Model Reduction for Maxwells Equations with Uncertain Parameters - PowerPoint PPT Presentation

Summer School on Numerical Linear Algebra for Dynamical and High-Dimensional Problems, October 12, 2011 Model Reduction for Maxwell’s Equations with Uncertain Parameters Judith Schneider Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg, Germany MAX PLANCK INSTITUTE FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS MoreSim 4 Nano MAGDEBURG Max Planck Institute Magdeburg J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters 1/6

Maxwell’s Equations Uncertainty Quantification Model Order Reduction Motivation Analyze the influence of the electromagnetic field on semiconductors. Max Planck Institute Magdeburg J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters 2/6

Maxwell’s Equations Uncertainty Quantification Model Order Reduction Motivation Analyze the influence of the electromagnetic field on semiconductors. Magnetic influence is described by Maxwell’s equations ∂ t ( ǫ E ) = ∇ × H − σ E − J Source ∂ t ( µ H ) = −∇ × E ∇ · ( ǫ E ) = ρ ∇ · ( µ H ) = 0 , with the electric field intensity E , the magnetic field intensity H , the charge density ρ , the impressed current source J Source , and material parameters ǫ, µ, σ . Max Planck Institute Magdeburg J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters 2/6

Maxwell’s Equations Uncertainty Quantification Model Order Reduction Motivation Analyze the influence of the electromagnetic field on semiconductors. Consider geometric variations (e.g. feature structure size) as uncertain parameters. Dependence on geometric parameters occurs in H , E , J Source , and ρ ∂ t ( ǫ E ( γ γ γ )) = ∇ × H ( γ γ γ ) − σ E ( γ γ γ ) − J Source ( γ γ γ ) ∂ t ( µ H ( γ γ γ )) = −∇ × E ( γ γ γ ) ∇ · ( ǫ E ( γ γ γ )) = ρ ( γ γ γ ) ∇ · ( µ H ( γ γ γ )) = 0 , γ ∈ R g is the vector containing the different geometric pa- where γ γ rameters. Max Planck Institute Magdeburg J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters 2/6

Maxwell’s Equations Uncertainty Quantification Model Order Reduction Motivation Analyze the influence of the electromagnetic field on semiconductors. Consider geometric variations (e.g. feature structure size) as uncertain parameters. Dependence on geometric parameters occurs in H , E , J Source , and ρ ∂ t ( ǫ E ( γ γ γ )) = ∇ × H ( γ γ γ ) − σ E ( γ γ γ ) − J Source ( γ γ γ ) ∂ t ( µ H ( γ γ γ )) = −∇ × E ( γ γ γ ) ∇ · ( ǫ E ( γ γ γ )) = ρ ( γ γ γ ) ∇ · ( µ H ( γ γ γ )) = 0 , γ ∈ R g is the vector containing the different geometric pa- where γ γ rameters. The last two equations can be eliminated by using initial values E 0 ( γ γ γ ) and H 0 ( γ γ γ ) that fulfill them. Max Planck Institute Magdeburg J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters 2/6

Maxwell’s Equations Uncertainty Quantification Model Order Reduction Discretized Parametrized System Discretization via Finite Element Method (FEM) or Boundary Ele- ment Method (BEM) leads to M ǫ ( γ γ γ )˙ e = − M σ ( γ γ γ ) e + K H ( γ γ γ ) h + B E ( γ γ γ ) u γ )˙ M µ ( γ γ h = − K E ( γ γ γ ) e + B H ( γ γ γ ) u y = C E ( γ γ γ ) e + C H ( γ γ γ ) h , with mass matrices M ǫ , M σ , M µ , stiffness matrices K H , K E , input matrices B E , B H and output matrices C E , C H , all depending on the parameter vector γ γ γ. Here, u and y are the input and output vectors, respectively. Max Planck Institute Magdeburg J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters 3/6

Maxwell’s Equations Uncertainty Quantification Model Order Reduction Discretized Parametrized System Discretization via Finite Element Method (FEM) or Boundary Ele- ment Method (BEM) leads to M 1 ( p )˙ e = − M 2 ( p ) e + K H ( p ) h + B E ( p ) u M 3 ( p )˙ h = − K E ( p ) e + B H ( p ) u y = C E ( p ) e + C H ( p ) h , with mass matrices M 1 , M 2 , M 3 , stiffness matrices K H , K E , input matrices B E , B H and output matrices C E , C H , all depending on the parameter vector p ∈ R q the vector of all geometric and material parameters. Max Planck Institute Magdeburg J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters 3/6

