Model Order Reduction Slides adopted from Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Joel Phillips, Cadence Berkeley Labs Jacob White, Massachusetts Instit. of Technology

Overview Reduction via moment matching: (Projection Framework) general Krylov Subspace methods case 1: Arnoldi case 2: PVL case 3: multipoint moment matching Importance of preserving passivity PRIMA

Projection Framework: Change of variables reduced state Note: q << N 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 multiplying by VqT If V and U biorthogonal

Projection Framework (cont.) qxn qxq nxn nxq

Projection Framework Change of variables Equation Testing

Approaches for picking V and U Use Eigenvectors Use Time Series Data Compute Use the SVD to pick q < k important vectors Use Frequency Domain Data Use Singular Vectors of System Grammians? Use Krylov Subspace Vectors?

Overview Reduction via moment matching: (Projection Framework) general Krylov Subspace methods case 1: Arnoldi case 2: PVL case 3: multipoint moment matching Importance of preserving passivity PRIMA

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

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’ Approximations and Krylov Subspace Projection Framework moment matching at single DC point numerically very ill-conditioned!!! Krylov Subspace Projection Framework: multipoint moment matching numerically very stable!!!

Aside on Krylov Subspaces - Definition The order k Krylov subspace generated from matrix A and vector b is defined as

Projection Framework: Moment Matching Theorem (E. Grimme 97) If and Then

Overview Reduction via moment matching: (Projection Framework) general Krylov Subspace methods case 1: Arnoldi case 2: PVL case 3: multipoint moment matching Importance of preserving passivity PRIMA

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 will 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

Orthonormalization of U: The Arnoldi Algorithm For i = 1 to k Generates k+1 vectors! Orthogonalize new vector For j = 1 to i Normalize new vector

Overview Reduction via moment matching: (Projection Framework) general Krylov Subspace methods case 1: Arnoldi case 2: PVL case 3: multipoint moment matching Importance of preserving passivity PRIMA

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: Substitute: Iq 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

Overview Reduction via moment matching: (Projection Framework) general Krylov Subspace methods case 1: Arnoldi case 2: PVL case 3: multipoint moment matching Importance of preserving passivity PRIMA

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 standard form

Case #3: Intuitive view of Krylov subspace choice for general expansion points (cont.) Hence choosing Krylov subspace s1=0 s1 s2 s3 matches first kj of transfer function around each expansion point sj

Overview Reduction via moment matching: (Projection Framework) general Krylov Subspace methods case 1: Arnoldi case 2: PVL case 3: multipoint moment matching Importance of preserving passivity PRIMA

Interconnected Systems In reality, reduced models are only useful when connected together with other models and circuit elements in a composite simulation Consider a state-space model connected to external circuitry (possibly with feedback!) ROM Can we assure that the simulation of the composite system will be well-behaved? At least preclude non-physical behavior of the reduced model?

Passivity Passive systems do not generate energy. We cannot extract out more energy than is stored. A passive system does not provide energy that is not in its storage elements. If the reduced model is not passive it can generate energy from nothingness and the simulation will explode

Interconnecting Passive Systems The interconnection of stable models is not necessarily stable BUT the interconnection of passive models is a passive model: Q D C - + Q D C - + Q D C - + Q D C - +

Positive Real Functions A positive real function is a function internally stable with non-negative real part (no unstable poles) (real response) (no negative resistors) Hermittian=conjugate and transposed It means its real part is a positive semidefinite matrix at all frequencies

Positive Realness & Passivity For systems with immittance (impedance or admittance) matrix representation, positive-realness of the transfer function is equivalent to passivity ROM + + - -

Necessary conditions for passivity for Poles/Zeros The positive-real condition on the matrix rational function implies that: If H(s) is positive-real also its inverse is positive real If H(s) is positive-real it has no poles in the RHP, and hence also no zeros there. Occasional misconception : “if the system function has no poles and no zeros in the RHP the system is passive”. It is necessary that a positive-real function have no poles or zeros in the RHP, but not sufficient.

Sufficient conditions for passivity Note that these are NOT necessary conditions (common misconception)

Congruence Transformations Preserve Positive Semidefinitness Def. congruence transformation same matrix Note: case #1 in the projection framework V=U produces congruence transformations Property: a congruence transformation preserves the positive semidefiniteness of the matrix Proof. Just rename Note:

Overview Reduction via moment matching: (Projection Framework) general Krylov Subspace methods case 1: Arnoldi case 2: PVL case 3: multipoint moment matching Importance of preserving passivity PRIMA

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 while may not be. 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:

Compare methods number of moments matched by model of order q preserving passivity case #1 (Arnoldi, V=U, UTU=I on sA-1Ex=x+Bu) q no PRIMA (Arnoldi, V=U, UTU=I on sEx=Ax+Bu) yes necessary when model is used in a time domain simulator case #2 (PVL, Lanczos,V≠U, VTU=I on sA-1Ex=x+Bu) 2q more efficient (good only if model is used in frequency domain)

Overview Reduction via moment matching: (Projection Framework) general Krylov Subspace methods case 1: Arnoldi case 2: PVL case 3: multipoint moment matching Importance of preserving passivity PRIMA Summary and Conclusions

Summary: Conventional Design Flow Funct. Spec Logic Synth. Gate-level Net. RTL Layout Floorplanning Place & Route Front-end Back-end Behav. Simul. Gate-Lev. Sim. Stat. Wire Model Parasitic Extrac.

Summary: Parasitic Extraction thousands of wires e.g. critical path e.g. gnd/vdd grid Parasitic Extraction identify some ports produce equivalent circuit that models response of wires at those ports tens of circuit elements for gate level spice simulation

Summary: Model Order Reduction (the second step of parasitic extraction) Electromagnetic Analysis (Tuesday) thin volume filaments with constant current small surface panels with constant charge million of elements Model Order Reduction (Today) tens of elements

Conclusions Reduction via moment matching: Krylov Subspace Projection Framework allows multipoint expansion moment matching (wider frequency band) numerically very robust use PVL is mode efficient for model in frequency domain use PRIMA to preserve passivity if model is for time domain simulator