emgr—The Empirical Gramian Framework
Abstract
:Code Meta Data
name (shortname) | EMpirical GRamian Framework (emgr) |
version (release-date) | 5.4 (2018-05-05) |
identifier (type) | doi:10.5281/zenodo.1241532 (doi) |
authors (ORCIDs) | Christian Himpe (0000-0003-2194-6754) |
topic (type) | Model Reduction (toolbox) |
license (type) | 2-Clause BSD (open) |
repository (type) | git:github.com/gramian/emgr (git) |
languages | Matlab |
dependencies | OCTAVE >= 4.2, MATLAB >= 2016b |
systems | Linux, Windows |
website | http://gramian.de |
keywords | empirical-gramians, cross-gramian, combined-reduction |
1. Introduction
1.1. Aim
- Non-symmetric cross Gramian variant,
- linear cross Gramian variant,
- distributed cross Gramian variant and interface,
- inner product kernel interface,
- time-integrator interface,
- time-varying system compatibility,
- tensor-based trajectory storage,
- functional paradigm software design.
1.2. Outline
2. Mathematical Preliminaries
Model Reduction
- Projection-Based Combined Reduction
- Gramian-Based Model Reduction
3. Empirical Gramians
3.1. State-Space Empirical Gramians
3.1.1. Empirical Controllability Gramian
3.1.2. Empirical Observability Gramian
3.1.3. Empirical Linear Cross Gramian
3.1.4. Empirical Cross Gramian
3.1.5. Empirical Non-Symmetric Cross Gramians
3.2. Parameter-Space Empirical Gramians
3.2.1. Empirical Sensitivity Gramian
3.2.2. Empirical Identifiability Gramian
3.2.3. Empirical Cross-Identifiability Gramian
3.3. Notes on Empirical Gramians
4. Implementation Details
4.1. Design Principles
4.2. Parallelization
4.2.1. Shared Memory Parallelization
4.2.2. Heterogeneous Parallelization
4.2.3. Distributed Memory Parallelization
5. Interface
5.1. Mandatory Arguments
- f
- handle to a function with the signature xdot = f(x,u,p,t) representing the system’s vector-field and expecting the arguments: current state x, current input u, (current) parameter p and current time t.
- g
- handle to a function with the signature y = g(x,u,p,t) representing the system’s output functional and expecting the arguments: current state x, current input u, (current) parameter p and current time t.If g = 1, the identity output functional $g(t,x(t),u(t),\theta )=x\left(t\right)$ is assumed.
- s
- three component vector s = [M,N,Q] setting the dimensions of the input $M\phantom{\rule{3.33333pt}{0ex}}:=\phantom{\rule{3.33333pt}{0ex}}\mathrm{dim}\left(u\right(t\left)\right)$, state $N\phantom{\rule{3.33333pt}{0ex}}:=\phantom{\rule{3.33333pt}{0ex}}\mathrm{dim}\left(x\right(t\left)\right)$ and output $Q\phantom{\rule{3.33333pt}{0ex}}:=\phantom{\rule{3.33333pt}{0ex}}\mathrm{dim}\left(y\right(t\left)\right)$.
- t
- two component vector t = [h,T] specifying the time-step width h and time horizon T.
- w
- character selecting the empirical Gramian type; for details see Section 5.2.
5.2. Features
- ′c′
- Empirical controllability Gramian (see Section 3.1.1),emgr returns a matrix:
- $N\times N$ empirical controllability Gramian matrix ${W}_{C}$.
- ′o′
- Empirical observability Gramian (see Section 3.1.2),emgr returns a matrix:
- $N\times N$ empirical observability Gramian matrix ${W}_{O}$.
- ′x′
- Empirical cross Gramian (see Section 3.1.4),emgr returns a matrix:
- $N\times N$ empirical cross Gramian matrix ${W}_{X}$.
- ′y′
- Empirical linear cross Gramian (see Section 3.1.3),emgr returns a matrix:
- $N\times N$ empirical linear cross Gramian matrix ${W}_{Y}$.
- ′s′
- Empirical sensitivity Gramian (see Section 3.2.1),emgr returns a cell array. holding:
- $N\times N$ empirical controllability Gramian matrix ${W}_{C}$,
- $P\times 1$ empirical sensitivity Gramian diagonal ${W}_{S}$.
