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Introduction to Model Order Reduction II.2 The Projection Framework Methods

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## Introduction to Model Order Reduction II.2 The Projection Framework Methods

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**Introduction to Model Order Reduction II.2 The Projection**Framework Methods Luca Daniel Massachusetts Institute of Technology with contributions from: Alessandra Nardi, Joel Phillips, Jacob White**Projection Framework:Non invertible Change of Coordinates**Note: q << N reduced state original state**Projection Framework**• Original System • Substitute • Note: now few variables (q<<N) in the state, but still thousands of equations (N)**Projection Framework (cont.)**Reduction of number of equations: test by multiplying byVqT • If VqT and UqT are chosen biorthogonal**qxn**nxn nxq nxq Projection Framework (graphically) qxq**Projection Framework**Equation Testing (Projection) Non-invertible change of coordinates (Projection)**Approaches for picking V and U**• Use Eigenvectors of the system matrix (modal analysis) • Use Frequency Domain Data • Compute • Use the SVD to pick q < k important vectors • Use Time Series Data • Compute • Use the SVD to pick q < k important vectors Point Matching II.2.b POD Principal Component Analysis or SVD Singular Value Decomposition or KLD Karhunen-Lo`eve Decomposition or PCA Principal Component Analysis**Approaches for picking V and U**• Use Eigenvectors of the system matrix • POD or SVD or KLD or PCA. • Use Krylov Subspace Vectors (Moment Matching) • Use Singular Vectors of System Grammians Product (Truncated Balance Realizations)**A canonical form for model order reduction**Assuming A is non-singular we can cast the dynamical linear system into a canonical form for moment matching model order reduction Note: this step is not necessary, it just makes the notation simple for educational purposes**Intuitive view of Krylov subspace choice for change of base**projection matrix Taylor series expansion: • change base and use only the first few vectors of the Taylor series expansion: equivalent to match first derivatives around expansion point U**Aside on Krylov Subspaces - Definition**The order k Krylov subspace generated from matrix A and vector b is defined as**Moment matching around non-zero frequencies**• In stead of expanding around only s=0 we can expand around another points • For each expansion point the problem can then be put again in the canonical form**Projection Framework: Moment Matching Theorem (E. Grimme 97)**If and Then Total of 2q moment of the transfer function will match**Combine point and moment matching: multipoint moment**matching • Multipole expansion points give larger band • Moment (derivates) matching gives more • accurate behavior in between expansion points**Compare Pade’ Approximationsand Krylov Subspace Projection**Framework • Pade approximations: • moment matching at • single DC point • numerically very • ill-conditioned!!! • Krylov Subspace Projection Framework: • multipoint moment • matching • AND numerically very • stable!!!**Approaches for picking V and U**• Use Eigenvectors of the system matrix • POD or SVD or KLD or PCA. • Use Krylov Subspace Vectors (Moment Matching) • general Krylov Subspace methods • case 1: Arnoldi • case 2: PVL • case 3: multipoint moment matching • moment matching preserving passivity: PRIMA • Use Singular Vectors of System Grammians Product (Truncated Balance Realizations)**Special simple case #1: expansion at s=0,V=U, orthonormal**UTU=I If U and V are such that: Then the first q moments (derivatives) of the reduced system match**Algebraic proof of case #1: expansion at s=0, V=U,**orthonormal UTU=I apply k times lemma in next slide**Lemma: .**Note in general: BUT... Substitute: Iq U is orthonormal**Need for Orthonormalization of U**Vectors{b,Eb,...,Ek-1b}cannot be computed directly Vectors will quickly line up with dominant eigenspace!**Need for Orthonormalization of U (cont.)**• In "change of base matrix" U transforming to the new reduced state space, we can use ANY columns that span the reduced state space • In particular we can ORTHONORMALIZE the Krylov subspace vectors**For i = 1 to q**Generates new Krylov subspace vector For j = 1 to i Orthogonalize new vector Normalize new vector Orthonormalization of U: The Arnoldi Algorithm Computational Complexity Normalize first vector O(n) sparse: O(n) dense:O(n2) O(q2n) O(n)**Generating vectors for the Krylov subspace**• Most of the computation cost is spent in calculating: • Set up and solve a linear system using GCR • If we have a good preconditioners and a fast matrix vector product each new vector is calculated in O(n) • The total complexity for calculating the projection matrix Uq is O(qn)**What about computing the reduced matrix**? Orthonormalization of the i-th column ofUq Orthonormalization of all columns ofUq So we don’t need to compute the reduced matrix. We have it already:**Approaches for picking V and U**• Use Eigenvectors of the system matrix • POD or SVD or KLD or PCA. • Use Krylov Subspace Vectors (Moment Matching) • general Krylov Subspace methods • case 1: Arnoldi • case 2: PVL • case 3: multipoint moment matching • moment matching preserving passivity: PRIMA • Use Singular Vectors of System Grammians Product (Truncated Balance Realizations)**Special case #2: expansion at s=0, biorthogonal VTU=I**If U and V are such that: Then the first 2q moments of reduced system match**Proof of special case #2: expansion at s=0, biorthogonal**VTU=UTV=Iq (cont.) apply k times the lemma in next slide**Lemma: .