Application Of Model Order Reduction Research Articles

REDUCED-ORDER MODELS FOR THE IMPLIED VARIANCE UNDER LOCAL VOLATILITY

Implied volatility is a key value in financial mathematics. We discuss some of the pros and cons of the standard ways to compute this quantity, i.e. numerical inversion of the well-known Black–Scholes formula or asymptotic expansion approximations, and propose a new way to directly calculate the implied variance in a local volatility framework based on the solution of a quasilinear degenerate parabolic partial differential equation. Since the numerical solution of this equation may lead to large nonlinear systems of equations and thus high computation times compared to the classical approaches, we apply model order reduction techniques to achieve computational efficiency. Our method of choice for the derivation of a reduced-order model (ROM) will be proper orthogonal decomposition (POD). This strategy is additionally combined with the discrete empirical interpolation method (DEIM) to deal with the nonlinear terms. Numerical results prove the quality of our approach compared to other methods.

Adaptive-order rational Arnoldi-type methods in computational electromagnetism

In this paper, we will consider moment-matching methods for the application of model order reduction to Maxwell’s equations. Since the main difficulty of moment-matching methods results from the adequate determination of a set of expansion points, we will introduce an adaptive expansion point selection on the basis of a suitable approximation of an upper bound of the output moment-matching error. Furthermore, we give some remarks about structure- and passivity preserving adaptive-order rational Arnoldi methods for Maxwell’s equations. Numerical examples indicate the reliability of the proposed algorithm.

A semi-explicit formulation of a coupled electromagnetic field/circuit problem

The demand of combining circuit simulation directly with complex device models to refine critical circuit parts becomes more and more important, since the classical circuit simulation can no longer supply sufficiently accurate results. The simulation of such coupled problems leads to large systems and therefore to high computing times. We consider a set of differential–algebraic equations, which arise from an electric circuit modeled by the modified nodal analysis coupled with electromagnetic devices. While the normal circuit elements are zero dimensional elements the electromagnetic devices are given by a three dimensional model. Therefore the number of variables can easily go beyond millions, if we refine the spatial discretization. We analyze the structure of the discretized coupled system and present a way to transform it into a semi-explicit system of differential–algebraic equations. In the process we make use of a new decoupling method for DAEs which results from a mix of the Strangeness Index and the Tractability Index. This remodeling allows us to prove a global unique solvability result for the coupled circuit/field problem and it is also a crucial step if we want to apply model order reduction techniques or semi-explicit solvers.

Model reduction for optimal control problems in field‐flow fractionation

AbstractWe discuss the application of model order reduction to optimal control problems governed by coupled systems of the Stokes‐Brinkman and advection‐diffusion equations. Such problems arise in field‐flow fractionation processes for the efficient and fast separation of particles of different size in microfluidic flows. Our approach is based on a combination of balanced truncation and tangential interpolation for model reduction of the semidiscretized optimality system. Numerical results demonstrate the properties of this approach. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Nonlinear modelling of the middle ear as an elastic multibody system — Applying model order reduction to acousto-structural coupled systems

In this study, modelling of the human hearing is considered. Due to the nonlinearity of the middle ear, the sound transfer changes as the equilibrium position of the middle ear structure varies. For the description of the middle ear a nonlinear elastic multibody system is derived. The tympanic membrane and the air in the ear canal as well as in the tympanic cavity are considered as elastic bodies. They are first modelled using the finite element method. The large number of degrees of freedom makes a following reduction step of the acousto-structural finite element model inevitable. The second-order structure of the system matrices is preserved by applying reduction techniques based on Petrov–Galerkin projection. The nonlinearity of the tympanic membrane is included following the approach of parametric model order reduction by matrix interpolation assuming that the nonlinearity can be represented by the relative pressure between the ear canal and the tympanic cavity. Finally the static and dynamic behaviour of the simulation model is reviewed for different static pressure loads of the middle ear.

Basic principles and aims of model order reduction in compliant mechanisms

Abstract. Model order reduction appears to be beneficial for the synthesis and simulation of compliant mechanisms due to computational costs. Model order reduction is an established method in many technical fields for the approximation of large-scale linear time-invariant dynamical systems described by ordinary differential equations. Based on system theory, underlying representations of the dynamical system are introduced from which the general reduced order model is derived by projection. During the last years, numerous new procedures were published and investigated appropriate to simulation, optimization and control. Singular value decomposition, condensation-based and Krylov subspace methods representing three order reduction methods are reviewed and their advantages and disadvantages are outlined in this paper. The convenience of applying model order reduction in compliant mechanisms is quoted. Moreover, the requested attributes for order reduction as a future research direction meeting the characteristics of compliant mechanisms are commented.

