Computation of Maxwell eigenmodes in an axisymmetric cavity using hierarchical vector finite elements is presented. The use of curl conforming vector basis functions, which span the null space of the curl operator, leads to the appearance of spurious modes with zero eigenvalues. Such spurious modes lead to electric flux solution with non-zero divergence. Constraining the solution space in the variational statement for the eigenvalue problem by weakly enforcing the flux to be divergence-free leads to the elimination of such modes. Discrete equivalent of such a constraint equation is developed for axisymmetric problems solved using hierarchical vector and scalar basis functions of orders complete to p=2. The discrete constraint equation, developed individually for Fourier modes m=0 and m[ge]1, is efficiently integrated with a subspace iteration-based eigenvalue solution technique such as the Lanczos/Arnoldi method. The resulting solution technique is free of spurious modes added with an advantage of seeking a solution of a positive definite matrix during each iteration of the eigenvalue solver. Convergence in solution is demonstrated for orders up to p=2, while the proposed technique can be extended to basis functions of arbitrary order. Copyright © 2009 John Wiley & Sons, Ltd.
martes, 15 de diciembre de 2009
Efficient computation of Maxwell eigenmodes in axisymmetric cavities using hierarchical vector finite elements
Computation of Maxwell eigenmodes in an axisymmetric cavity using hierarchical vector finite elements is presented. The use of curl conforming vector basis functions, which span the null space of the curl operator, leads to the appearance of spurious modes with zero eigenvalues. Such spurious modes lead to electric flux solution with non-zero divergence. Constraining the solution space in the variational statement for the eigenvalue problem by weakly enforcing the flux to be divergence-free leads to the elimination of such modes. Discrete equivalent of such a constraint equation is developed for axisymmetric problems solved using hierarchical vector and scalar basis functions of orders complete to p=2. The discrete constraint equation, developed individually for Fourier modes m=0 and m[ge]1, is efficiently integrated with a subspace iteration-based eigenvalue solution technique such as the Lanczos/Arnoldi method. The resulting solution technique is free of spurious modes added with an advantage of seeking a solution of a positive definite matrix during each iteration of the eigenvalue solver. Convergence in solution is demonstrated for orders up to p=2, while the proposed technique can be extended to basis functions of arbitrary order. Copyright © 2009 John Wiley & Sons, Ltd.
Accelerated gradient descent methods with line search
Abstract We introduced an algorithm for unconstrained optimization based on the transformation of the Newton method with the line search
into a gradient descent method. Main idea used in the algorithm construction is approximation of the Hessian by an appropriate
diagonal matrix. The steplength calculation algorithm is based on the Taylor’s development in two successive iterative points
and the backtracking line search procedure. The linear convergence of the algorithm is proved for uniformly convex functions
and strictly convex quadratic functions satisfying specified conditions.
into a gradient descent method. Main idea used in the algorithm construction is approximation of the Hessian by an appropriate
diagonal matrix. The steplength calculation algorithm is based on the Taylor’s development in two successive iterative points
and the backtracking line search procedure. The linear convergence of the algorithm is proved for uniformly convex functions
and strictly convex quadratic functions satisfying specified conditions.