**Substitute: biorthonormality Iq Substitute: biorthonormality Iq**PVL: Pade Via Lanczos[P. Feldmann, R. W. Freund TCAD95]**• PVL is an implementation of the biorthogonal case 2: Use Lanczos process to biorthonormalize the columns of U and V: gives very good numerical stability**Example: Simulation of voltage gain of a filter with PVL**(Pade Via Lanczos)**Approaches for picking V and U**• Use Eigenvectors of the system matrix • POD or SVD or KLD or PCA. • Use Krylov Subspace Vectors (Moment Matching) • general Krylov Subspace methods • case 1: Arnoldi • case 2: PVL • case 3: multipoint moment matching • moment matching preserving passivity: PRIMA • Use Singular Vectors of System Grammians Product (Truncated Balance Realizations)**Case #3: Intuitive view of subspace choice for general**expansion points • In stead of expanding around only s=0 we can expand around another points • For each expansion point the problem can then be put again in the canonical form**Case #3: Intuitive view of Krylov subspace choice for**general expansion points (cont.) Hence choosing Krylov subspace s2 s1 matches first kj of transfer function around each expansion point sj s1=0 s3**Generating vectors for the Krylov subspace**• Most of the computation cost is spent in calculating: • Set up and solve a linear system using GCR • If we have a good preconditioners and a fast matrix vector product each new vector is calculated in O(n) • The total complexity for calculating the projection matrix Uq is O(qn)**Approaches for picking V and U**• Use Eigenvectors of the system matrix • POD or SVD or KLD or PCA. • Use Krylov Subspace Vectors (Moment Matching) • general Krylov Subspace methods • case 1: Arnoldi • case 2: PVL • case 3: multipoint moment matching • moment matching preserving passivity: PRIMA • Use Singular Vectors of System Grammians Product (Truncated Balance Realizations)**Sufficient conditions for passivity**• Sufficient conditions for passivity: i.e. A is negative semidefinite • Note that these are NOT necessary conditions (common misconception)**Heat In**Example Finite Difference System from on Poisson Equation (heat problem) We already know the Finite Difference matrices is positive semidefinite. Hence A or E=A-1 are negative semidefinite.**Sufficient conditions for passivity**• Sufficient conditions for passivity: i.e. E is negative semidefinite • Note that these are NOT necessary conditions (common misconception)**Congruence Transformations Preserve Negative (or positive)**Semidefinitness • Def. congruence transformation same matrix • Note: case #1 in the projection framework V=U produces congruence transformations • Lemma: a congruence transformation preserves the negative (or positive) semidefiniteness of the matrix • Proof. Just rename**qxn**nxn nxq nxq Congruence Transformation Preserves Negative Definiteness of E (hence passivity and stability) If we use • Then we loose half of the degrees of freedom • i.e. we match only q moments instead of 2q • But if the original matrix E is negative semidefinite • so is the reduced, hence the system is passive and stable**Sufficient conditions for passivity**• Sufficient conditions for passivity: i.e. E is positive semidefinite i.e. A is negative semidefinite • Note that these are NOT necessary conditions (common misconception)**+**+ - - Example. hState-Space Model from MNA of R, L, C circuits Lemma: A is negative semidefinite if and only if When using MNA For immittance systems in MNA form A is Negative Semidefinite E is Positive Semidefinite**PRIMA (for preserving passivity) (Odabasioglu, Celik,**Pileggi TCAD98) A different implementation of case #1: V=U, UTU=I, Arnoldi Krylov Projection Framework: Use Arnoldi: Numerically very stable**PRIMA preserves passivity**• The main difference between and case #1 and PRIMA: • case #1 applies the projection framework to • PRIMA applies the projection framework to • PRIMA preserves passivity because • uses Arnoldi so that U=V and the projection becomes a congruence transformation • E and -A produced by electromagnetic analysis are typically positive semidefinite • input matrix must be equal to output matrix**Algebraic proof of moment matching for PRIMA expansion at**s=0, V=U, orthonormal UTU=I Used Lemma: If U is orthonormal (UTU=I) and b is a vector such that**Proof of lemma**Proof:**Conclusions**• Reduction via eigenmodes • expensive and inefficient • Reduction via rational function fitting (point matching) • inaccurate in between points, numerically ill-conditioned • Reduction via Quasi-Convex Optimization • quite efficient and accurate • Reduction via moment matching: Pade approximations • better behavior but covers small frequency band • numerically very ill-conditioned • Reduction via moment matching: Krylov Subspace Projection Framework • allows multipoint expansion moment matching (wider frequency band) • numerically very robust and computationally very efficient • use PVL is more efficient for model in frequency domain • use PRIMA to preserve passivity if model is for time domain simulator**Case study: Passive Reduced Models from an Electromagnetic**Field Solver long coplanar T-line, shorted on other side dielectric layer