40,102 research outputs found

    The Reproducing Kernel Hilbert Space Method for Solving System of Linear Weakly Singular Volterra Integral Equations

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    The exact solutions of a system of linear weakly singular Volterra integral equations (VIE) have been a difficult to find.  The aim of this paper is to apply reproducing kernel Hilbert space (RKHS) method to find the approximate solutions to this type of systems. At first, we used Taylor's expansion to omit the singularity.  From an expansion the given system of linear weakly singular VIE is transform into a system of linear ordinary differential equations (LODEs).   The approximate solutions are represent in the form of series in the reproducing kernel space . By comparing with the exact solutions of two examples, we saw that RKHS is a powerful, easy to apply and full efficiency in scientific applications to build a solution without linearization and turbulence or discretization.&nbsp

    Learning High-Dimensional Nonparametric Differential Equations via Multivariate Occupation Kernel Functions

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    Learning a nonparametric system of ordinary differential equations (ODEs) from nn trajectory snapshots in a dd-dimensional state space requires learning dd functions of dd variables. Explicit formulations scale quadratically in dd unless additional knowledge about system properties, such as sparsity and symmetries, is available. In this work, we propose a linear approach to learning using the implicit formulation provided by vector-valued Reproducing Kernel Hilbert Spaces. By rewriting the ODEs in a weaker integral form, which we subsequently minimize, we derive our learning algorithm. The minimization problem's solution for the vector field relies on multivariate occupation kernel functions associated with the solution trajectories. We validate our approach through experiments on highly nonlinear simulated and real data, where dd may exceed 100. We further demonstrate the versatility of the proposed method by learning a nonparametric first order quasilinear partial differential equation.Comment: 22 pages, 3 figures, submitted to Neurips 202

    Meshless Galerkin method based on RBFs and reproducing Kernel for quasi-linear parabolic equations with dirichlet boundary conditions

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    The main aim of this paper is to present a hybrid scheme of both meshless Galerkin and reproducing kernel Hilbert space methods. The Galerkin meshless method is a powerful tool for solving a large class of multi-dimension problems. Reproducing kernel Hilbert space method is an extremely efficient approach to obtain an analytical solution for ordinary or partial differential equations appeared in vast areas of science and engineering. The error analysis and convergence show that the proposed mixed method is very efficient. Since the solution space spanned by radial basis functions do not directly satisfy essential boundary conditions, an auxiliary parameterized technique is employed. Theoretical studies indicate that this new method is very stable, though a parameterized problem is employed instead of the main problem

    Spin Calogero Particles and Bispectral Solutions of the Matrix KP Hierarchy

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    Pairs of n×nn\times n matrices whose commutator differ from the identity by a matrix of rank rr are used to construct bispectral differential operators with r×rr\times r matrix coefficients satisfying the Lax equations of the Matrix KP hierarchy. Moreover, the bispectral involution on these operators has dynamical significance for the spin Calogero particles system whose phase space such pairs represent. In the case r=1r=1, this reproduces well-known results of Wilson and others from the 1990's relating (spinless) Calogero-Moser systems to the bispectrality of (scalar) differential operators. This new class of pairs (L,Λ)(L, \Lambda) of bispectral matrix differential operators is different than those previously studied in that LL acts from the left, but Λ\Lambda from the right on a common r×rr\times r eigenmatrix.Comment: 16 page

    Statistical inference in mechanistic models: time warping for improved gradient matching

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    Inference in mechanistic models of non-linear differential equations is a challenging problem in current computational statistics. Due to the high computational costs of numerically solving the differential equations in every step of an iterative parameter adaptation scheme, approximate methods based on gradient matching have become popular. However, these methods critically depend on the smoothing scheme for function interpolation. The present article adapts an idea from manifold learning and demonstrates that a time warping approach aiming to homogenize intrinsic length scales can lead to a significant improvement in parameter estimation accuracy. We demonstrate the effectiveness of this scheme on noisy data from two dynamical systems with periodic limit cycle, a biopathway, and an application from soft-tissue mechanics. Our study also provides a comparative evaluation on a wide range of signal-to-noise ratios

    Stability in the instantaneous Bethe-Salpeter formalism: harmonic-oscillator reduced Salpeter equation

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    A popular three-dimensional reduction of the Bethe-Salpeter formalism for the description of bound states in quantum field theory is the Salpeter equation, derived by assuming both instantaneous interactions and free propagation of all bound-state constituents. Numerical (variational) studies of the Salpeter equation with confining interaction, however, observed specific instabilities of the solutions, likely related to the Klein paradox and rendering (part of the) bound states unstable. An analytic investigation of this problem by a comprehensive spectral analysis is feasible for the reduced Salpeter equation with only harmonic-oscillator confining interactions. There we are able to prove rigorously that the bound-state solutions correspond to real discrete energy spectra bounded from below and are thus free of any instabilities.Comment: 23 pages, 3 figures, extended conclusions, version to appear in Phys. Rev.
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