54,213 research outputs found
Exponential Runge-Kutta methods for stiff kinetic equations
We introduce a class of exponential Runge-Kutta integration methods for
kinetic equations. The methods are based on a decomposition of the collision
operator into an equilibrium and a non equilibrium part and are exact for
relaxation operators of BGK type. For Boltzmann type kinetic equations they
work uniformly for a wide range of relaxation times and avoid the solution of
nonlinear systems of equations even in stiff regimes. We give sufficient
conditions in order that such methods are unconditionally asymptotically stable
and asymptotic preserving. Such stability properties are essential to guarantee
the correct asymptotic behavior for small relaxation times. The methods also
offer favorable properties such as nonnegativity of the solution and entropy
inequality. For this reason, as we will show, the methods are suitable both for
deterministic as well as probabilistic numerical techniques
A RBF partition of unity collocation method based on finite difference for initial-boundary value problems
Meshfree radial basis function (RBF) methods are popular tools used to
numerically solve partial differential equations (PDEs). They take advantage of
being flexible with respect to geometry, easy to implement in higher
dimensions, and can also provide high order convergence. Since one of the main
disadvantages of global RBF-based methods is generally the computational cost
associated with the solution of large linear systems, in this paper we focus on
a localizing RBF partition of unity method (RBF-PUM) based on a finite
difference (FD) scheme. Specifically, we propose a new RBF-PUM-FD collocation
method, which can successfully be applied to solve time-dependent PDEs. This
approach allows to significantly decrease ill-conditioning of traditional
RBF-based methods. Moreover, the RBF-PUM-FD scheme results in a sparse matrix
system, reducing the computational effort but maintaining at the same time a
high level of accuracy. Numerical experiments show performances of our
collocation scheme on two benchmark problems, involving unsteady
convection-diffusion and pseudo-parabolic equations
Cardiac Electromechanics: The effect of contraction model on the mathematical problem and accuracy of the numerical scheme
Models of cardiac electromechanics usually contain a contraction model determining the active tension induced at the cellular level, and the equations of nonlinear elasticity to determine tissue deformation in response to this active tension. All contraction models are dependent on cardiac electro-physiology, but can also be dependent on\ud
the stretch and stretch-rate in the fibre direction. This fundamentally affects the mathematical problem being solved, through classification of the governing PDEs, which affects numerical schemes that can be used to solve the governing equations. We categorise contraction models into three types, and for each consider questions such as classification and the most appropriate choice from two numerical methods (the explicit and implicit schemes). In terms of mathematical classification, we consider the question of strong ellipticity of the total strain energy (important for precluding ‘unnatural’ material behaviour) for stretch-rate-independent contraction models; whereas for stretch-rate-dependent contraction models we introduce a corresponding third-order problem and explain how certain choices of boundary condition could lead to constraints on allowable initial condition. In terms of suitable numerical methods, we show that an explicit approach (where the contraction model is integrated in the timestep prior to the bulk deformation being computed) is: (i) appropriate for stretch-independent contraction models; (ii) only conditionally-stable, with the stability criterion independent of timestep, for contractions models which just depend on stretch (but not stretch-rate), and (iii) inappropriate for stretch-rate-dependent models
A rational deferred correction approach to parabolic optimal control problems
The accurate and efficient solution of time-dependent PDE-constrained optimization problems is a challenging task, in large part due to the very high dimension of the matrix systems that need to be solved. We devise a new deferred correction method for coupled systems of time-dependent PDEs, allowing one to iteratively improve the accuracy of low-order time stepping schemes. We consider two variants of our method, a splitting and a coupling version, and analyze their convergence properties. We then test our approach on a number of PDE-constrained optimization problems. We obtain solution accuracies far superior to that achieved when solving a single discretized problem, in particular in cases where the accuracy is limited by the time discretization. Our approach allows for the direct reuse of existing solvers for the resulting matrix systems, as well as state-of-the-art preconditioning strategies
Contracting Nonlinear Observers: Convex Optimization and Learning from Data
A new approach to design of nonlinear observers (state estimators) is
proposed. The main idea is to (i) construct a convex set of dynamical systems
which are contracting observers for a particular system, and (ii) optimize over
this set for one which minimizes a bound on state-estimation error on a
simulated noisy data set. We construct convex sets of continuous-time and
discrete-time observers, as well as contracting sampled-data observers for
continuous-time systems. Convex bounds for learning are constructed using
Lagrangian relaxation. The utility of the proposed methods are verified using
numerical simulation.Comment: conference submissio
A One Step Method for the Solution of General Second Order Ordinary Differential Equations
In this paper, an implicit one step method for the numerical solution of second order initial value problems of ordinary differential equations has been developed by collocation and interpolation technique. The introduction of an o step point guaranteed the zero stability and consistency of the method. The implicit method developed was implemented as a block which gave simultaneous solutions, as well as their rst derivatives, at both o step and the step point. A comparison of our method to the predictor-corrector method after solving some sample problems reveals that our method performs better
Deformations of minimal Lagrangian submanifolds with boundary
Let be a special Lagrangian submanifold of a compact, Calabi-Yau manifold
with boundary lying on the symplectic, codimension 2 submanifold . It is
shown how deformations of which keep the boundary of confined to
can be described by an elliptic boundary value problem, and two results about
minimal Lagrangian submanifolds with boundary are derived using this fact. The
first is that the space of minimal Lagrangian submanifolds near with
boundary on is found to be finite dimensional and is parametrised over the
space of harmonic 1-forms of satisfying Neumann boundary conditions. The
second is that if is a symplectic, codimension 2 submanifold sufficiently
near , then under suitable conditions, there exists a minimal Lagrangian
submanifold near with boundary on .Comment: Final version; to appear in Proceedings of the American Mathematical
Society. The presentation is somewhat cleaner in places and the result is
restated for a general Calabi-Yau settin
- …