410 research outputs found
A Symbolic Transformation Language and its Application to a Multiscale Method
The context of this work is the design of a software, called MEMSALab,
dedicated to the automatic derivation of multiscale models of arrays of micro-
and nanosystems. In this domain a model is a partial differential equation.
Multiscale methods approximate it by another partial differential equation
which can be numerically simulated in a reasonable time. The challenge consists
in taking into account a wide range of geometries combining thin and periodic
structures with the possibility of multiple nested scales.
In this paper we present a transformation language that will make the
development of MEMSALab more feasible. It is proposed as a Maple package for
rule-based programming, rewriting strategies and their combination with
standard Maple code. We illustrate the practical interest of this language by
using it to encode two examples of multiscale derivations, namely the two-scale
limit of the derivative operator and the two-scale model of the stationary heat
equation.Comment: 36 page
Computer-Aided Derivation of Multi-scale Models: A Rewriting Framework
We introduce a framework for computer-aided derivation of multi-scale models.
It relies on a combination of an asymptotic method used in the field of partial
differential equations with term rewriting techniques coming from computer
science.
In our approach, a multi-scale model derivation is characterized by the
features taken into account in the asymptotic analysis. Its formulation
consists in a derivation of a reference model associated to an elementary
nominal model, and in a set of transformations to apply to this proof until it
takes into account the wanted features. In addition to the reference model
proof, the framework includes first order rewriting principles designed for
asymptotic model derivations, and second order rewriting principles dedicated
to transformations of model derivations. We apply the method to generate a
family of homogenized models for second order elliptic equations with periodic
coefficients that could be posed in multi-dimensional domains, with possibly
multi-domains and/or thin domains.Comment: 26 page
Lazy AC-Pattern Matching for Rewriting
We define a lazy pattern-matching mechanism modulo associativity and
commutativity. The solutions of a pattern-matching problem are stored in a lazy
list composed of a first substitution at the head and a non-evaluated object
that encodes the remaining computations. We integrate the lazy AC-matching in a
strategy language: rewriting rule and strategy application produce a lazy list
of terms.Comment: In Proceedings WRS 2011, arXiv:1204.531
Formal methods for Multiscale models derivation
We are currently developing a software dedicated to multiscale and multiphysics model derivation oriented to arrays of micro and nanosystems. The software is based on the two-scale transform, together with formal specification and verification techniques in computer science. It helps to derive multiphysics models for complex geometries including thin or periodic structures or combinations of them. Final models are correct by construction since human errors are avoided and model derivation effort is dramatically reduced
Prospects for Declarative Mathematical Modeling of Complex Biological Systems
Declarative modeling uses symbolic expressions to represent models. With such
expressions one can formalize high-level mathematical computations on models
that would be difficult or impossible to perform directly on a lower-level
simulation program, in a general-purpose programming language. Examples of such
computations on models include model analysis, relatively general-purpose
model-reduction maps, and the initial phases of model implementation, all of
which should preserve or approximate the mathematical semantics of a complex
biological model. The potential advantages are particularly relevant in the
case of developmental modeling, wherein complex spatial structures exhibit
dynamics at molecular, cellular, and organogenic levels to relate genotype to
multicellular phenotype. Multiscale modeling can benefit from both the
expressive power of declarative modeling languages and the application of model
reduction methods to link models across scale. Based on previous work, here we
define declarative modeling of complex biological systems by defining the
operator algebra semantics of an increasingly powerful series of declarative
modeling languages including reaction-like dynamics of parameterized and
extended objects; we define semantics-preserving implementation and
semantics-approximating model reduction transformations; and we outline a
"meta-hierarchy" for organizing declarative models and the mathematical methods
that can fruitfully manipulate them
Nested Integrals and Rationalizing Transformations
A brief overview of some computer algebra methods for computations with
nested integrals is given. The focus is on nested integrals over integrands
involving square roots. Rewrite rules for conversion to and from associated
nested sums are discussed. We also include a short discussion comparing the
holonomic systems approach and the differential field approach. For
simplification to rational integrands, we give a comprehensive list of
univariate rationalizing transformations, including transformations tuned to
map the interval bijectively to itself.Comment: manuscript of 25 February 2021, in "Anti-Differentiation and the
Calculation of Feynman Amplitudes", Springe
An iterative semi-implicit scheme with robust damping
An efficient, iterative semi-implicit (SI) numerical method for the time
integration of stiff wave systems is presented. Physics-based assumptions are
used to derive a convergent iterative formulation of the SI scheme which
enables the monitoring and control of the error introduced by the SI operator.
This iteration essentially turns a semi-implicit method into a fully implicit
method. Accuracy, rather than stability, determines the timestep. The scheme is
second-order accurate and shown to be equivalent to a simple preconditioning
method. We show how the diffusion operators can be handled so as to yield the
property of robust damping, i.e., dissipating the solution at all values of the
parameter \mathcal D\dt, where is a diffusion operator and \dt
the timestep. The overall scheme remains second-order accurate even if the
advection and diffusion operators do not commute. In the limit of no physical
dissipation, and for a linear test wave problem, the method is shown to be
symplectic. The method is tested on the problem of Kinetic Alfv\'en wave
mediated magnetic reconnection. A Fourier (pseudo-spectral) representation is
used. A 2-field gyrofluid model is used and an efficacious k-space SI operator
for this problem is demonstrated. CPU speed-up factors over a CFL-limited
explicit algorithm ranging from to several hundreds are obtained,
while accurately capturing the results of an explicit integration. Possible
extension of these results to a real-space (grid) discretization is discussed.Comment: Submitted to the Journal of Computational Physics. Clarifications and
caveats in response to referees, numerical demonstration of convergence rate,
generalized symplectic proo
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