An s-stage Runge-Kutta-type iterative method with the convex penalty for solving nonlinear ill-posed problems is proposed and analyzed in this paper. The approach is developed by using a family of Runge-Kutta-type methods to solve the asymptotical regularization method, which can be seen as an ODE with the initial value. The convergence and regularity of the proposed method are obtained under certain conditions. The reconstruction results of the proposed method for some special cases are studied through numerical experiments on both parameter identification in inverse potential problem and diffuse optical tomography. The numerical results indicate that the developed methods yield stable approximations to true solutions, especially the implicit schemes have obvious advantages on allowing a wider range of step length, reducing the iterative numbers, and saving computation time.

It is known that discontinuous finite element methods use more unknown variables but have the same convergence rate comparing to their continuous counterpart. In this paper, a novel conforming discontinuous Galerkin (CDG) finite element method is introduced for Poisson equation using discontinuous $P_k$ elements on triangular and tetrahedral meshes. Our new CDG method maximizes the potential of discontinuous $P_k$ element in order to improve the convergence rate. Superconvergence of order two for the CDG finite element solution is proved in an energy norm and in the $L^2$ norm. A local post-process is defined which lifts a $P_k$ CDG solution to a discontinuous $P_{k+2}$ solution. It is proved that the lifted $P_{k+2}$ solution converges at the optimal order. The numerical tests confirm the theoretic findings. Numerical comparison is provided in 2D and 3D, showing the $P_k$ CDG finite element is as good as the $P_{k+2}$ continuous Galerkin finite element.

We propose a method combining boundary integral equations and neural networks (BINet) to solve (parametric) partial differential equations (PDEs) and operator problems in both bounded and unbounded domains. For PDEs with explicit fundamental solutions, BINet learns to solve, as a proxy, associated boundary integral equations using neural networks. The benefits are three-fold. Firstly, only the boundary conditions need to be fitted since the PDE can be automatically satisfied with single or double layer potential according to the potential theory. Secondly, the dimension of the boundary integral equations is less by one, and as such, the sample complexity can be reduced significantly. Lastly, in the proposed method, all differential operators have been removed, hence the numerical efficiency and stability are improved. Adopting neural tangent kernel (NTK) techniques, we provide proof of the convergence of BINets in the limit that the width of the neural network goes to infinity. Extensive numerical experiments show that, without calculating high-order derivatives, BINet is much easier to train and usually gives more accurate solutions, especially in the cases that the boundary conditions are not smooth enough. Further, BINet outperforms strong baselines for both one single PDE and parameterized PDEs in the bounded and unbounded domains.

This paper is devoted to the numerical computation of transmission eigenvalues arising in the inverse acoustic scattering theory. This problem is first reformulated as a two-by-two system of boundary integral equations. Next, we develop a Schur complement operator with regularization to obtain a reduced system of boundary integral equations. The Nyström discretization is then used to obtain an eigenvalue problem for a matrix. In conjunction with the recursive integral method, the numerical computation of the matrix eigenvalue problem produces the indicator for finding the transmission eigenvalues. Numerical implementations are presented and archetypal examples are provided to demonstrate the effectiveness of the proposed method.

We present a well improved surface segmentation algorithm for 3-axis/3+2-axis CNC subtractive fabrication. For a free-form surface (represented by the triangular mesh), to avoid collision with the cutter during complex surface machining, it is essential to segment it into several patches. We transform the surface segmentation problem into a mathematical problem based on energy minimization according to several fabrication constraints, and solved by establishing a weighted graph and searching the minimum cut. Our algorithm has simple structure and is easy to implement. Moreover, the algorithm guarantees correctness and completeness in theory, that is, we prove that the weight of the minimum cut is equivalent to the minimum value of the energy function. Experimental results are provided to illustrate and clarify our method.

In this paper, we provide an optimal rate convergence analysis and error estimate for a structure-preserving numerical scheme for the Poisson-Nernst-Planck-Cahn-Hilliard (PNPCH) system. The numerical scheme is based on the Energetic Variational Approach of the physical model, which is reformulated as a non-constant mobility gradient flow of a free-energy functional that consists of singular logarithmic energy potentials arising from the PNP theory and the Cahn-Hilliard surface diffusion process. The mobility function is explicitly updated, while the logarithmic and the surface diffusion terms are computed implicitly. The primary challenge in the development of theoretical analysis on optimal error estimate has been associated with the nonlinear parabolic coefficients. To overcome this subtle difficulty, an asymptotic expansion of the numerical solution is performed, so that a higher order consistency order can be obtained. The rough error estimate leads to a bound in maximum norm for concentrations, which plays an essential role in the nonlinear analysis. Finally, the refined error estimate is carried out, and the desired convergence estimate is accomplished. Numerical results are presented to demonstrate the convergence order and performance of the numerical scheme in preserving physical properties and capturing ionic steric effects in concentrated electrolytes.

We consider a diffusion and a wave equations

$$∂^k_tu(x,t) =∆u(x,t)+\mu(t)f(x), x∈Ω, t>0, k=1,2$$

with the zero initial and boundary conditions, where $Ω ⊂\mathbb{R}^d$ is a bounded domain. We establish uniqueness and/or stability results for inverse problems of determining $\mu(t), 0<t< T$ with given $f(x),$ determining $f(x), x ∈ Ω$ with given $\mu(t),$ by data of $u: u(x_0,·)$ with fixed point $x_0 ∈ Ω$ or Neumann data on subboundary over time interval. In our inverse problems, data are taken over time interval $T_1<t<T_2,$ by assuming that $T < T_1 < T_2$ and $\mu(t) =0$ for $t≥ T,$ which means that the source stops to be active after the time $T$ and the observations are started only after $T.$ This assumption is practical by such a posteriori data after incidents, although inverse problems had been well studied in the case of $T = 0.$ We establish the non-uniqueness, the uniqueness and conditional stability for a diffusion and a wave equations. The proofs are based on eigenfunction expansions of the solutions $u(x,t),$ and we rely on various knowledge of the generalized Weierstrass theorem on polynomial approximation, almost periodic functions, Carleman estimate, non-harmonic Fourier series.