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MAX PLANCK INSTITUTE FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS MAGDEBURG

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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 Magdeburg

• J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters

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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

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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(ǫE ) = ∇ × H − σE − JSource ∂t(µH ) = −∇ × E ∇ · (ǫE ) = ρ ∇ · (µH ) = 0, with the electric field intensity E, the magnetic field intensity H, the charge density ρ, the impressed current source JSource, and material parameters ǫ, µ, σ.

Max Planck Institute Magdeburg

• J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters

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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, JSource, and ρ ∂t(ǫE(γ γ γ)) = ∇ × H(γ γ γ) − σE(γ γ γ) − JSource(γ γ γ) ∂t(µH(γ γ γ)) = −∇ × E(γ γ γ) ∇ · (ǫE(γ γ γ)) = ρ(γ γ γ) ∇ · (µH(γ γ γ)) = 0, where γ γ γ ∈ Rg is the vector containing the different geometric pa- 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, JSource, and ρ ∂t(ǫE(γ γ γ)) = ∇ × H(γ γ γ) − σE(γ γ γ) − JSource(γ γ γ) ∂t(µH(γ γ γ)) = −∇ × E(γ γ γ) ∇ · (ǫE(γ γ γ)) = ρ(γ γ γ) ∇ · (µH(γ γ γ)) = 0, where γ γ γ ∈ Rg is the vector containing the different geometric pa-

• rameters. The last two equations can be eliminated by using initial

values E0(γ γ γ) and H0(γ γ γ) that fulfill them.

Max Planck Institute Magdeburg

• J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters

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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 Mǫ(γ γ γ)˙ e = −Mσ(γ γ γ)e + KH(γ γ γ)h + BE(γ γ γ)u Mµ(γ γ γ)˙ h = −KE(γ γ γ)e + BH(γ γ γ)u y = CE(γ γ γ)e + CH(γ γ γ)h, with mass matrices Mǫ, Mσ, Mµ, stiffness matrices KH, KE, input matrices BE, BH and output matrices CE, CH, all depending on the parameter vector γ γ γ. Here, u and y are the input and output vectors, respectively.

Max Planck Institute Magdeburg

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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 M1(p)˙ e = −M2(p)e + KH(p)h + BE(p)u M3(p)˙ h = −KE(p)e + BH(p)u y = CE(p)e + CH(p)h, with mass matrices M1, M2, M3, stiffness matrices KH, KE, input matrices BE, BH and output matrices CE, CH, all depending on the parameter vector p ∈ Rq the vector of all geometric and material parameters.

Max Planck Institute Magdeburg

• J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters

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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 A(p) =

n

• i=1

fi(p)Ai, where the fi 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

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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 A(p) =

n

• i=1

fi(p)Ai, where the fi are independent of the spacial variable and n is not too large. System has to be discretized only once, though Ai 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 A(p) =

n

• i=1

fi(p)Ai, where the fi are independent of the spacial variable and n is not too large. System has to be discretized only once, though Ai 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

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Maxwell’s Equations Uncertainty Quantification Model Order Reduction

Stochastic Collocation

Collocation methods are interpolation based. The fi’s are evaluated at a number of realizations of the uncertain parameters to interpolate A(p).

Max Planck Institute Magdeburg

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Maxwell’s Equations Uncertainty Quantification Model Order Reduction

Stochastic Collocation

Collocation methods are interpolation based. The fi’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 fi’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

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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:

f1(p) ˜ M1 ˙ e = −f2(p) ˜ M2e + f3(p) ˜ KHh + f4(p) ˜ BEu f5(p) ˜ M3 ˙ h = −f6(p) ˜ KEe + f7(p) ˜ BHu y = f8(p) ˜ CEe + f9(p) ˜ CHh.

Max Planck Institute Magdeburg

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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:

f1(p) ˆ ˜ M1 ˙ ˆ e = −f2(p) ˆ ˜ M2ˆ e + f3(p) ˆ ˜ KHˆ h + f4(p) ˆ ˜ BE ˆ u f5(p) ˆ ˜ M3 ˙ ˆ h = −f6(p) ˆ ˜ KEˆ e + f7(p) ˆ ˜ BHˆ u ˆ y = f8(p) ˆ ˜ CEˆ e + f9(p) ˆ ˜ CHˆ h.

Max Planck Institute Magdeburg

• J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters

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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:

f1(p) ˆ ˜ M1 ˙ ˆ e = −f2(p) ˆ ˜ M2ˆ e + f3(p) ˆ ˜ KHˆ h + f4(p) ˆ ˜ BE ˆ u f5(p) ˆ ˜ M3 ˙ ˆ h = −f6(p) ˆ ˜ KEˆ e + f7(p) ˆ ˜ BHˆ u ˆ y = f8(p) ˆ ˜ CEˆ e + f9(p) ˆ ˜ CHˆ h. The collocation methods then are applied to the reduced system to

• btain the interpolation of the parametrized system.

Max Planck Institute Magdeburg

• J. Schneider, MOR for Maxwell’s Equations with Uncertain Parameters

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