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Interpolation-based model reduction of
nonlinear systems
CSC Seminar 2016
MPI Magdeburg, 09.02.2016
Maria Cruz Varona
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Maria Cruz Varona, M.Sc. 2
Outline
1. Nonlinear Model Order Reduction
► Motivation
► Projective reduction
► Challenges
2. State-of-the-Art Nonlinear Model Reduction Approaches
► Overview
► Proper Orthogonal Decomposition (POD)
► Trajectory piecewise-linear approximation (TPWL)
3. Model Reduction for Bilinear Systems
► Carleman bilinearization
► Output response and transfer functions
► Multimoment-Matching and -optimal reduction
► pseudo-optimal reduction
4. Summary and Outlook
► Discussion
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Maria Cruz Varona, M.Sc. 3
Motivation for Nonlinear Model Order Reduction
Given a large-scale nonlinear control system of the form
with and
MOR
Simulation, design, control and optimization cannot be done efficiently!
Reduced order model
with and
Goal:
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Maria Cruz Varona, M.Sc. 4
Projective nonlinear MOR
Procedure:
1. Replace by its approximation
2. Reduce the number of equations (via projection with )
3. Petrov-Galerkin condition
Approximation in the subspace
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Maria Cruz Varona, M.Sc. 5
Model Order Reduction (MOR)
Large-scale nonlinear model
Reduced order model (ROM)
MOR
Projection
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Maria Cruz Varona, M.Sc. 6
• Nonlinear systems can exhibit complex behaviours
– Multiple equilibria
– Stable, unstable or semi-stable limit cycles
– Chaotic behaviours
• Input-output behaviour of nonlinear systems cannot be described with the help of
transfer functions, the state-transition matrix or the convolution (only possible for
special cases)
• Choice of the reduced order basis
– Projection bases should comprise the most dominant directions of the state-space
– Existing approaches:
Simulation-based methods
Volterra-based approaches
Quadratic-bilinear-based techniques
• Expensive evaluation of the full-order vector of nonlinearities
– Approximation by so-called hyper-reduction techniques: EIM, DEIM, Gappy-POD,
GNAT, ECSW, …
Challenges of Nonlinear Model Reduction
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Maria Cruz Varona, M.Sc. 7
• Classification in
1. Simulation- or trajectory-based methods
2. Volterra-based approaches (bilinear)
3. Polynomialization- and variational analysis-based techniques (quadratic-bilinear)
or
a) Time domain approaches (Simulation- or trajectory-based approaches)
b) Frequency domain approaches (Interpolation-based methods: bilinear & QBMOR)
or
i. Strong nonlinear approaches (POD, NL-BT, Empirical Gramians, TPWL, QBMOR)
ii. Weakly nonlinear approaches (Bilinear models)
• Methods:
1. POD, Nonlinear Balanced Truncation (NL-BT), Empirical Gramians, TPWL
2. Bilinear systems (BT, bilinear RK, BIRKA, Loewner Framework,…)
3. Quadratic-bilinear (BT, RK)
Overview of existing nonlinear model reduction methods
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Maria Cruz Varona, M.Sc. 8
Overview
Simulation-based methods
Proper Orthogonal Decomposition (POD)
Nonlinear BalancedTruncation
Empirical Gramians
Trajectory piecewiselinear approximation
(TPWL)
Bilinear Rational Krylov
Bilinear IRKA
Bilinear LoewnerFramework
Bilinear BalancedTruncation
Volterra-based methods Quadratic-bilinear methods
Balanced Truncationfor QBDAEs
Two-sided Rational Krylov for QBDAEs
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Maria Cruz Varona, M.Sc. 9
Starting point:
1. Choose suitable training input signals
2. Take snapshots from simulated full-order state trajectories
3. Perform singular value decomposition (SVD) of the snapshot matrix
4. Reduced order basis:
Proper Orthogonal Decomposition (POD)
Advantages Drawbacks
• Straightforward data-driven method
• Error bound for approximation error
• Optimal in least squares sense:
• Simulation of full-order model for
different input signals required
• SVD of large snapshot matrix
• Training input dependency
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Maria Cruz Varona, M.Sc. 10
Starting point:
1. Linearize original nonlinear model along simulated state trajectory
2. Reduce linearized models with well-known linear model reduction techniques
(e.g. POD, Balanced Truncation, Rational Krylov, …)
3. Construct reduced order model as weighted sum of linearized reduced models:
Trajectory Piecewise-Linear Approximation (TPWL)
Jacobi matrixWeighted sum oflinearized models
Weighting functions
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Maria Cruz Varona, M.Sc. 11
Offline stage
1. Simulation of full-order model for several appropriate training input signals
2. Selection of linearization points (number and distance ) and linearization at
selected points
3. Reduction of all linearized models
4. Choice of weighting function (e.g. Gaussian, squared, trapezoidal,
