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We develop a novel a posteriori error estimator for the L2 error committed by the finite element discretization of the solution of the fractional Laplacian. Our a posteriori error estimator takes advantage of the semi-discretization scheme using a rational approximation which allows to reformulate the fractional problem into a family of non-fractional parametric problems. The estimator involves applying the implicit Bank-Weiser error estimation strategy to each parametric non-fractional problem and reconstructing the fractional error through the same rational approximation used to compute the solution to the original fractional problem. We provide several numerical examples in both two and three-dimensions demonstrating the effectivity of our estimator for varying fractional powers and its ability to drive an adaptive mesh refinement strategy.

Multi-material problems often exhibit complex geometries along with physical responses presenting large spatial gradients or discontinuities. In these cases, providing high-quality body-fitted finite element analysis meshes and obtaining accurate solutions remain challenging. Immersed boundary techniques provide elegant solutions for such problems. Enrichment methods alleviate the need for generating conforming analysis grids by capturing discontinuities within mesh elements. Additionally, increased accuracy of physical responses and geometry description can be achieved with higher-order approximation bases. In particular, using B-splines has become popular with the development of IsoGeometric Analysis. In this work, an eXtended IsoGeometric Analysis (XIGA) approach is proposed for multi-material problems. The computational domain geometry is described implicitly by level set functions. A novel generalized Heaviside enrichment strategy is employed to accommodate an arbitrary number of materials without artificially stiffening the physical response. Higher-order B-spline functions are used for both geometry representation and analysis. Boundary and interface conditions are enforced weakly via Nitsche's method, and a new face-oriented ghost stabilization methodology is used to mitigate numerical instabilities arising from small material integration subdomains. Two- and three-dimensional heat transfer and elasticity problems are solved to validate the approach. Numerical studies provide insight into the ability to handle multiple materials considering sharp-edged and curved interfaces, as well as the impact of higher-order bases and stabilization on the solution accuracy and conditioning.

Physical systems are usually modeled by differential equations, but solving these differential equations analytically is often intractable. Instead, the differential equations can be solved numerically by discretization in a finite computational domain. The discretized equation is reduced to a large linear system, whose solution is typically found using an iterative solver. We start with an initial guess, x_0, and iterate the algorithm to obtain a sequence of solution vectors, x_m. The iterative algorithm is said to converge to solution $x$ if and only if x_m converges to $x$. Accuracy of the numerical solutions is important, especially in the design of safety critical systems such as airplanes, cars, or nuclear power plants. It is therefore important to formally guarantee that the iterative solvers converge to the "true" solution of the original differential equation. In this paper, we first formalize the necessary and sufficient conditions for iterative convergence in the Coq proof assistant. We then extend this result to two classical iterative methods: Gauss-Seidel iteration and Jacobi iteration. We formalize conditions for the convergence of the Gauss--Seidel classical iterative method, based on positive definiteness of the iterative matrix. We then formally state conditions for convergence of Jacobi iteration and instantiate it with an example to demonstrate convergence of iterative solutions to the direct solution of the linear system. We leverage recent developments of the Coq linear algebra and mathcomp library for our formalization.

Elementary function operations such as sin and exp cannot in general be computed exactly on today's digital computers, and thus have to be approximated. The standard approximations in library functions typically provide only a limited set of precisions, and are too inefficient for many applications. Polynomial approximations that are customized to a limited input domain and output accuracy can provide superior performance. In fact, the Remez algorithm computes the best possible approximation for a given polynomial degree, but has so far not been formally verified. This paper presents Dandelion, an automated certificate checker for polynomial approximations of elementary functions computed with Remez-like algorithms that is fully verified in the HOL4 theorem prover. Dandelion checks whether the difference between a polynomial approximation and its target reference elementary function remains below a given error bound for all inputs in a given constraint. By extracting a verified binary with the CakeML compiler, Dandelion can validate certificates within a reasonable time, fully automating previous manually verified approximations.

This paper studies the inverse problem of determination the history for a stochastic diffusion process, by means of the value at the final time $T$. By establishing a new Carleman estimate, the conditional stability of the problem is proven. Based on the idea of Tikhonov method, a regularized solution is proposed. The analysis of the existence and uniqueness of the regularized solution, and proof for error estimate under an a-proior assumption are present. Numerical verification of the regularization, including numerical algorithm and examples are also illustrated.

We consider non-cooperative binding, so-called 'temperature 1', in deterministic or directed (called here confluent) tile self-assembly systems in two dimensions and show a necessary and sufficient condition for such system to have an ultimately periodic assembly path. We prove that an infinite maximal assembly has an ultimately periodic assembly path if and only if it contains an infinite assembly path that does not intersect a periodic path in the Z2 grid. Moreover we show that every infinite assembly must satisfy this condition, and therefore, contains an ultimately periodic path. This result is obtained through a super-position and a combination of two paths that produce a new path with desired properties, a technique that we call co-grow of two paths. The paper is an updated and improved version of the first part of arXiv 1901.08575.