- Content Type Journal Article
- Category Original Paper
- DOI 10.1007/s11075-009-9350-8
- Authors
- Predrag S. Stanimirović, University of Niš Department of Mathematics, Faculty of Science Višegradska 33 18000 Niš Serbia
- Marko B. Miladinović, University of Niš Department of Mathematics, Faculty of Science Višegradska 33 18000 Niš Serbia
- Journal Numerical Algorithms
- Online ISSN 1572-9265
- Print ISSN 1017-1398
A Central Discontinuous Galerkin Method for Hamilton-Jacobi Equations
Abstract In this paper, a central discontinuous Galerkin method is proposed to solve for the viscosity solutions of Hamilton-Jacobi
equations. Central discontinuous Galerkin methods were originally introduced for hyperbolic conservation laws. They combine
the central scheme and the discontinuous Galerkin method and therefore carry many features of both methods. Since Hamilton-Jacobi
equations in general are not in the divergence form, it is not straightforward to design a discontinuous Galerkin method to
directly solve such equations. By recognizing and following a “weighted-residual” or “stabilization-based” formulation of
central discontinuous Galerkin methods when applied to hyperbolic conservation laws, we design a high order numerical method
for Hamilton-Jacobi equations. The L
2 stability and the error estimate are established for the proposed method when the Hamiltonians are linear. The overall performance
of the method in approximating the viscosity solutions of general Hamilton-Jacobi equations are demonstrated through extensive
numerical experiments, which involve linear, nonlinear, smooth, nonsmooth, convex, or nonconvex Hamiltonians.
equations. Central discontinuous Galerkin methods were originally introduced for hyperbolic conservation laws. They combine
the central scheme and the discontinuous Galerkin method and therefore carry many features of both methods. Since Hamilton-Jacobi
equations in general are not in the divergence form, it is not straightforward to design a discontinuous Galerkin method to
directly solve such equations. By recognizing and following a “weighted-residual” or “stabilization-based” formulation of
central discontinuous Galerkin methods when applied to hyperbolic conservation laws, we design a high order numerical method
for Hamilton-Jacobi equations. The L
2 stability and the error estimate are established for the proposed method when the Hamiltonians are linear. The overall performance
of the method in approximating the viscosity solutions of general Hamilton-Jacobi equations are demonstrated through extensive
numerical experiments, which involve linear, nonlinear, smooth, nonsmooth, convex, or nonconvex Hamiltonians.
- Content Type Journal Article
- DOI 10.1007/s10915-009-9340-y
- Authors
- Fengyan Li, Rensselaer Polytechnic Institute Department of Mathematical Sciences Troy NY 12180 USA
- Sergey Yakovlev, Rensselaer Polytechnic Institute Department of Mathematical Sciences Troy NY 12180 USA
- Journal Journal of Scientific Computing
- Online ISSN 1573-7691
- Print ISSN 0885-7474
Convergence of the stochastic Euler scheme for locally Lipschitz coefficients. (arXiv:0912.2609v1 [math.NA])
Stochastic differential equations are often simulated with the Monte Carlo
Euler method. Convergence of this method is well understood in the case of
globally Lipschitz continuous coefficients of the stochastic differential
equation. The important case of superlinearly growing coefficients, however,
remained an open question for a long time now. The main difficulty is that
numerically weak convergence fails to hold in many cases of superlinearly
growing coefficients. In this paper we overcome this difficulty and establish
convergence of the Monte Carlo Euler method for a large class of
one-dimensional stochastic differential equations whose drift functions have at
most polynomial growth.
Synchrosqueezed Wavelet Transforms: a Tool for Empirical Mode Decomposition. (arXiv:0912.2437v1 [math.NA])
The EMD algorithm, first proposed in [11], made more robust as well as more
versatile in [12], is a technique that aims to decompose into their building
blocks functions that are the superposition of a (reasonably) small number of
components, well separated in the time-frequency plane, each of which can be
viewed as approximately harmonic locally, with slowly varying amplitudes and
frequencies. The EMD has already shown its usefulness in a wide range of
applications including meteorology, structural stability analysis, medical
studies -- see, e.g. [13]. On the other hand, the EMD algorithm contains
heuristic and ad-hoc elements that make it hard to analyze mathematically. In
this paper we describe a method that captures the flavor and philosophy of the
EMD approach, albeit using a different approach in constructing the components.
We introduce a precise mathematical definition for a class of functions that
can be viewed as a superposition of a reasonably small number of approximately
harmonic components, and we prove that our method does indeed succeed in
decomposing arbitrary functions in this class. We provide several examples, for
simulated as well as real data.
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