High Frequency Wave Propagation Using the Level Set Method

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High Frequency Wave Propagation Using the Level
Set Method

• Stan Osher
• Joint with
• L.-T. Cheng, M. Kang, H. Shim and Y. Tsai
• Supported by AFOSR

Paper(s) Available www.levelset.com
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GRID BASED GEOMETRIC OPTICS
Why? Problems with ray tracing
Diverging Rays Miss large regions
PDEs on grids have advantages e.g. other
physics can be easily attached, Self
interpolation, accurate finite difference schemes
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e.g. Scalar Wave Equation
Use ansatz
get, for
eikonal equation.
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We want the multivalued solution to this eikonal
equation. The eikonal equation is nonlinear, but
solutions to the wave equation superpose
linearly.
cross through each other
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This can be computed by the method of
characteristics from
Ray Tracing
Diverging Rays cause problems
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Grid Based PDE Approaches (Vidale,
Fatemi-Engquist-Osher, Benamou) Viscosity
solution of Hamilton-Jacobi eikonal equation
(Crandall-Lions,)
single valued, loses
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Patch together somehow uses upwind H.-J.
monotone, ENO, WENO schemes (O-Shu, O-Sethian,
Jiang-Peng) Self interpolates - Loses
multivaluedness need to patch things together.
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Level Set Method
Represented by
with
Implicit surface, use grid Key idea move
with velocity , get
Easy geometry extraction, topological changes
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Vector Level Set Method
e.g.
Intersection of 2 level sets can evolve by moving
, So curve in
can evolve by moving , .
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New idea, based on using vector valued level set
method to move high codimension stuff, e.g.
curves in (VVLSM invented by Burchard,
Cheng, Merriman O, 1999) (suggested as a
theoretical device by Ambrosio-Soner)
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Geometry of
(curvature) normal , on
All geometry comes simply from and
on their zero level sets restricted to
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Use Liouville (Vlasov??) equation Solve for
, with ,
, 2d 1 independent variables
linear equation, characteristics
same as for the eikonal equation along these
rays in 2D space is constant along
characteristic! To represent a curve
in space we let
is projection on space of

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Can lower the dimension by one
Thus we can use angle variables for d 2. Need
only
angle of normal to
Equations in 2D become
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Two linear, decoupled equations
NOTE if has disjoint components in
, , space at , they
never intersect at later in this (cotangent
space) representation. Also Note propagation
speed involves
so different time step restriction, generally a
bit more restrictive.
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In 3D we have spherical coordinates
Reduce it to a 5 dimension time
problem. System of 3 equations
Trouble at (north south
poles) if at such points.
Easily fixed.
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Complexity seems high. But we can use local
level set. In principle we are looking at a
manifold of dimension d-1 time. Should be
complexity to update,
also low storage. We now have that. Solve only
near where .
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Earlier work Engquist, Runborg,
Tornberg Use segment projection needs logic
many many segments can develop. Complexity is
the same, intricate programming Our method
Review for d 2
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Intensity
Easy (passive) calculation. Also 3D.
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Initialization
Given an initial surface via a level set
function Initialize In 3D
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Example
2D Start with an ellipse Say
Then
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Curve , is
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Other example in
2D Suppose initial data for eikonal equation is
given at Then simple
initialization
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3D example
Initially ellipse Can initialize

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Reflection let

Can solve reflection refraction easily in this
framework
, is level set function for
inwards normal to
, (in 2D).
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Reflection Treat as an
initial boundary value problem The value of u
corresponding to , which is
incoming value of u at is the angle of
the outwards normal. Discrete values of
are given so we interpolate in
. Transmission via Snells Law solve
initial boundary value problem and
Transmitted wave where in
.
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• More about
  reflection
• Give (x, y) lying on boundary of and any
  ,
• This translates into
• No boundary conditions needed for incoming ray
• (upwind differencing takes care of this)
• Reflection boundary conditions as above for
  reflected rays.
• Since we use x, y differencing, we
• Interpolate if an incoming ray upwinds the
  wrong way at the boundary
• use a subgrid interpolant to go to the boundary
  for the reflected wave
• Make sure that we stop at least away from
  this subgrid point
• Limit the ENO stencils so as not to cross the
  boundary
• Works well in 2 and 3D.

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Alternative for transmission Just solve
directly Have approximate delta function
coefficients of , just solve By smoothing
c slightly and restricting Works, but slower,
actually works even with
!! Automatically get Snells Law. 3D also
simple.
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Exciting New Stuff Computing Multiphase
Semiclassical Limits of Schrödinger Equation S.
Osher, S. Jin, Y.-H. Tsai, H. Liu and L.-T.
Cheng
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Define Use
vector level set method, get linear Liouville
system Solve for multivalued (Li) Let
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Compute single valued density
Then also satisfies
Liouville equation Can desingularize
problem Compute gives mass given ith
component of momentum New Approach No moments No
Wigner Transform Works in multi-dimension
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