Outline. Level Set Methods. For Inverse Obstacle Problems 4. Introduction. Introduction. Martin Burger
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1 For Inverse Obstacle Problems Martin Burger Outline Introduction Optimal Geometries Inverse Obstacle Problems & Shape Optimization Sensitivity Analysis based on Gradient Flows Numerical Methods University of California Los Angeles University Linz, Austria For Inverse Obstacle Problems 2 Introduction Many applications deal with the reconstruction and optimization of geometries (shapes, topologies); e.g.: Identification of piecewise constant parameters Inverse obstacle scattering Inclusion detection Structural optimization Optimal design of photonic bandgap structures... Introduction In such applications, there is no natural a- priori information on shapes or topological structures of the solution (number of connected components, star-shapedness, convexity,...) flexible representations of the shapes needed! For Inverse Obstacle Problems 3 For Inverse Obstacle Problems 4
2 Osher & Sethian, JCP 1987 Osher & Fedkiw, Springer, 2002 Basic idea: implicit shape representation Evolution of a curve with velocity Implicit representation with continuous level-set function For Inverse Obstacle Problems 5 For Inverse Obstacle Problems 6 Geometric Motion Tangential velocity corresponds to change of parametrization only, i.e. Geometric Motion Normal can be computed from level set function : Restriction to normal velocities is natural: For Inverse Obstacle Problems 7 For Inverse Obstacle Problems 8
3 Geometric Motion Evolution becomes nonlinear transport equation: In general, normal velocity may depend on the geometric properties of, e.g. Geometric Motion is homogeneous extension. Fully nonlinear parabolic equation For Inverse Obstacle Problems 9 For Inverse Obstacle Problems 10 Geometric Motion Classical geometric motions: Eikonal equation Geometric Motion Mean curvature flow computes minimal arrival times in a velocity field v For Inverse Obstacle Problems 11 For Inverse Obstacle Problems 12
4 Viscosity Solutions In general, nonlinear parabolic and Hamilton-Jacobi equations do not have classical solutions. Standard notion of weak solutions are viscosity solutions. First-order Hamilton-Jacobi (Crandall-Lions) For Inverse Obstacle Problems 13 Viscosity Solutions Viscosity subsolution Viscosity supersolution Viscosity solution = subsolution + supersolution For Inverse Obstacle Problems 14 Viscosity Solutions Typical type of regularity Viscosity Solutions Comparison Theorems For Inverse Obstacle Problems 15 For Inverse Obstacle Problems 16
5 Properties of Level Sets Level sets are independent of chosen initial value: Properties of Level Sets Comparison: In particular For Inverse Obstacle Problems 17 For Inverse Obstacle Problems 18 Higher-Order Evolutions Comparison results still hold for second order evolutions like mean curvature. Higher-Order Evolutions Mullins-Sekerka: No comparison results for higher order evolutions, e.g. surface diffusion (4th order) (3rd order) No global level set method! For Inverse Obstacle Problems 19 For Inverse Obstacle Problems 20
6 Computing Viscosity Solutions First-order equations Explicit time discretization Computing Viscosity Solutions As in numerical schemes for conservation laws, first-order Hamilton-Jacobi equations are solved by a scheme of the form Stability bound CFL-condition with approximate numerical flux - analogous to conservation laws (Godunov, Lax- Friedrichs, ENO, WENO) For Inverse Obstacle Problems 21 For Inverse Obstacle Problems 22 Computing Viscosity Solutions Mean curvature type equation Computing Viscosity Solutions Discretization with linear finite elements ( piecewise constant) Set Convergence to viscosity solution as (Deckelnick, Dziuk, 2002) For Inverse Obstacle Problems 23 For Inverse Obstacle Problems 24
