• Abstract

Reconstructing faithfully causal networks from observed time series data is fundamental to revealing the intrinsic nature of complex systems. With the increase of the network scale, indirect causal relations will arise due to causation transitivity but existing methods suffer from dimension curse in eliminating such indirect influences. In this paper, we propose a novel technique to overcome the difficulties by integrating the idea of randomly distributed embedding into conditional Granger causality. Validated by both benchmark and synthetic data sets, our method demonstrates potential applicability in reconstructing high-dimensional causal networks based only on a short-term time series.

• Abstract

We consider a Partial Differential Equation model combining second and fourth-order operators for solving the geometry inpainting and denoising problems. The model allows the accurate recovery of curvatures and the singular set of the reconstructed image (edges, corners). The approach proposed permits a dynamical modelling by constructing a family of simple discrete energies that admit as a $Γ$-limit Mumford-Shah-Euler like functional. The approximation functionals are build within an adaptive strategy, based on two ingredients: a fine location of the singular set using mesh refinement, and second, a local choice of the diffusion coefficients which modify the reconstruction operator. Unlike the usual methods, mostly based on prior guess on the continuous solution and leading to complex and nonlinear systems of PDEs, our method consists in solving linear problems and updating the diffusion coefficients. The high order of the operator allows us to perform simultaneously efficient filtering of the data and the interpolation in the damaged regions. The method turns out to be superior to any second-order model in restoring large gap connections and curvy features. In order to validate this approach, we compare the results of our method with those of some existing one in the fields of geometry-oriented inpainting and we present several numerical examples.

• Abstract

In this paper, we compute the stationary states of the multicomponent phase-field crystal model by formulating it as a block constrained minimization problem. The original infinite-dimensional non-convex minimization problem is approximated by a finite-dimensional constrained non-convex minimization problem after an appropriate spatial discretization. To efficiently solve the above optimization problem, we propose a so-called adaptive block Bregman proximal gradient (AB-BPG) algorithm that fully exploits the problem’s block structure. The proposed method updates each order parameter alternatively, and the update order of blocks can be chosen in a deterministic or random manner. Besides, we choose the step size by developing a practical linear search approach such that the generated sequence either keeps energy dissipation or has a controllable subsequence with energy dissipation. The convergence property of the proposed method is established without the requirement of global Lipschitz continuity of the derivative of the bulk energy part by using the Bregman divergence. The numerical results on computing stationary ordered structures in binary, ternary, and quinary component coupled-mode Swift-Hohenberg models have shown a significant acceleration over many existing methods.

• Abstract

In this paper, we propose and develop a novel saturation component based fuzzy Mumford-Shah model for color image segmentation. The main feature of this model is that we determine different segments by using the saturation component in hue, saturation, and value (HSV) color space instead of the original red, green and blue (RGB) color space. The proposed model is formulated for multiphase segmentation of color images with the assumption that a piecewise smooth function is approximated by the product of a piecewise constant function and a smooth function. The piecewise constant function and the smooth function are used to represent different segments and to estimate the bias field respectively in the color image. The approximation is calculated based on the saturation component which is particularly useful to distinguish edges and capture the inherent correlation among red, green and blue channels in color images. Experimental results are presented to demonstrate that the segmentation performance of the proposed model is much better than existing color image segmentation methods.

• Abstract

Boundary and initial conditions are important for the wellposedness of partial differential equations (PDEs). Numerically, these conditions can be enforced exactly in classical numerical methods, such as finite difference method and finite element method. Recent years, we have witnessed growing interests in solving PDEs by deep neural networks (DNNs), especially in the high-dimensional case. However, in the generic situation, a careful literature review shows that boundary conditions cannot be enforced exactly for DNNs, which inevitably leads to a modeling error. In this work, based on the recently developed deep mixed residual method (MIM), we demonstrate how to make DNNs satisfy boundary and initial conditions automatically by using distance functions and explicit constructions. As a consequence, the loss function in MIM is free of the penalty term and does not have any modeling error. Using numerous examples, including Dirichlet, Neumann, mixed, Robin, and periodic boundary conditions for elliptic equations, and initial conditions for parabolic and hyperbolic equations, we show that enforcing exact boundary and initial conditions not only provides a better approximate solution but also facilitates the training process.

• Abstract

The paper presents a holomorphic operator function approach for the Laplace eigenvalue problem using the discontinuous Galerkin method. We rewrite the problem as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. The convergence for the discontinuous Galerkin method is proved by using the abstract approximation theory for holomorphic operator functions. We employ the spectral indicator method to compute the eigenvalues. Extensive numerical examples are presented to validate the theory.