triangular, …)
Online stage
1. Calculation of the weights according to the current state
2. Computation of reduced model as convex combination of linearized reduced
models
Trajectory Piecewise-Linear Approximation (TPWL)
Advantages Drawbacks
• Strong nonlinear approach
• Linear model reduction techniques
can be used
• No hyper-reduction step necessary
• Simulation, linearization and
reduction of full-order models
• Many degrees of freedom ( )
• Training input dependency
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Maria Cruz Varona, M.Sc. 12
Variations and extensions of the TPWL approach
• Fast approximate simulation
– select the linearization points using the linearized or the reduced trajectory
• Reduction of the linearized models
– Using global projection matrices:
– Using local projection matrices:
Computation of state transformations to common subspace are necessary
• Generation of stable TPWL reduced models
• Reduction of nonlinear, parametric models using TPWL + pMOR by Matrix
Interpolation
• Reduction of nonlinear DAE models (e.g electrostatic beam, IMTEK) using TPWL
Trajectory Piecewise-Linear Approximation (TPWL)
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Maria Cruz Varona, M.Sc. 13
Overview
Simulation-based methods
Proper Orthogonal Decomposition (POD)
Nonlinear BalancedTruncation
Empirical Gramians
Trajectory piecewiselinear approximation
(TPWL)
Bilinear Rational Krylov
Bilinear IRKA
Bilinear LoewnerFramework
Bilinear BalancedTruncation
Volterra-based methods Quadratic-bilinear methods
Balanced Truncationfor QBDAEs
Two-sided Rational Krylov for QBDAEs
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Maria Cruz Varona, M.Sc. 14
Starting point:
Goal: Approximation of (weakly) nonlinear systems by Carleman linearization
• Taylor series representation:
• State-space model:
Carleman linearization
Assumptions:
• X
• k
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Maria Cruz Varona, M.Sc. 15
Starting point:
Goal: Bilinear model
• Consider differential equations for
• Bilinear model: with
Carleman bilinearization
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Maria Cruz Varona, M.Sc. 16
Starting point:
Bilinear model:
Carleman bilinearization: example
Carleman
linearization
Carleman
bilinearization
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Maria Cruz Varona, M.Sc. 17
Consider bilinear SISO systems of the form
with and .
• Many (weakly) nonlinear systems can be approximated by bilinear systems
through Carleman bilinearization
Drawback: Dimension of the bilinear model is significantly higher than the original
state dimension only applicable for medium-sized (weakly) nonlinear systems
• Linear in input and linear in state, but not jointly linear in state and input
• Advantage: Close relation to linear systems, a lot of well-known concepts can be
extended, e.g. transfer functions, Gramians, Sylvester and Lyapunov equations.
State-Space Representation of Bilinear Systems
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Maria Cruz Varona, M.Sc. 18
Some background on Volterra theory
• Output response expressed by Volterra series:
• Multivariable Laplace-transform:
Output response and Transfer Functions of Bilinear Systems
Impulse response / kernel of th degree
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Maria Cruz Varona, M.Sc. 19
• [Phillips ’00], [Bai/Skoogh ’06], [Breiten/Damm ’10]
• Multimoment-Matching for bilinear systems
Model Reduction of Bilinear Systems
Bilinear Rational Krylov
Bilinear IRKA
Bilinear LoewnerFramework
Bilinear BalancedTruncation
Volterra-based methods
• [Al-Baiyat ’93], [Benner/Damm ’11]
• Solution of two bilinear Lyapunov equations
• [Zhang/Lam ’02], [Benner/Breiten ’12], [Flagg ’12]
• H2-optimal model reduction for bilinear systems
• [Flagg ’12], [Antoulas ’14]
• Data-driven interpolation-based approach
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Maria Cruz Varona, M.Sc. 20
Multimoments for bilinear systems: [Bai/Skoogh ’06], [Breiten/Damm ’10]
• Transfer function:
• Multimoments:
• Markov parameters:
MOR for Bilinear Systems: Multimoment-Matching
• Make use of Neumann expansion
• Expansion in a multivariable Maclaurin series
with
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Maria Cruz Varona, M.Sc. 21
Multimoment-Matching: [Bai/Skoogh ’06], [Feng/Benner ’07], [Breiten/Damm ’10]
1. Calculation of the Krylov subspaces:
2. Computation of the reduced order model:
Example:
• 1st subsystem:
• 2nd subsystem:
MOR for Bilinear Systems: Multimoment-Matching
for
for
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Maria Cruz Varona, M.Sc. 22
Multimoment-Matching: [Bai/Skoogh ’06], [Feng/Benner ’07], [Breiten/Damm ’10]
1. Calculation of the Krylov subspaces:
2. Computation of the reduced order model:
Open questions/problems:
• How to choose the expansion points?
Optimal expansion points via -optimal model reduction (bilinear IRKA)
• How many moments should be matched per subsystem?