7 Redistancing In order to prevent fattening and for initial values, should be close to signed distance function. is limit of solving Redistancing Upwind scheme, first order as 1994) (Osher, Sussman, Smereka, For Inverse Obstacle Problems 25 For Inverse Obstacle Problems 26 Redistancing Velocity Extension In many cases, natural velocity is given on the interface only. Simplest extension is constant in normal direction: Extension velocity is the limit of the linear transport equation For Inverse Obstacle Problems 27 For Inverse Obstacle Problems 28
8 Velocity Extension Velocity Extension Upwind scheme, first order For Inverse Obstacle Problems 29 For Inverse Obstacle Problems 30 Example: Eikonal Equation Optimal Geometries For Inverse Obstacle Problems 31 For Inverse Obstacle Problems 32
9 Optimal Geometries Classical problem for optimal geometry: PLATEAU PROBLEM (MINIMAL SURFACE PROBLEM) Optimal Geometries Minimal surface (L.T.Cheng, PhD 2002) Minimize area of surface between fixed boundary curves. For Inverse Obstacle Problems 33 For Inverse Obstacle Problems 34 Optimal Geometries Wulff-Shapes: crystals tend to minimize energy at fixed volume. Pure surface energy: Optimal Geometries Wulff-Shapes: Pb[111] in Cu[111] Surnev et al, J.Vacuum Sci. Tech. A, 1998 is the normal on given anisotropic surface tension For Inverse Obstacle Problems 35 For Inverse Obstacle Problems 36
10 Optimal Geometries Isotropic case: Minimization of perimeter, yields ball Optimal Geometries Free discontinuity problems: find the set of discontinuity from a noisy observation of a function. Mumford-Shah functional Again, solves partial differential equation with interface condition on. For Inverse Obstacle Problems 37 For Inverse Obstacle Problems 38 Optimal Geometries Structural topology optimization Optimal Geometries Inverse Obstacle Problems E.g., inclusion detection Design of Photonic Crystals, Semiconductor Design, Electromagnetic Design,... For Inverse Obstacle Problems 39 Inverse Obstacle Scattering, Impedance Tomography, Identification of Discontinuities in PDE Coefficients,... For Inverse Obstacle Problems 40
11 Gradient Flows Physical Processes tend to minimize energy by a gradient flow: Gradient Flows Gradient flow can be obtained as limit of variational problems E.g., heat diffusion, thermal energy (Fife 1978; Minimizing movements, De Giorgi 1974) Scales of gradient flows are obtained by changing the norm. For Inverse Obstacle Problems 41 For Inverse Obstacle Problems 42 Geometric Gradient Flows For geometric motion, there is no natural Hilbert space setting, generalized notion of gradient flow needed. Natural velocity replacing is normal velocity on. Geometric Gradient Flows Scale of geometric gradient flows obtained, in the limit by using different Hilbert spaces for the velocity. Variational form for : Where is the shape obtained by the motion of with normal velocity (Almgren-Taylor 1994) For Inverse Obstacle Problems 43 where is the shape derivative For Inverse Obstacle Problems 44
12 Geometric Gradient Flows Geometric Gradient Flows... mean curvature, Eikonal Equation, mean-curvature flow For Inverse Obstacle Problems 45 For Inverse Obstacle Problems 46 Geometric Gradient Flows Geometric Gradient Flows volume-conserving mean curvature flow For Inverse Obstacle Problems 47 surface diffusion For Inverse Obstacle Problems 48
13 Geometric Gradient Flows Geometric Gradient Flows Mullins-Sekerka Problem, Bulk diffusion For Inverse Obstacle Problems 49 For Inverse Obstacle Problems 50 Geometric Gradient Flows Inverse Obstacle Problems & Shape Optimization For Inverse Obstacle Problems 51 For Inverse Obstacle Problems 52
14 Inverse Obstacle Problem... Set of shapes... Hilbert space... Nonlinear operator Given noisy measurement for find a shape approximating. Associated Least-Squares Problem Inverse Obstacle Problem In general, minimization of is ill-posed: without convergence of a subsequence possible. No stable dependence of minimizer (if existing) on the data. For Inverse Obstacle Problems 53 For Inverse Obstacle Problems 54 Inverse Obstacle Problem Ill-posedness causes need for Regularization. (i) Tikhonov-type regularization with regularization functional Inverse Obstacle Problem (ii) Iterative regularization concept Apply iterative (level set) method directly to, use appropriate stopping criterium, e.g. stop at the first iteration where residual is less than (noise level),. For Inverse Obstacle Problems 55 For Inverse Obstacle Problems 56