This paper applies a discontinuous Galerkin finite element method to the Kelvin-Voigt viscoelastic fluid motion equations when the forcing function is in $L^\infty({\bf L}^2)$-space. Optimal a priori error estimates in $L^\infty({\bf L}^2)$-norm for the velocity and in $L^\infty(L^2)$-norm for the pressure approximations for the semi-discrete discontinuous Galerkin method are derived here. The main ingredients for establishing the error estimates are the standard elliptic duality argument and a modified version of the Sobolev-Stokes operator defined on appropriate broken Sobolev spaces. Further, under the smallness assumption on the data, it has been proved that these estimates are valid uniformly in time. Then, a first-order accurate backward Euler method is employed to discretize the semi-discrete discontinuous Galerkin Kelvin-Voigt formulation completely. The fully discrete optimal error estimates for the velocity and pressure are established. Finally, using the numerical experiments, theoretical results are verified. It is worth highlighting here that the error results in this article for the discontinuous Galerkin method applied to the Kelvin-Voigt model using finite element analysis are the first attempt in this direction.

In this paper we propose a novel approach to discretize linear port-Hamiltonian systems while preserving the underlying structure. We present a finite element exterior calculus formulation that is able to mimetically represent conservation laws and cope with mixed open boundary conditions using a single computational mesh. The possibility of including open boundary conditions allows for modular composition of complex multi-physical systems whereas the exterior calculus formulation provides a coordinate-free treatment. Our approach relies on a dual-field representation of the physical system that is redundant at the continuous level but eliminates the need of mimicking the Hodge star operator at the discrete level. By considering the Stokes-Dirac structure representing the system together with its adjoint, which embeds the metric information directly in the codifferential, the need for an explicit discrete Hodge star is avoided altogether. By imposing the boundary conditions in a strong manner, the power balance characterizing the Stokes-Dirac structure is then retrieved at the discrete level via symplectic Runge-Kutta integrators based on Gauss-Legendre collocation points. Numerical experiments validate the convergence of the method and the conservation properties in terms of energy balance both for the wave and Maxwell equations in a three dimensional domain. For the latter example, the magnetic and electric fields preserve their divergence free nature at the discrete level.

Many papers in the field of integer linear programming (ILP, for short) are devoted to problems of the type $\max\{c^\top x \colon A x = b,\, x \in \mathbb{Z}^n_{\geq 0}\}$, where all the entries of $A,b,c$ are integer, parameterized by the number of rows of $A$ and $\|A\|_{\max}$. This class of problems is known under the name of ILP problems in the standard form, adding the word "bounded" if $x \leq u$, for some integer vector $u$. Recently, many new sparsity, proximity, and complexity results were obtained for bounded and unbounded ILP problems in the standard form. In this paper, we consider ILP problems in the canonical form $$\max\{c^\top x \colon b_l \leq A x \leq b_r,\, x \in \mathbb{Z}^n\},$$ where $b_l$ and $b_r$ are integer vectors. We assume that the integer matrix $A$ has the rank $n$, $(n + m)$ rows, $n$ columns, and parameterize the problem by $m$ and $\Delta(A)$, where $\Delta(A)$ is the maximum of $n \times n$ sub-determinants of $A$, taken in the absolute value. We show that any ILP problem in the standard form can be polynomially reduced to some ILP problem in the canonical form, preserving $m$ and $\Delta(A)$, but the reverse reduction is not always possible. More precisely, we define the class of generalized ILP problems in the standard form, which includes an additional group constraint, and prove the equivalence to ILP problems in the canonical form. We generalize known sparsity, proximity, and complexity bounds for ILP problems in the canonical form. Additionally, sometimes, we strengthen previously known results for ILP problems in the canonical form, and, sometimes, we give shorter proofs. Finally, we consider the special cases of $m \in \{0,1\}$. By this way, we give specialised sparsity, proximity, and complexity bounds for the problems on simplices, Knapsack problems and Subset-Sum problems.

Semidefinite programming (SDP) is a powerful tool for tackling a wide range of computationally hard problems such as clustering. Despite the high accuracy, semidefinite programs are often too slow in practice with poor scalability on large (or even moderate) datasets. In this paper, we introduce a linear time complexity algorithm for approximating an SDP relaxed $K$-means clustering. The proposed sketch-and-lift (SL) approach solves an SDP on a subsampled dataset and then propagates the solution to all data points by a nearest-centroid rounding procedure. It is shown that the SL approach enjoys a similar exact recovery threshold as the $K$-means SDP on the full dataset, which is known to be information-theoretically tight under the Gaussian mixture model. The SL method can be made adaptive with enhanced theoretic properties when the cluster sizes are unbalanced. Our simulation experiments demonstrate that the statistical accuracy of the proposed method outperforms state-of-the-art fast clustering algorithms without sacrificing too much computational efficiency, and is comparable to the original $K$-means SDP with substantially reduced runtime.