Maxwell’s Equations Uncertainty Quantification Model Order Reduction Parameter Handling For simplicity, we assume that each system matrix A ( p ) has an affine representation of the form n � A ( p ) = f i ( p ) A i , i =1 where the f i are independent of the spacial variable and n is not too large. Max Planck Institute Magdeburg J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters 4/6

Maxwell’s Equations Uncertainty Quantification Model Order Reduction Parameter Handling For simplicity, we assume that each system matrix A ( p ) has an affine representation of the form n � A ( p ) = f i ( p ) A i , i =1 where the f i are independent of the spacial variable and n is not too large. System has to be discretized only once, though A i independent of p . Max Planck Institute Magdeburg J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters 4/6

Maxwell’s Equations Uncertainty Quantification Model Order Reduction Parameter Handling For simplicity, we assume that each system matrix A ( p ) has an affine representation of the form n � A ( p ) = f i ( p ) A i , i =1 where the f i are independent of the spacial variable and n is not too large. System has to be discretized only once, though A i independent of p . Uncertainties can be handled with the help of stochastic collocation methods. Max Planck Institute Magdeburg J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters 4/6

Maxwell’s Equations Uncertainty Quantification Model Order Reduction Stochastic Collocation Collocation methods are interpolation based. The f i ’s are evaluated at a number of realizations of the uncertain parameters to interpolate A ( p ) . Max Planck Institute Magdeburg J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters 5/6

Maxwell’s Equations Uncertainty Quantification Model Order Reduction Stochastic Collocation Collocation methods are interpolation based. The f i ’s are evaluated at a number of realizations of the uncertain parameters to interpolate A ( p ) . There are different possibilities for the choice of the interpolation points. One can use Monte-Carlo integrators, tensor-product integration rules, sparse grid points, or Stroud integration rules. Max Planck Institute Magdeburg J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters 5/6

Maxwell’s Equations Uncertainty Quantification Model Order Reduction Stochastic Collocation Collocation methods are interpolation based. The f i ’s are evaluated at a number of realizations of the uncertain parameters to interpolate A ( p ) . There are different possibilities for the choice of the interpolation points. One can use Monte-Carlo integrators, tensor-product integration rules, sparse grid points, or Stroud integration rules. The most accurate are tensor-product integration rules, but sparse grids and Stroud integration rules are much more efficient and provide enough accuracy. Max Planck Institute Magdeburg J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters 5/6

Maxwell’s Equations Uncertainty Quantification Model Order Reduction Model Order Reduction While the system matrices after the “splitting” are independent of the uncertain parameters, we can use, e.g., Balanced Truncation to reduce the discretized system. To simplify matters, we assume A ( p ) = f ( p )˜ A for each system matrix. The discretized system then is of the form: f 1 ( p ) ˜ e = − f 2 ( p ) ˜ M 2 e + f 3 ( p ) ˜ K H h + f 4 ( p ) ˜ M 1 ˙ B E u f 5 ( p ) ˜ M 3 ˙ h = − f 6 ( p ) ˜ K E e + f 7 ( p ) ˜ B H u y = f 8 ( p ) ˜ C E e + f 9 ( p ) ˜ C H h . Max Planck Institute Magdeburg J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters 6/6

Maxwell’s Equations Uncertainty Quantification Model Order Reduction Model Order Reduction While the system matrices after the “splitting” are independent of the uncertain parameters, we can use, e.g., Balanced Truncation to reduce the discretized system. To simplify matters, we assume A ( p ) = f ( p )˜ A for each system matrix. The reduced system then is of the form: f 1 ( p ) ˆ e = − f 2 ( p ) ˆ e + f 3 ( p ) ˆ h + f 4 ( p ) ˆ M 1 ˙ ˜ ˜ K H ˆ ˜ ˜ ˆ M 2 ˆ B E ˆ u f 5 ( p ) ˆ M 3 ˙ h = − f 6 ( p ) ˆ e + f 7 ( p ) ˆ ˜ ˆ ˜ ˜ K E ˆ B H ˆ u y = f 8 ( p ) ˆ e + f 9 ( p ) ˆ ˜ C H ˆ ˜ ˆ C E ˆ h . Max Planck Institute Magdeburg J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters 6/6