15 Inverse Obstacle Problem Regularization functional must be defined on general class of shapes (multiply connected, no fixed parametrization with respect to reference shape,...). Popular choice: Perimeter Inverse Obstacle Problem Other possibilities for regularization functionals: based on distance function. Reference (starting) shape :... Indicator function of For Inverse Obstacle Problems 57 For Inverse Obstacle Problems 58 Shape Optimization shape functional equality constraints in Banach space inequality constraints in ordered Banach space Shape Optimization In general, existence of minimizer not guaranteed (except simple 2D cases, e.g. Chambolle 2001) Perimeter constraint or penalization by perimeter For Inverse Obstacle Problems 59 For Inverse Obstacle Problems 60
16 Metrics on Shapes For analysis of inverse obstacle and shape optimization problems, metrics on classes of shapes are needed! Transformation-metrics -metric Hausdorff-metric Transformation Metrics Transformation metrics are based on cost of transformation of shapes: subject to For Inverse Obstacle Problems 61 For Inverse Obstacle Problems 62 Transformation Metrics use any appropriate Hilbert space norm -metric -metric measures distance of shapes via their indicator functions: used e.g. for conformal mappings restricts class of admissible shapes Many shape functionals are lower semicontinuous with respect to - metric, typically if For Inverse Obstacle Problems 63 For Inverse Obstacle Problems 64
17 -metric Perimeter is weakly lower semicontinuous with respect to -metric Hausdorff Metric Natural metric of shapes (?) respect to is pre-compact with -metric Perimeter is lower-semicontinuous with respect to on the class of compact sets in with finite number of connected components (Golab s Theorem) For Inverse Obstacle Problems 65 For Inverse Obstacle Problems 66 Hausdorff Metric Neumann-Problems for elliptic partial differential equations are lower semicontinuous with respect to on the class of compact sets in with finite number of connected components (Chambolle 2002, DalMaso-Toader 2002) Regularization by Perimeter Assumptions: Let be minimizer of For Inverse Obstacle Problems 67 For Inverse Obstacle Problems 68
18 Regularization by Perimeter Respectively Regularization by Perimeter Uniqueness of the limit problem implies Then there exists supsequence such that as. where solves s.t. For Inverse Obstacle Problems 69 For Inverse Obstacle Problems 70 Sensitivity Analysis Shape Sensitivity Analysis As usual for optimization problems sensitivities (derivatives) are needed. Gateâux-Derivatives in Banach spaces For Inverse Obstacle Problems 71 For Inverse Obstacle Problems 72
19 Sensitivity Analysis Alternative definition: Speed Method Analogous to Gateâux-Derivative define Shape Derivative of Speed Method For Inverse Obstacle Problems 73 For Inverse Obstacle Problems 74 Speed Method Classical definition for smooth shapes and velocities extension via level set method Volume Functionals Let Level-set formulation, with Heaviside function. For Inverse Obstacle Problems 75 For Inverse Obstacle Problems 76
20 Volume Functionals Formal derivative: Surface Functionals Let Level-set formulation, Formal derivative with Dirac -distribution For Inverse Obstacle Problems 77 For Inverse Obstacle Problems 78 Surface Functionals Surface Functionals Use Gauss-Theorem: and Extension of, arbitrary on use constant normal extension For Inverse Obstacle Problems 79 For Inverse Obstacle Problems 80
21 Gradient Flows Based on Gradient Flows In the above framework of gradient flows, we can derive equations for velocity by minimizing with respect to and For Inverse Obstacle Problems 81 For Inverse Obstacle Problems 82 Gradient Methods Variational equation for yields continuous time evolution for. Classical gradient method is explicit time discretization of gradient flow. Gradient Methods Set Loop: Set Compute time Select time step Solve, initial value from variational equation at in For Inverse Obstacle Problems 83 For Inverse Obstacle Problems 84
