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Register a new userA Reduced Radial Basis Function Method for Partial Differential Equations on irregular domains
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We propose and test the first Reduced Radial Basis Function Method (R$^2$BFM) for solving parametric partial differential equations on irregular domains. The two major ingredients are a stable Radial Basis Function (RBF) solver that has an optimized set of centers chosen through a reduced-basis-type greedy algorithm, and a collocation-based model reduction approach that systematically generates a reduced-order approximation whose dimension is orders of magnitude smaller than the total number of RBF centers. The resulting algorithm is efficient and accurate as demonstrated through two- and three-dimensional test problems.
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In mathematical physics, the space-fractional diffusion equations are of particular interest in the studies of physical phenomena modelled by L{e}vy processes, which are sometimes called super-diffusion equations. In this article, we develop the differential quadrature (DQ) methods for solving the 2D space-fractional diffusion equations on irregular domains. The methods in presence reduce the original equation into a set of ordinary differential equations (ODEs) by introducing valid DQ formulations to fractional directional derivatives based on the functional values at scattered nodal points on problem domain. The required weighted coefficients are calculated by using radial basis functions (RBFs) as trial functions, and the resultant ODEs are discretized by the Crank-Nicolson scheme. The main advantages of our methods lie in their flexibility and applicability to arbitrary domains. A series of illustrated examples are finally provided to support these points.
Linear kinetic transport equations play a critical role in optical tomography, radiative transfer and neutron transport. The fundamental difficulty hampering their efficient and accurate numerical resolution lies in the high dimensionality of the physical and velocity/angular variables and the fact that the problem is multiscale in nature. Leveraging the existence of a hidden low-rank structure hinted by the diffusive limit, in this work, we design and test the angular-space reduced order model for the linear radiative transfer equation, the first such effort based on the celebrated reduced basis method (RBM). Our method is built upon a high-fidelity solver employing the discrete ordinates method in the angular space, an asymptotic preserving upwind discontinuous Galerkin method for the physical space, and an efficient synthetic accelerated source iteration for the resulting linear system. Addressing the challenge of the parameter values (or angular directions) being coupled through an integration operator, the first novel ingredient of our method is an iterative procedure where the macroscopic density is constructed from the RBM snapshots, treated explicitly and allowing a transport sweep, and then updated afterwards. A greedy algorithm can then proceed to adaptively select the representative samples in the angular space and form a surrogate solution space. The second novelty is a least-squares density reconstruction strategy, at each of the relevant physical locations, enabling the robust and accurate integration over an arbitrarily unstructured set of angular samples toward the macroscopic density. Numerical experiments indicate that our method is highly effective for computational cost reduction in a variety of regimes.
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At present, deep learning based methods are being employed to resolve the computational challenges of high-dimensional partial differential equations (PDEs). But the computation of the high order derivatives of neural networks is costly, and high order derivatives lack robustness for training purposes. We propose a novel approach to solving PDEs with high order derivatives by simultaneously approximating the function value and derivatives. We introduce intermediate variables to rewrite the PDEs into a system of low order differential equations as what is done in the local discontinuous Galerkin method. The intermediate variables and the solutions to the PDEs are simultaneously approximated by a multi-output deep neural network. By taking the residual of the system as a loss function, we can optimize the network parameters to approximate the solution. The whole process relies on low order derivatives. Numerous numerical examples are carried out to demonstrate that our local deep learning is efficient, robust, flexible, and is particularly well-suited for high-dimensional PDEs with high order derivatives.
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