Efficient simulation of hard disk drive operational shock response using model order reduction

The complex structure, coupled mechanical and fluidic energy domains, and inherent nonlinearity of air bearing between slider and disk involved in the hard disk drive (HDD) are normally presented as a large scale problem which will result in very heavy computational costs in terms of intensive computation and time consuming for HDD research communities and industries to carry out the transient dynamic simulation for HDD design verification, performance analysis, and optimization by using the traditional full-order models, such as finite element model (FEM). This paper presents a method of application of model order reduction (MOR) technique to dramatically reduce the computation time for HDD transient shock performance analysis while capturing the behaviors of original problem faithfully. The reduced models are obtained by performing MOR directly to the FEMs through Krylov subspace and Arnoldi algorithm. The transient operational shock response results of the reduced models of a head suspension assembly (HSA) subjected to half-sine shock pulse demonstrate that the reduced models can dramatically reduce total computation by at least three orders and have very good agreement with those simulated from the original large problem by full-order FEM.

Reduction of Finite Element Mesh and Model Order for Fast Dynamic Analysis of Global/Local Problem

MEMS structures and machines fabricated by photolithography and/or etching have often repeated unit microstructure. We need to design not only the global parameters but also microscopic and local parameters. In pursuit of this, a dynamic analysis of global/local problem is essential, and especially fast computational method and convenient modeling technique are both required for the MEMS design. Hence, this paper presents the application of model order reduction (MOR) for fast dynamic analysis combined with the finite element mesh superposition (FEMS) technique for practical and convenient modeling of the above-mentioned microstructures. Through two examples, the accuracy is carefully investigated for both global and local responses associated with the base vectors in MOR algorithm. The first example is analyzed by a kind of homogenization technique and MOR. The number of DOFs was finally reduced by a factor of approximately 1/6 (47,362/296,268). In addition, MOR enabled us to reduce the coefficient matrix size to only 300. The second example is analyzed by both FEMS and MOR, and it was found that MOR is also applicable to an unusual matrices generated by FEMS.

PEEC capacitance extraction of 3-D interconnects

As feature size of electronic devices decreases, fast and accurate capacitance extraction has become increasingly critical for verification and analysis as regard as electromagnetic compat- ibility and signal integrity issues. The partial element equivalent circuit method is implemented for three-dimensional capacitance extraction of interconnects and very large-scale integration circuits with multiple dielectrics. The electric field coupling due to free and bound charges are analysed and modelled separately thus allowing to distinguish their contribution. The proposed approach provides physical insight, totally compatible fast methods for accelerating matrix-vector products, allows an easy treatment of lossy and dispersive dielectrics and is well suited for applying model order reduction techniques. Modern electronic design is surely extremely challenging. It is known that power distribution and management, digital computing and radio frequency wireless communication have very different bandwidths, as well as voltage and current levels. In addition, today's mobile devices share the same printed circuit board and/or packaging substrate in a very dense design. The consequence of this fact is that parasitics and interference poses significant design challenges and deserves a special attention to electromag- netic compatibility (EMC) engineers to meet design requirements. Parasitics may compromise the signal fidelity and/or level and dominate the coupling in the case of elec- tromagnetic interference and immunity. Typically, coupling takes place through the electric field (capacitive coupling), magnetic field (inductive coupling) or through a common current return path. It is understood that design meeting EMC criteria has a critical need for methodologies and computer-aided design tools, which allow to predict para- sitic effects and capture coupling physics. Also it is extre- mely desirable to extract parasitic lumped element circuit models from a layout geometry. As the feature size of electronics in mobile devices has dramatically decreased, the interconnect is becoming a dominant factor in system delay and signal integrity (SI). An accurate and efficient electromagnetic modelling of three-dimensional (3-D) high speed interconnects and very large-scale integration (VLSI) circuits embedded in mul- tiple dielectrics is extremely important for the design and verification of nowadays systems on package and systems on-chip for communication systems. Circuit performances such as SI, EMC, delay, radiation, power consumption, reliability are strongly influenced by the electrical charac- teristics of interconnects. Among the other effects, timing verification requires accurate interconnect modelling of parasitic parameters especially as regard as the parasitic

Application of Model Order Reduction to Compressor Aeroelastic Models

Abstract A model order reduction technique that yields low-order models of blade row unsteady aerodynamics is introduced. The technique is applied to linearized unsteady Euler CFD solutions in such a way that the resulting blade row models can be linked to their surroundings through their boundary conditions. The technique is applied to a transonic compressor aeroelastic analysis, in which the high-fidelity CFD forced-response results are better captured than with models that use single-frequency influence coefficients. A low-speed compressor stage is also modeled to demonstrate the multistage capability of the method. These examples demonstrate how model order reduction can be used to systematically improve the versatility, fidelity, and range of applicability of the low-order aerodynamic models typically used for incorporation of CFD results into aeroelastic analyses.

Application of model order reduction to a hydropneumatic vehicle suspension

The simulation, analysis and controller design of technical systems are frequently complicated by the complexity of the corresponding (mostly nonlinear) system models. The method presented here can simplify these tasks by reducing the number of state equations describing the system. Starting from a special state space representation the nonlinear terms are taken over into the reduced model and all couplings of state variables, input variables and nonlinear functions are redetermined. A dominance analysis helps in choosing the system order and the dominant state variables. The order of the treated model of a vehicle suspension is reduced from order 10 to 7 or 5, shortening the simulation time by a factor of 10 or 12, respectively.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>