22 Gradient Methods Lemma: is descent direction, i.e. If, sufficiently small. Proof: For Inverse Obstacle Problems 85 For Inverse Obstacle Problems 86 Define adjoint state via independent of Adjoint Method For Inverse Obstacle Problems 87 For Inverse Obstacle Problems 88
23 Alternative: For Inverse Obstacle Problems 89 For Inverse Obstacle Problems 90 Residual - Norm - Norm For Inverse Obstacle Problems 91 For Inverse Obstacle Problems 92
24 - error Example 2 For Inverse Obstacle Problems 93 For Inverse Obstacle Problems 94 Example 2 Example 2 Adjoint state defined by For Inverse Obstacle Problems 95 For Inverse Obstacle Problems 96
25 Example 2 Example 2 - Norm - Norm For Inverse Obstacle Problems 97 For Inverse Obstacle Problems 98 Example 2 Residual Example 2 - error For Inverse Obstacle Problems 99 For Inverse Obstacle Problems 100
26 Levenberg-Marquardt Levenberg-Marquardt obtained from firstorder expansion of together with penalty on velocity Variational equation Levenberg-Marquardt Set Loop: Set Compute time Select time step, initial value from variational equation at Solve in For Inverse Obstacle Problems 101 For Inverse Obstacle Problems 102 Levenberg-Marquardt update becomes For Inverse Obstacle Problems 103 For Inverse Obstacle Problems 104
27 Define Lagrange parameter Primal-Dual formulation For Inverse Obstacle Problems 105 For Inverse Obstacle Problems 106 Noise level 1% For Inverse Obstacle Problems 107 For Inverse Obstacle Problems 108
28 Noise level 4% For Inverse Obstacle Problems 109 For Inverse Obstacle Problems 110 Example 2 Example 2 Primal-Dual formulation For Inverse Obstacle Problems 111 For Inverse Obstacle Problems 112
29 Example 2 Multiple State Equations Example For Inverse Obstacle Problems 113 For Inverse Obstacle Problems 114 Example 2 Noise level 0.1 % Example 2 Comparison with Gradient method For Inverse Obstacle Problems 115 For Inverse Obstacle Problems 116
30 Example 2 Newton-Type Methods Basic structure: compute second derivative For Inverse Obstacle Problems 117 For Inverse Obstacle Problems 118 Newton-Type Methods Compute velocity by solving Numerical Methods Hintermüller, Ring, SIAP 2003 For Inverse Obstacle Problems 119 For Inverse Obstacle Problems 120
31 Numerical Methods Besides computational techniques for level set evolution, (Hamilton-Jacobi solver, redistancing, velocity extension), we need numerical methods to solve partial differential equations with/on interfaces (state/adjoint equation, Newton equation,...). Numerical Methods Possibilities for elliptic PDEs with interfaces: 1. Resolve interface by mesh (e.g. finite elements) accurately Remeshing at each iteration step is needed. Expensive in particular in 3D, difficult. For Inverse Obstacle Problems 121 For Inverse Obstacle Problems 122 Numerical Methods 2. Use adaptive refinement of basic mesh (fixed during iteration). 3. Moving meshes Problems with too strong change of obstacle. For Inverse Obstacle Problems 123 Numerical Methods 4. Immersed interface method: finite difference discretization on fixed grid with local connections to system matrix, around interface. 5. Partion of Unity FE/Extended Finite Element. FE analogous to immersed interface method, fixed triangular grid + discontinuous elements around interface. For Inverse Obstacle Problems 124
32 Numerical Methods 6. Fictitious Domain Methods: extend problem to larger domain, use Lagrange parameter for correction. Numerical Methods 7. Averaged fictitious domain methods: use weighted average over several level sets. So far, all methods require construction of the zero level set Expensive in 3D! For Inverse Obstacle Problems 125 For Inverse Obstacle Problems 126 Numerical Methods Let State equation, weak form Linearized state equation, weak form Use adaptize finite element method on fixed grid on. For Inverse Obstacle Problems 127 For Inverse Obstacle Problems 128
33 Example 2 State equation, weak form Example 2 Linearized state equation Ersatz material, stiffness For Inverse Obstacle Problems 129 For Inverse Obstacle Problems 130
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