Model order reduction (MOR) is a technique for reducing the
computational complexity of
mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
s in
numerical simulation
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be dete ...
s. As such it is closely related to the concept of
metamodeling
A metamodel or surrogate model is a model of a model, and metamodeling is the process of generating such metamodels. Thus metamodeling or meta-modeling is the analysis, construction and development of the frames, rules, constraints, models and ...
, with applications in all areas of
mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
ling.
Overview
Many modern
mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
s of real-life processes pose challenges when used in
numerical simulation
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be dete ...
s, due to complexity and large size (dimension). Model order reduction aims to lower the computational complexity of such problems, for example, in simulations of large-scale
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s and
control system
A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial c ...
s. By a reduction of the model's associated
state space
A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory.
For instance, the toy ...
dimension or
degrees of freedom, an approximation to the original model is computed which is commonly referred to as a reduced order model.
Reduced order models are useful in settings where it is often unfeasible to perform
numerical simulation
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be dete ...
s using the complete full order model. This can be due to limitations in
computational resource
In computational complexity theory, a computational resource is a resource used by some computational models in the solution of computational problems.
The simplest computational resources are computation time, the number of steps necessary t ...
s or the requirements of the simulations setting, for instance
real-time simulation
Real-time simulation refers to a computer model of a physical system that can execute at the same rate as actual "wall clock" time. In other words, the computer model runs at the same rate as the actual physical system. For example, if a tank tak ...
settings or many-query settings in which a large number of simulations needs to be performed.
Examples of Real-time simulation settings include
control systems
A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial c ...
in electronics and
visualization
Visualization or visualisation may refer to:
*Visualization (graphics), the physical or imagining creation of images, diagrams, or animations to communicate a message
* Data visualization, the graphic representation of data
* Information visualiz ...
of model results while examples for a many-query setting can include
optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
problems and design exploration. In order to be applicable to real-world problems, often the requirements of a reduced order model are:
* A small
approximation error compared to the full order model.
* Conservation of the properties and characteristics of the full order model (E.g. stability and
passivity in electronics).
* Computationally efficient and robust reduced order modelling techniques.
It is interesting to note that in some cases (e.g. constrained lumping of polynomial differential equations) it is possible to have a null approximation error, resulting in an exact model order reduction.
Methods
Contemporary model order reduction techniques can be broadly classified into 5 classes:
*
Proper orthogonal decomposition
The proper orthogonal decomposition is a numerical method that enables a reduction in the complexity of computer intensive simulations such as computational fluid dynamics and structural analysis (like crash simulations). Typically in fluid dynam ...
methods.
* Reduced basis methods.
* Balancing methods
* simplified physics
or operational based reduction methods.
* Nonlinear manifold methods.
The simplified physics approach can be described to be analogous to the traditional
mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
ling approach, in which a less complex description of a system is constructed based on assumptions and simplifications using physical insight or otherwise derived information. However, this approach is not often the topic of discussion in the context of model order reduction as it is a general method in science, engineering, and mathematics.
The remaining listed methods fall into the category of projection-based reduction. Projection-based reduction relies on the projection of either the model equations or the solution onto a basis of reduced dimensionality compared to the original solution space. Methods that also fall into this class but are perhaps less common are:
*
Proper generalized decomposition
The proper generalized decomposition (PGD) is an iterative numerical method for solving boundary value problems (BVPs), that is, partial differential equations constrained by a set of boundary conditions, such as the Poisson's equation or the Lapla ...
* Matrix interpolation
* Transfer function interpolation
* Piecewise tangential interpolation
* Loewner framework
* (Empirical)
cross Gramian
*
Krylov subspace methods
Implementations
* RBmatlab: A MATLAB library containing all reduced simulation approaches for linear and nonlinear, affine or arbitrarily parameter dependent evolution problems with finite element, finite volume or local discontinuous Galerkin discretizations. Further information can be found on th
download and documentation page
* pyMOR: pyMOR is a software library for building model order reduction applications with the Python programming language. Its main focus lies on the application of reduced basis methods to parameterized partial differential equations. All algorithms in pyMOR are formulated in terms of abstract interfaces for seamless integration with external high-dimensional PDE solvers. Moreover, pure Python implementations of finite element and finite volume discretizations using the NumPy/SciPy scientific computing stack are provided for getting started quickly. For more information, visit http://pymor.org
* emgr: Empirical Gramian Framework. Empirical gramians can be computed for linear and nonlinear control systems for purposes of model order reduction, uncertainty quantification or system identification. The emgr framework is a compact open source toolbox for gramian-based model reduction and compatible with OCTAVE and MATLAB. More at: http://gramian.de
* KerMor: An object-oriented MATLAB© library providing routines for model order reduction of nonlinear dynamical systems. Reduction can be achieved via subspace projection and approximation of nonlinearities via kernels methods or DEIM. Standard procedures like the POD-Greedy method are readily implemented as well as advanced a-posteriori error estimators for various system configurations. KerMor also includes several working examples and some demo files to quickly get familiarized with the provided functionality. More information can be found at http://www.morepas.org/software/kermor/
* JaRMoS: JaRMoS stands for “Java Reduced Model Simulations” and aims to enable import and simulation of various reduced models from multiple sources on any java-capable platform. So far support for RBmatlab, KerMor and rbMIT reduced models is present, where we can only import the rbMIT models that have previously been published with the rbAppMIT Android application. Extensions so far are a desktop-version to run reduced models and initial support for KerMor kernel-based reduced models is on the way. More information can be found at http://www.morepas.org/software/jarmos/
* MORLAB: Model Order Reduction Laboratory. This toolbox is a collection of MATLAB/OCTAVE routines for model order reduction of linear dynamical systems based on the solution of matrix equations. The implementation is based on spectral projection methods, e.g., methods based on the matrix sign function and the matrix disk function. For more details on this software, see: https://www.mpi-magdeburg.mpg.de/projects/morlab
* Dune-RB: A module for the Dune library (
www.dune-project.org, http://dune.mathematik.uni-freiburg.de), which realizes C++ template classes for use in snapshot generation and RB offline phases for various discretizations. Apart from single-core algorithms, the package also aims at using parallelization techniques for efficient snapshot generation. More at: http://users.dune-project.org/projects/dune-rb/wiki
* libROM: Collection of C++ classes that compute model order reduction and hyper-reduction for systems of partial and ordinary differential equations. libROM includes scalable and parallel, adaptive methods for proper orthogonal decomposition, parallel, non-adaptive methods for hyper-reduction, and randomized singular value decomposition. libROM also includes the dynamic mode decomposition capability. libROM has physics-informed greedy sampling capability. Source codes can be found at: https://github.com/LLNL/libROM. Webpage can be found at: https://www.librom.net, where you can find many examples, e.g., reduced order models for Lagrangian hydrodynamics with shock-moving wave.