- ′i′
- Empirical identifiability Gramian (see Section 3.2.2),emgr returns a cell array holding:
- $N\times N$ empirical observability Gramian matrix ${W}_{O}$,
- $P\times P$ empirical identifiability Gramian matrix ${W}_{I}$.
- ′j′
- Empirical joint Gramian (see Section 3.2.3),emgr returns a cell array holding:
- $N\times N$ empirical cross Gramian matrix ${W}_{X}$,
- $P\times P$ empirical cross-identifiability Gramian matrix ${W}_{\ddot{I}}$.
5.2.1. Non-Symmetric Cross Gramian
5.2.2. Parametric Systems
5.2.3. Time-Varying Systems
5.3. Optional Arguments
- pr system parameters (Default value: 0)
- vector a column vector holding the parameter components,
- matrix a set of parameters, each column holding one parameter.
- nf twelve component vector encoding the option flags, for details see Section 5.4.
- ut input function (Default value: 1)
- handle function handle expecting a signature u_t = u(t),
- 0 pseudo-random binary input,
- 1 delta impulse input,
- ∞ decreasing frequency exponential chirp.
- us steady-state input (Default value: 0)
- scalar set all M steady-state input components to argument,
- vector set steady-state input to argument of expected dimension $M\times 1$.
- xs steady-state (Default value: 0)
- scalar set all N steady-state components to argument,
- vector set steady-state to argument of expected dimension $N\times 1$.
- um input scales (Default value: 1)
- scalar set all M maximum input scales to argument,
- vector set maximum input scales to argument of expected dimension $M\times 1$,
- matrix set scales to argument with M rows; used as is.
- xm initial state scales (Default value: 1)
- scalar set all N maximum initial state scales to argument,
- vector set maximum steady-state scales to argument of expected dimension $N\times 1$,
- matrix set scales to argument with N rows; used as is.
- dp inner product interface via a handle to a function with the signature z = dp(x,y) defining the dot product for the Gramian matrix computation (Default value: []).
Inner Product Interface
- x
- matrix of dimension $N\times \frac{T}{h}$,
- y
- matrix of dimension $\frac{T}{h}\times n$ for $n\le N$.
5.4. Option Flags
- nf(1)
- Time series centering:
- = 0
- No centering,
- = 1
- Steady-state (for empirical covariance matrices),
- = 2
- Final state,
- = 3
- Arithmetic average over time (for empirical Gramians),
- = 4
- Root-mean-square over time,
- = 5
- Mid-range over time.
- nf(2)
- Input scale sequence:
- = 0
- Single scale: um ← um,
- = 1
- Linear scale subdivision: um ← um * [0.25, 0.5, 0.75, 1.0],
- = 2
- Geometric scale subdivision: um ← um * [0.125, 0.25, 0.5, 1.0],
- = 3
- Logarithmic scale subdivision: um ← um * [0.001, 0.01, 0.1, 1.0],
- = 4
- Sparse scale subdivision: um ← um * [0.01, 0.5, 0.99, 1.0].
- nf(3)
- Initial state scale sequence:
- = 0
- Single scale: xm ← xm,
- = 1
- Linear scale subdivision: xm ← xm * [0.25, 0.5, 0.75, 1.0],
- = 2
- Geometric scale subdivision: xm ← xm * [0.125, 0.25, 0.5, 1.0],
- = 3
- Logarithmic scale subdivision: xm ← xm * [0.001, 0.01, 0.1, 1.0],
- = 4
- Sparse scale subdivision: xm ← xm * [0.01, 0.5, 0.99, 1.0].
- nf(4)
- Input directions:
- = 0
- Positive and negative: um ← [-um, um],
- = 1
- Only positive: um ← um.
- nf(5)
- Initial state directions:
- = 0
- Positive and negative: xm ← [-xm, xm],
- = 1
- Only positive: xm ← xm.
- nf(6)
- Normalizing:
- nf(7)
- Non-Symmetric Cross Gramian, only ${W}_{X}$, ${W}_{Y}$, ${W}_{J}$:
- = 0
- Regular cross Gramian,
- = 1
- Non-symmetric cross Gramian.
