For curves or general interfaces, Euler’s elastica energy has a wide range of applications in computer vision and image processing. It is however difficult to minimize the functionals related to the elastica energy due to its non-convexity, nonlinearity and higher order with derivatives. In this paper, we propose a very simple way to combine level set and binary representations for interfaces and then use a fast algorithm to minimize the functionals involving the elastica energy. The proposed algorithm essentially just needs to solve a total variation type minimization problem and a re-distance problem. Nowadays, there are many fast algorithms to solve these two problems and thus the overall efficiency of the proposed algorithm is very high. We then apply the new Euler’s elastica minimization algorithm to image segmentation, image inpainting and illusory shape reconstruction problems. Extensive experimental results are finally conducted to validate the effectiveness of the proposed algorithm.

This article presents an error analysis of a Hermite cubic immersed finite element (IFE) method for solving interface problems of the differential equation modeling a Euler-Bernoulli beam made up of multiple materials together with suitable jump conditions at material interfaces. The analysis consists of three essential groups. The first group is about IFE functions including bounds for the IFE shape functions and inverse inequalities. The second group is about error bounds for IFE interpolation derived with a multi-point Taylor expansion technique. The last group, and perhaps the most important group, is for proving the optimal convergence of the IFE solution generated by the usual Galerkin scheme based on the Hermite cubic IFE space considered in this article.

We deal with a multivalued second order dynamical system involving a Clarke subdifferential of a locally Lipschitz functional. We apply a time discretization procedure to construct a sequence of solutions to a family of the approximate problems and show its convergence to a solution of the exact problem as the time step size vanishes. We consider a nonautonomous problem in which both the viscosity and the multivalued operators depend on time explicitly. The time discretization method we use, is the $\theta$-scheme with $\theta \in [\frac{1}{2}, 1]$, thus, in particular, the Crank-Nicolson scheme and the implicit Euler scheme are included. We apply our result to a class of dynamic hemivariational inequalities.

We consider a quadrilateral ‘mini’ finite element for approximating the solution of Stokes equations using a quadrilateral mesh. We use the standard bilinear finite element space enriched with element-wise defined bubble functions for the velocity and the standard bilinear finite element space for the pressure space. With a simple modification of the standard bubble function we show that a single bubble function is sufficient to ensure the inf-sup condition. We have thus improved an earlier result on the quadrilateral ‘mini’ element, where more than one bubble function are used to get the stability.

In this paper, we propose a novel adaptive finite volume method (AFVM) for elliptic equations. As a standard adaptive method, a loop of our method involves four steps: Solve $\rightarrow$ Estimate $\rightarrow$ Mark $\rightarrow$ Refine. The novelty of our method is that we do not have the traditional "completion" procedure in the Refine step. To guarantee the conformity, a triangular element with a hanging node is treated as a quadrilateral element, and the corresponding function space consists of the bilinear functions. The optimal computational complexity of our AFVM is validated by numerical examples.

This paper studies a regularization approach for simultaneously reconstructing space-time dependent Robin coefficient $\gamma(\rm{x}, t)$ and heat flux $q(\rm{x}, t)$. The differentiability results and adjoint systems are established. A standard finite element method (FEM) is employed to discretize the constrained optimization problem which is reduced to a sequence of unconstrained optimization problem by adding regularization terms. We propose an improved algorithm for the introduced problem based on modified conjugate gradient method (MCGM) for quadratic minimization. Numerical experiments present the efficiency, accuracy, and robustness of the proposed algorithm.

Most numerical solvers and libraries nowadays are implemented to use mathematical models created with language-specific built-in data types (e.g. real in Fortran or double in C) and their respective elementary algebra implementations. However, the built-in elementary algebra typically has limited functionality and often restricts the flexibility of mathematical models and the analysis types that can be applied to those models. To overcome this limitation, a number of domain-specific languages such as gPROMS or Modelica with more feature-rich built-in data types have been proposed. In this paper, we argue that if numerical libraries and solvers are designed to use abstract elementary algebra rather than the language-specific built-in algebra, modern mainstream languages can be as effective as any domain-specific language. We illustrate our ideas using the example of sparse Jacobian matrix computation. We implement an automatic differentiation method that takes advantage of sparse system structures and is straightforward to parallelize in a distributed memory setting. Furthermore, we show that the computational cost scales linearly with the size of the system.

By combining the Minimal Residual Method and the Primal-Dual Active Set algorithm, we derive an efficient scheme for solving a class of PDE-constrained optimization problems with inequality constraints. The approach studied in this paper addresses box constraints on the control function, and leads to an iterative scheme in which linear optimality systems must be solved in each iteration. We prove that the spectra of the associate saddle point operators, appearing in each iteration, are well behaved: Almost all the eigenvalues are contained in three bounded intervals, not containing zero. In fact, for severely ill-posed problems, the number of eigenvalues outside these three intervals are of order $O(ln(α^{-1}))$ as $α → 0$, where $α$ is the parameter employed in the Tikhonov regularization. Krylov subspace methods are well known to handle such systems of algebraic equations very well, and we thus obtain a fast method for PDE-constrained optimization problems with box constraints. In contrast to previous papers, our investigation is not targeted at analyzing a specific model, but instead covers a rather large class of problems. Our theoretical findings are illuminated by several numerical experiments. An example covered by our theoretical findings, as well as cases not fulfilling all the assumptions needed in the analysis, is presented. Also, in addition to computations only involving synthetic data, we briefly explore whether these new techniques can be applied to real world problems. More specifically, the algorithm is tested on a medical imaging problem with clinical patient data. These tests suggest that the method is fast and reliable.