5,090 research outputs found
How AD Can Help Solve Differential-Algebraic Equations
A characteristic feature of differential-algebraic equations is that one
needs to find derivatives of some of their equations with respect to time, as
part of so called index reduction or regularisation, to prepare them for
numerical solution. This is often done with the help of a computer algebra
system. We show in two significant cases that it can be done efficiently by
pure algorithmic differentiation. The first is the Dummy Derivatives method,
here we give a mainly theoretical description, with tutorial examples. The
second is the solution of a mechanical system directly from its Lagrangian
formulation. Here we outline the theory and show several non-trivial examples
of using the "Lagrangian facility" of the Nedialkov-Pryce initial-value solver
DAETS, namely: a spring-mass-multipendulum system, a prescribed-trajectory
control problem, and long-time integration of a model of the outer planets of
the solar system, taken from the DETEST testing package for ODE solvers
Index Reduction for Differential-Algebraic Equations with Mixed Matrices
Differential-algebraic equations (DAEs) are widely used for modeling of
dynamical systems. The difficulty in solving numerically a DAE is measured by
its differentiation index. For highly accurate simulation of dynamical systems,
it is important to convert high-index DAEs into low-index DAEs. Most of
existing simulation software packages for dynamical systems are equipped with
an index-reduction algorithm given by Mattsson and S\"{o}derlind.
Unfortunately, this algorithm fails if there are numerical cancellations.
These numerical cancellations are often caused by accurate constants in
structural equations. Distinguishing those accurate constants from generic
parameters that represent physical quantities, Murota and Iri introduced the
notion of a mixed matrix as a mathematical tool for faithful model description
in structural approach to systems analysis. For DAEs described with the use of
mixed matrices, efficient algorithms to compute the index have been developed
by exploiting matroid theory.
This paper presents an index-reduction algorithm for linear DAEs whose
coefficient matrices are mixed matrices, i.e., linear DAEs containing physical
quantities as parameters. Our algorithm detects numerical cancellations between
accurate constants, and transforms a DAE into an equivalent DAE to which
Mattsson--S\"{o}derlind's index-reduction algorithm is applicable. Our
algorithm is based on the combinatorial relaxation approach, which is a
framework to solve a linear algebraic problem by iteratively relaxing it into
an efficiently solvable combinatorial optimization problem. The algorithm does
not rely on symbolic manipulations but on fast combinatorial algorithms on
graphs and matroids. Furthermore, we provide an improved algorithm under an
assumption based on dimensional analysis of dynamical systems.Comment: A preliminary version of this paper is to appear in Proceedings of
the Eighth SIAM Workshop on Combinatorial Scientific Computing, Bergen,
Norway, June 201
The Exponentially Faster Stick-Slip Dynamics of the Peeling of an Adhesive Tape
The stick-slip dynamics is considered from the nonlinear
differential-algebraic equation (DAE) point of view and the peeling dynamics is
shown to be a switching differential index DAE model. In the stick-slip regime
with bifurcations, the differential index can be arbitrarily high. The time
scale of the peeling velocity, the algebraic variable, in this regime is shown
to be exponentially faster compared to the angular velocity of the spool and/or
the stretch rate of the tape. A homogenization scheme for the peeling velocity
which is characterized by the bifurcations is discussed and is illustrated with
numerical examples.Comment: 7 figures, 24 page
Medical image denoising using convolutional denoising autoencoders
Image denoising is an important pre-processing step in medical image
analysis. Different algorithms have been proposed in past three decades with
varying denoising performances. More recently, having outperformed all
conventional methods, deep learning based models have shown a great promise.
These methods are however limited for requirement of large training sample size
and high computational costs. In this paper we show that using small sample
size, denoising autoencoders constructed using convolutional layers can be used
for efficient denoising of medical images. Heterogeneous images can be combined
to boost sample size for increased denoising performance. Simplest of networks
can reconstruct images with corruption levels so high that noise and signal are
not differentiable to human eye.Comment: To appear: 6 pages, paper to be published at the Fourth Workshop on
Data Mining in Biomedical Informatics and Healthcare at ICDM, 201
Agricultural Policy Adjustments in East Asia: The Korean Rice Economy
The processes of agricultural policy decision-making in the East Asian economies of Japan, Taiwan, and Korea are characterized by the complex interactions of competing special interest groups in their respective political arenas. An analytical model is designed to investigate how adjustments in rice price policies are endogenously related to the political economic forces associated with macroeconomic changes. The model is tested against the case of Korean rice price policy, which has not been satisfactorily explained in earlier modeling efforts. A political preference function is used to estimate the relative political weights of producers, consumers, and government over a 25-year period (1961-1985) during which major macroeconomic changes occurred. A simultaneous 12-equation model is then constructed to explore the effects of the macroeconomic changes on the rice pricing decisions through the estimated political weights. Highly significant econometric results are obtained to explain the pattern of increasing political influence of producers relative to that of consumers and government. Simulation experiments show how the patterns in the rice economy are linked to macroeconomic change. The results obtained, thus far, open the way for methodological improvements to further our functional understanding of agricultural policy adjustments in Korea and other East Asian economies.endogenous rice policy, political macroeconomy, political preference function, political weights, econometric simulation, East Asia, Korea, Agricultural and Food Policy, Crop Production/Industries,
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
Constraint-consistent Runge-Kutta methods for one-dimensional incompressible multiphase flow
New time integration methods are proposed for simulating incompressible
multiphase flow in pipelines described by the one-dimensional two-fluid model.
The methodology is based on 'half-explicit' Runge-Kutta methods, being explicit
for the mass and momentum equations and implicit for the volume constraint.
These half-explicit methods are constraint-consistent, i.e., they satisfy the
hidden constraints of the two-fluid model, namely the volumetric flow
(incompressibility) constraint and the Poisson equation for the pressure. A
novel analysis shows that these hidden constraints are present in the
continuous, semi-discrete, and fully discrete equations.
Next to constraint-consistency, the new methods are conservative: the
original mass and momentum equations are solved, and the proper shock
conditions are satisfied; efficient: the implicit constraint is rewritten into
a pressure Poisson equation, and the time step for the explicit part is
restricted by a CFL condition based on the convective wave speeds; and
accurate: achieving high order temporal accuracy for all solution components
(masses, velocities, and pressure). High-order accuracy is obtained by
constructing a new third order Runge-Kutta method that satisfies the additional
order conditions arising from the presence of the constraint in combination
with time-dependent boundary conditions.
Two test cases (Kelvin-Helmholtz instabilities in a pipeline and liquid
sloshing in a cylindrical tank) show that for time-independent boundary
conditions the half-explicit formulation with a classic fourth-order
Runge-Kutta method accurately integrates the two-fluid model equations in time
while preserving all constraints. A third test case (ramp-up of gas production
in a multiphase pipeline) shows that our new third order method is preferred
for cases featuring time-dependent boundary conditions
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