* Pressio: Pressio is an open-source project aimed at alleviating the intrusive nature of projection-based reduced-order models for large-scale codes. The core of the project is a header-only C++ library that leverages generic programming to interface with shared or distributed memory applications using arbitrary data-types. Pressio provides numerous functionalities and solvers for performing model reduction, such as Galerkin and least-squares Petrov–Galerkin projections. The Pressio ecosystem also offers: (1) ''pressio4py'', a Python binding library for ease of prototyping, (2) ''pressio-tutorials'', a library also offering end-to-end demos that one can easily play with, which can be found at https://pressio.github.io/pressio-tutorials/, (3) ''pressio-tools'', a library for large-scale SVD, QR and sample mesh, and (4) ''pressio-demoapps,'' a suite of 1d, 2d and 3d demo applications for testing ROMs and hyper-reduction. The ecosystem main website can be found at https://pressio.github.io/, the C++ library documentation at https://pressio.github.io/pressio/.
Applications
Model order reduction finds application within all fields involving mathematical modelling and many reviews
exist for the topics of
electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
,
fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
,
hydrodynamics,
structural mechanics,
Boltzmann equation
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerne ...
,
and
design optimization.
Fluid mechanics
Current problems in fluid mechanics involve large
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s representing many effects on many different scales.
Computational fluid dynamics
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate th ...
studies often involve models solving the
Navier–Stokes equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
with a number of
degrees of freedom in the order of magnitude upwards of
. The first usage of model order reduction techniques dates back to the work of Lumley in 1967, where it was used to gain insight into the mechanisms and intensity of
turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
and
large coherent structures present in fluid flow problems. Model order reduction also finds modern applications in
aeronautics
Aeronautics is the science or art involved with the study, design, and manufacturing of air flight–capable machines, and the techniques of operating aircraft and rockets within the atmosphere. The British Royal Aeronautical Society identifies ...
to model the flow over the body of aircraft. An example can be found in Lieu et al in which the full order model of an
F16 fighter-aircraft with over 2.1 million degrees of freedom, was reduced to a model of just 90 degrees of freedom. Additionally reduced order modeling has been applied to study
rheology
Rheology (; ) is the study of the flow of matter, primarily in a fluid (liquid or gas) state, but also as "soft solids" or solids under conditions in which they respond with Plasticity (physics), plastic flow rather than deforming Elasticity (phy ...
in
hemodynamics
Hemodynamics or haemodynamics are the dynamics of blood flow. The circulatory system is controlled by homeostatic mechanisms of autoregulation, just as hydraulic circuits are controlled by control systems. The hemodynamic response continuously m ...
and the
fluid–structure interaction Fluid–structure interaction (FSI) is the interaction of some movable or deformable structure with an internal or surrounding fluid flow. Fluid–structure interactions can be stable or oscillatory. In oscillatory interactions, the strain induced i ...
between the blood flowing through the vascular system and the vascular walls.
See also
*
Dimension reduction
Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally ...
*
Metamodeling
A metamodel or surrogate model is a model of a model, and metamodeling is the process of generating such metamodels. Thus metamodeling or meta-modeling is the analysis, construction and development of the frames, rules, constraints, models and ...
*
Principal component analysis
Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...
*
Singular value decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is re ...
*
Nonlinear dimensionality reduction
*
System identification
*
Iterative rational Krylov algorithm (IRKA)
References
Further reading
*
*
*
*
*
*
*{{Cite book
, last1 = Benner
, first1 = Peter
, last2 = Cohen
, first2 = Albert
, last3 = Ohlberger
, first3 = Mario
, last4 = Willcox
, first4 = Karen
, title = Model Reduction and Approximation: Theory and Algorithms
, doi = 10.1137/1.9781611974829
, year = 2017
, publisher = SIAM Publications
, isbn = 978-1-611974-81-2
External links
Model Order Reduction WikiModel Reduction for Parametrized SystemsEuropean Model Reduction Network
Numerical analysis
Mathematical modeling