• How many subsystems are necessary for a good approximation?
• Error bounds?
MOR for Bilinear Systems: Multimoment-Matching
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Maria Cruz Varona, M.Sc. 23
• -norm of a MIMO bilinear system:
Alternative calculation via
where and are the solutions of the following bilinear Lyapunov equations:
• Error system:
MOR for Bilinear Systems: -optimal model reduction
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Maria Cruz Varona, M.Sc. 24
• -norm of error system:
where and are the solutions of the following bilinear Lyapunov equations:
Assume the reduced model is given by ist eigenvalue decomposition:
• Necessary conditions for -optimality:
MOR for Bilinear Systems: -optimal model reduction
Optimization
parameters
1
2
3
4
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Maria Cruz Varona, M.Sc. 25
Bilinear IRKA approach
MOR for Bilinear Systems: -optimal model reduction
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Maria Cruz Varona, M.Sc. 26
• Duality: Krylov subspaces with Sylvester equations
• -optimality vs. pseudo-optimality
MOR for Linear Systems: pseudo-optimal reduction
-optimality pseudo-optimality
• Problem:
• Necessary conditions for local -
optimality (SISO): (Meier-Luenberger)
• minimizes the error locally
within the set of all ROMs of order
• Problem:
• Necessary and sufficient condition
for global pseudo-optimality:
• Pseudo-optimal means optimal in a
certain subset
• minimizes the error globally
within the subset of all ROMs of
order with poles
: shifts
: tangential directions
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Maria Cruz Varona, M.Sc. 27
MOR for Linear Systems: pseudo-optimal reduction
Notation:
Gramian
Scalar product
Krylov
Projection
(known)
(unknown)
(known)
(known)
Let be a basis of a Krylov subspace. Let be the reduced model obtained
by projection with . Then, the following conditions are equivalent:
New conditions for pseudo-optimality [Wolf ’14]:
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Maria Cruz Varona, M.Sc. 28
MOR for Linear Systems: pseudo-optimal reduction
PORK: Pseudo-optimal rational Krylov
Advantages and properties of PORK:
• ROM is globally optimal within a subset:
• Eigenvalues of ROM:
choice of the shifts is twice as important
• Stability preservation in the ROM can be ensured
• Low numerical effort required: solution of a Lyapunov equation and a linear
system of equations, both of reduced order.
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Maria Cruz Varona, M.Sc. 29
• Duality: Bilinear Krylov subspaces with bilinear Sylvester equations [Flagg ’12]
Can we derive new conditions for pseudo-optimality for bilinear systems?
MOR for Bilinear Systems: pseudo-optimal reduction
: shifts
: tangential directions
: weights
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Maria Cruz Varona, M.Sc. 30
MOR for Bilinear Systems: pseudo-optimal reduction
Notation:
Gramian
Scalar product
Krylov
Projection
(known)
(unknown)
(known)
(known)
Let be a basis of a Krylov subspace. Let be the reduced model obtained by
projection with . Then, the following conditions are equivalent:
New conditions for pseudo-optimality for bilinear systems:
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Maria Cruz Varona, M.Sc. 31
MOR for Bilinear Systems: pseudo-optimal reduction
BIPORK: Bilinear pseudo-optimal rational Krylov
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Maria Cruz Varona, M.Sc. 32
Summary and Outlook
► Goal: Reduction of high dimensional nonlinear systems
► Simulation-based, Volterra-based and quadratic-bilinear-based approaches
► Model reduction for bilinear systems (BT, Krylov, BIRKA, Loewner)
► pseudo-optimal model reduction for bilinear systems
► Derivation of new conditions for pseudo-optimality for bilinear systems
► Bilinear pseudo-optimal Rational Krylov (BIPORK)
► Solution of bilinear Lyapunov equations with BIPORK:
► Cumulative reduction of bilinear systems
► Quadratic-bilinear MOR
► Stability-preserving two-sided rational Krylov for QBDAEs?
► IRKA for QBDAEs? Algorithm for choosing optimal expansion points?
Summary:
Outlook:
BI-LR-ADI = RKSM + BIPORK
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Maria Cruz Varona, M.Sc. 33
Feedback and hints relating the following topics are welcome:
• Numerical solvers (direct and/or indirect) for nonlinear matrix equations
a) Direct solvers
Direct solvers for bilinear Sylvester and Lyapunov equations
b) Indirect solvers
Bilinear low-rank ADI method
Bilinear Extended Krylov Subspace Method (EKSM)
Other Krylov-based iterative solvers, e.g. CG, PCG, BiCG, BiCGstab
• Error bounds for bilinear systems
– Existing approaches or literature?
• Nonlinear, parametric benchmarks
– Parametric Nonlinear RC-Ladder?
– Parametric Nonlinear Heat Transfer (IMTEK)?
– …
Discussion and open problems
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Maria Cruz Varona, M.Sc. 34
Thank you for your attention!