham cohomology group is denoted HdR(Ω) and is given by k HdR := ker dk /imdk−1 Lemma Let Ω be a compact oriented Riemannian n-manifold Ω without boundary and 0 ≤ k ≤ n. Then k k HdR(Ω) =∼ H . Abbreviated Proof: k k−1 k k−1 k HdR = d(Λ ) ⊕ H /d(Λ ) =∼ H h i university-logo Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 13/32 University Summary of Smooth Hodge Decomposition Facts Summary of results in the smooth setting: k-forms are meant to be integrated over k dimensional regions d : Λk → Λk+1 is characterized by Stokes’ theorem ⋆ : Λk → Λn−k is characterized by α ∧ ⋆β =(α, β)µ δ : Λk → Λk−1 is characterized by δ =(−1)nk+1 ⋆ d⋆ The Hodge Decomposition is Λk = dΛk−1 ⊕ δΛk+1 ⊕ Hk Solutions to the Poisson equation ∆ω = α lie in dΛk−1 ⊕ δΛk+1 Solutions to the Laplace equation ∆ω = 0 are exactly the harmonic forms Hk Each k-cohomology class of Ω has a unique harmonic k-form representative. How can we recreate these results on a discrete level for computational modeling? d0 d1 d2 d3 smooth: 0 o Λ0 o / Λ1 o / Λ2 o / Λ3 / 0 δ0 O δ1 O δ2 O δ3 O discrete: 0 / 0 / 1 / 2 / 3 / 0 C C C C university-logo Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 14/32 University Outline 1 The Hodge Decomposition for smooth differential forms 2 The Hodge Decomposition for discrete differential forms 3 Applications of the Hodge Decomposition to biological modeling university-logo Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 15/32 University Discrete Differential Forms k-forms are meant to be integrated over k dimensional regions Let T be a triangulation of a smooth compact n-manifold Ω.
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