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This paper introduces a factorization for the inverse of discrete Fourier integral operators that can be applied in quasi-linear time. The factorization starts by approximating the operator with the butterfly factorization. Next, a hierarchical matrix representation is constructed for the hermitian matrix arising from composing the Fourier integral operator with its adjoint. This representation is inverted efficiently with a new algorithm based on the hierarchical interpolative factorization. By combining these two factorizations, an approximate inverse factorization for the Fourier integral operator is obtained as a product of $O(\log N)$ sparse matrices of size $N\times N$. The resulting approximate inverse factorization can be used as a direct solver or as a preconditioner. Numerical examples on 1D and 2D Fourier integral operators, including a generalized Radon transform, demonstrate the performance of this new approach.

This note proposes an efficient preconditioner for solving linear and semi-linear parabolic equations. With the Crank-Nicholson time stepping method, the algebraic system of equations at each time step is solved with the conjugate gradient method, preconditioned with hierarchical interpolative factorization. Stiffness matrices arising in the discretization of parabolic equations typically have large condition numbers, and therefore preconditioning becomes essential, especially for large time steps. We propose to use the hierarchical interpolative factorization as the preconditioning for the conjugate gradient iteration. Computed only once, the hierarchical interpolative factorization offers an efficient and accurate approximate inverse of the linear system. As a result, the preconditioned conjugate gradient iteration converges in a small number of iterations. Compared to other classical exact and approximate factorizations such as Cholesky or incomplete Cholesky, the hierarchical interpolative factorization can be computed in linear time and the application of its inverse has linear complexity. Numerical experiments demonstrate the performance of the method and the reduction of conjugate gradient iterations.

This paper introduces a meta-learning approach for parameterized pseudo-differential operators with deep neural networks. With the help of the nonstandard wavelet form, the pseudo-differential operators can be approximated in a compressed form with a collection of vectors. The nonlinear map from the parameter to this collection of vectors and the wavelet transform are learned together from a small number of matrix-vector multiplications of the pseudo-differential operator. Numerical results for Green's functions of elliptic partial differential equations and the radiative transfer equations demonstrate the efficiency and accuracy of the proposed approach., Comment: 21 pages, 9 figures

While it is well-documented that climate change accepters and deniers have become increasingly polarized in the United States over time, there has been no large-scale examination of whether these individuals are prone to changing their opinions as a result of natural external occurrences. On the sub-population of Twitter users, we examine whether climate change sentiment changes in response to five separate natural disasters occurring in the U.S. in 2018. We begin by showing that relevant tweets can be classified with over 75% accuracy as either accepting or denying climate change when using our methodology to compensate for limited labeled data; results are robust across several machine learning models and yield geographic-level results in line with prior research. We then apply RNNs to conduct a cohort-level analysis showing that the 2018 hurricanes yielded a statistically significant increase in average tweet sentiment affirming climate change. However, this effect does not hold for the 2018 blizzard and wildfires studied, implying that Twitter users' opinions on climate change are fairly ingrained on this subset of natural disasters.

In recent years, deep learning has led to impressive results in many fields. In this paper, we introduce a multi-scale artificial neural network for high-dimensional non-linear maps based on the idea of hierarchical nested bases in the fast multipole method and the $\mathcal{H}^2$-matrices. This approach allows us to efficiently approximate discretized nonlinear maps arising from partial differential equations or integral equations. It also naturally extends our recent work based on the generalization of hierarchical matrices [Fan et al. arXiv:1807.01883] but with a reduced number of parameters. In particular, the number of parameters of the neural network grows linearly with the dimension of the parameter space of the discretized PDE. We demonstrate the properties of the architecture by approximating the solution maps of non-linear Schr{\"o}dinger equation, the radiative transfer equation, and the Kohn-Sham map., Comment: 21 figures, 25 pages. arXiv admin note: text overlap with arXiv:1807.01883

The hierarchical interpolative factorization for elliptic partial differential equations is a fast algorithm for approximate sparse matrix inversion in linear or quasilinear time. Its accuracy can degrade, however, when applied to strongly ill-conditioned problems. Here, we propose a simple modification that can significantly improve the accuracy at no additional asymptotic cost: applying a block Jacobi preconditioner before each level of skeletonization. This dramatically limits the impact of the underlying system conditioning and enables the construction of robust and highly efficient preconditioners even at quite modest compression tolerances. Numerical examples demonstrate the performance of the new approach.