- nf(8)
- Extra input for state and parameter perturbation trajectories, only ${W}_{O}$, ${W}_{X}$, ${W}_{S}$, ${W}_{I}$, ${W}_{J}$:
- = 0
- No extra input,
- = 1
- Apply extra input (see [83]).
- nf(9)
- Center parameter scales, only ${W}_{S}$, ${W}_{I}$, ${W}_{J}$:
- = 0
- No centering,
- = 1
- Center around arithmetic mean,
- = 2
- Center around logarithmic mean.
- nf(10)
- Parameter Gramian variant, only ${W}_{S}$, ${W}_{I}$, ${W}_{J}$:
- = 0
- Average input-to-state (${W}_{S}$), detailed Schur-complement (${W}_{I}$, ${W}_{J}$),
- = 1
- Average input-to-output (${W}_{S}$), approximate Schur-complement (${W}_{I}$, ${W}_{J}$).
- nf(11)
- Empirical cross Gramian partition width, only ${W}_{X}$, ${W}_{J}$:
- = 0
- Full cross Gramian computation, no partitioning.
- < N
- Maximum partition size in terms of cross Gramian columns.
- nf(12)
- Partitioned empirical cross Gramian running index, only ${W}_{X}$, ${W}_{J}$:
- = 0
- No partitioning.
- > 0
- Index of the set of cross Gramian columns to be computed.
5.4.1. Schur-Complement
5.4.2. Partitioned Computation
5.5. Solver Configuration
- f
- handle to a function with the signature xdot = f(x,u,p,t) representing the system’s vector-field and expecting the arguments: current state x, current input u, (current) parameter p and current time t.
- g
- handle to a function with the signature y = g(x,u,p,t) representing the system’s output functional and expecting the arguments: the current state x, current input u, (current) parameter p and current time t.
- t
- two component vector t = [h,T] specifying the time-step width h and time horizon T.
- x0
- column vector of dimension N setting the initial condition.
- u
- handle to a function with the signature u_t = u(t).
- p
- column vector of dimension P holding the (current) parameter.
5.6. Sample Usage
6. Numerical Examples
6.1. Linear Verification
6.2. Hyperbolic Evaluation
6.3. Nonlinear Validation
6.4. On Hyper-Reduction
7. Concluding Remark
Code Availability
Funding
Conflicts of Interest
Abbreviations
PDE | Partial Differential Equation |
ODE | Ordinary Differential Equation |
MOR | Model Order Reduction |
pMOR | parametric Model Order Reduction |
nMOR | nonlinear Model Order Reduction |
POD | Proper Orthogonal Decomposition |
bPOD | balanced Proper Orthogonal Decomposition |
SVD | Singular Value Decomposition |
HSV | Hankel Singular Values |
EVD | Eigenvalue Decomposition |
SISO | Single-Input-Single-Output |
MIMO | Multiple-Input-Multiple-Output |
BLAS | Basic Linear Algebra System |
LoC | Lines of Code |
SIMD | Single Instruction Multiple Data |
UMA | Unified Memory Access |
GPGPU | General Purpose Graphics Processing Unit |
GPU | Graphics Processing Unit |
UMM | Unified Memory Model |
hUMA | heterogeneous Unified Memory Access |
CPU | Central Processing Unit |
GEMM | GEneralized Matrix Multiplication |
HAPOD | Hierarchical Approximate Proper Orthgoonal Decomposition |
RKHS | Reproducing Kernel Hilbert Spaces |
SSP | Strong Stability Preserving |
SIMO | Single-Input-Multiple-Output |
MPE | Missing Point Estimation |
DEIM | Discrete Empirical Interpolation Method |
DMD | Dynamic Mode Decomposition |
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Himpe, C. emgr—The Empirical Gramian Framework. Algorithms 2018, 11, 91. https://doi.org/10.3390/a11070091
Himpe C. emgr—The Empirical Gramian Framework. Algorithms. 2018; 11(7):91. https://doi.org/10.3390/a11070091
Chicago/Turabian StyleHimpe, Christian. 2018. "emgr—The Empirical Gramian Framework" Algorithms 11, no. 7: 91. https://doi.org/10.3390/a11070091