Approximating Gradient Flows on Surfaces via MM + CPM
[Ginder, Elliott; Svadlenka, Karel; and Muramatsu, Takuma (2024). Approximation and Application of Minimizing Movements for Surface PDE. arXiv preprint arXiv:2409.03139. See Section 3.2, pp. 7–9. Available at https://arxiv.org/abs/2409.03139.]
MM - Minimizing Movements
CPM - Closest Point Method
Let
surface
closed
smooth
boundary-less
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(e.g. surfaces of sphere and torus are boundary-less surfaces)
bounded domain of Euclidean space
with suitable boundary (e.g. Lipschitz boundary)
u/uSu / u^S gives each point of the whole domain / surface scalar value in some time t∈[0,T]t\in\left[0,T\right]
temperature / energy
displacement
Heat Equation
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heat dissipation on flat\small\textbf{flat} 2D surface
Wave Equation
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wave propagation through flat\small\textbf{flat} 2D surface
(red - positive displacement, blue - negative)
thermal diffusivity
wave propagation coeff.
on Euclidean domain Ω\Omega
on curved surface SS
ΔS\Delta_S - Laplace-Beltrami operator
(surface version of Laplacian operator Δ\Delta)
Dirichlet Boundary Conditions
"all the values on the boundary of our domain are fixed"
homogenous
are equal to =0= 0
we need to define boundary values for bounded domain
Time discretization
Let
time step size
Taylor theorem
Lagrangian functional
for n-dimensional PDE
Lagrangian density
extremum of the functional (we are looking for uu with minimal evaluation)
Euler-Lagrange equation
Dirichlet energy functional
spatial term in gradient flows is always Dirichlet energy
Minimizing Movements
approximating gradient flow by iteratively minimizing functionals
F\mathcal{F} is functional - accepts function uu and returns its scalar evaluation
in MM, for next step n+1n+1 we are searching for uu that is of lowest such evaluation
it would be costly and tricky to numerically approximate field uSu^S across only the surface (we would need to create mesh etc.)
Closest Point Method
therefore for a given n-dimensional arbitrary surface
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we create a smooth extension, by extending the values of uSu^S in the direction of normal vectors coming from the surface
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for each point that belongs to a normal vector, we give same value as the vector's origin which also is its closest point
(if normal vectors intersect with each other, we can always make them shorter such that they do not intersect)
we therefore construct functionals for our band ΩS\Omega_S around the surface SS
notice that the new uu doesn't return the same values for points lying along surface normals in ΩS\Omega_S
but with each step CSC_S maps x\boldsymbol{x} to its nearest surface point for previous uu's
now we take Gateaux derivative of the functional at =0=0 to yield Euler-Lagrange equation = implicit time-step scheme for the time-discrete approximation
showing that it works with CSC_S - Closest Point Method
Gateaux derivative
infinitely differentiable on ΩS\Omega_S
with compact support in ΩS\Omega_S
when set to 0, allows us to find such uu at which F\mathcal{F} attains extremum (is stationary)
similarly as with classical derivative set to 0, we can find all xx in which the ff attains extremum
Divergence Product Rule
Divergence Theorem
total source flow in the volume = net flow through the surface (its boundary)
dVdV is often abbreviated as dxd\boldsymbol{x} and both are practically interchangeable
(we did projection of rate of change ∇u\nabla u onto outward normal vector ν\boldsymbol{\nu})
supp(ϕ)\text{supp}\left(\phi\right) is compact subset of ΩS\Omega_S
compact ⇒\Rightarrow closed
ΩS\Omega_S open
(definite integral ⇒\Rightarrow no CC constant)
Fundamental Lemma of the Calculus of Variations
Implicit approximations to the surface PDEs via MM + CPM ■\blacksquare
Viktor Futó
Viktor Futó