Screened Poisson Equation Research Articles

Box Constraints and Weighted Sparsity Regularization for Identifying Sources in Elliptic PDEs

We explore the possibility for using boundary data to identify sources in elliptic PDEs. Even though the associated forward operator has a large null space, it turns out that box constraints, combined with weighted sparsity regularization, can enable rather accurate recovery of sources with constant magnitude/strength. In addition, for sources with varying strength, the support of the inverse solution will be a subset of the support of the true source. We present both an analysis of the problem and a series of numerical experiments. Our work only addresses discretized problems. The reason for introducing the weighting procedure is that standard (unweighted) sparsity regularization fails to provide adequate results for the source identification task considered in this paper. This investigation is also motivated by applications, e.g. recovering mass distributions from measurements of gravitational fields and inverse scattering. We develop the methodology and the analysis in terms of Euclidean spaces, and our results can therefore be applied to many problems. For example, the results are equally applicable to models involving the screened Poisson equation as to models using the Helmholtz equation, with both large and small wave numbers.

A compressible solver for two phase-flows with phase change for bubble cavitation

The present work describes the development of a novel fully compressible solver for two-phase flows with liquid-vapour phase change. Phase change phenomena in compressible two-phase flows are driving mechanisms for a multitude of phenomena including cavitation, droplet evaporation in acoustic fields, phase change in closed environments. Those are fundamental for a large number of technological applications including combustion chambers, propellant management in space and submarine engine design. The accurate simulation of phase change phenomena in compressible flows requires a direct numerical solver thermodynamically consistent and able to describe the smallest driving scales and to handle capillary effects. The present solver has these characteristics. It solves the full system of conservation equations for real fluid compressible flows of single species written in primitive variables using a semi-implicit pressure based solver. It leads to the energy equation being a Screened Poisson equation for the pressure that can be solved as a linear system. For the sake of thermodynamic consistency, several cubic equations of state are proposed to describe liquid and vapour phases as well as evolving saturation conditions at the interface. The proposed solver can be considered as an extension of classical incompressible solvers for nucleate boiling based on sharp interface approaches towards configurations with temporally and spatially evolving saturation conditions at the interface. It enables a complete coupling between local liquid-vapour phase change at the interface and pressure variations due to local flow features or time evolving thermodynamic conditions. Its interest is demonstrated with several benchmarks of increasing complexity with an ultimate configuration involving bubble cavitation due to the pressure drop in a convergent pipe.

Solution of a Screened Poisson Equation with Dirichlet Boundary Condition with the Method of Iterative Extensions

Solution of a Screened Poisson Equation with Dirichlet Boundary Condition with the Method of Iterative Extensions

A real-time quality improvement method based on an adaptive dynamic screened Poisson equation for surveillance video in sand-dust weather

A real-time quality improvement method based on an adaptive dynamic screened Poisson equation for surveillance video in sand-dust weather

Continuous gap contact formulation based on the screened Poisson equation

We introduce a PDE-based node-to-element contact formulation as an alternative to classical, purely geometrical formulations. It is challenging to devise solutions to nonsmooth contact problem with continuous gap using finite element discretizations. We herein achieve this objective by constructing an approximate distance function (ADF) to the boundaries of solid objects, and in doing so, also obtain universal uniqueness of contact detection. Unilateral constraints are implemented using a mixed model combining the screened Poisson equation and a force element, which has the topology of a continuum element containing an additional incident node. An ADF is obtained by solving the screened Poisson equation with constant essential boundary conditions and a variable transformation. The ADF does not explicitly depend on the number of objects and a single solution of the partial differential equation for this field uniquely defines the contact conditions for all incident points in the mesh. Having an ADF field to any obstacle circumvents the multiple target surfaces and avoids the specific data structures present in traditional contact-impact algorithms. We also relax the interpretation of the Lagrange multipliers as contact forces, and the Courant–Beltrami function is used with a mixed formulation producing the required differentiable result. We demonstrate the advantages of the new approach in two- and three-dimensional problems that are solved using Newton iterations. Simultaneous constraints for each incident point are considered.

Quaternion Screened Poisson Equation for Low-Light Image Enhancement

Image enhancement is a technique to enhance the illumination of dark images while keeping the reality and the naturalness of the enhanced images at the same time. For color images, most methods tackle different color channels in a separate way, which overlooks the connection between the color channels. Therefore, in this paper, we consider a quaternion-based model to reserve the color connectivity, which integrates the color information of a pixel by a quaternion number. Moreover, we propose a regularizer based on the gamma-correction function and incorporate it into a screened Poisson equation for the image enhancement task. The uniqueness and existence of the solution of the proposed model are analyzed. The numerical results also prove the superiority of our scheme for color image enhancement.

Robust resistance to noise and outliers: Screened Poisson Surface Reconstruction using adaptive kernel density estimation

Screened Poisson Surface Reconstruction has a good performance among the state-of-art surface reconstruction algorithms in obtaining a triangle mesh from oriented points. In order to better deal with nonuniform point clouds, Screened Poisson Surface Reconstruction uses B-spline functions with a fixed support for kernel density estimation to construct a vector field for solving the screened Poisson equation. In this paper, an adaptive bandwidth Gaussian kernel density estimator is applied, which reduces the bandwidth where the density is low, and increases the bandwidth where the density is high. Experiments show that such an estimator that makes use of both global and local points distribution can effectively remove noise and outliers in the reconstruction.

Normal manipulation for bas-relief modeling

We introduce a normal-based modeling framework for bas-relief generation and stylization which is motivated by the recent advancement in this topic. Creating bas-relief from normal images has successfully facilitated bas-relief modeling in image space. However, the use of normal images in previous work is restricted to the cut-and-paste or blending operations of layers. These operations simply treat a normal vector as a pixel of a general color image. This paper is intended to extend normal-based methods by processing the normal image from a geometric perspective. Our method can not only generate a new normal image by combining various frequencies of existing normal images and details transferring, but also build bas-reliefs from a single RGB image and its edge-based sketch lines. In addition, we introduce an auxiliary function to represent a smooth base surface or generate a layered global shape. To integrate above considerations into our framework, we formulate the bas-relief generation as a variational problem which can be solved by a screened Poisson equation. One important advantage of our method is that it can generate more styles than previous methods and thus it expands the bas-relief shape space. We experimented our method on a range of normal images and it compares favorably to other popular classic and state-of-the-art methods.

Coupled finite-element/topology optimization of continua using the Newton-Raphson method

We show that conventional finite element technology can be applied to topology optimization of 2D and 3D continua. Nodal design variables, classical Newton-Raphson solution for the first-order KKT equations and the screened-Poisson equation are sufficient to produce smooth results without the use of level-sets or phase-field technologies. Side-constraints are addressed with variable transformation and a single Lagrange multiplier enforces the volume constraint. We also use our mesh-division Algorithm specialized to topology optimization as a form of producing compatible meshes. By means of 2D and 3D numerical experimentation, we make the case for this approach to density-based compliance minimization. Verification examples are shown, with the corresponding compliance and volume evolution exhibiting a sound behavior.

Gradient-enhanced Raviart-Thomas tetrahedron for finite-strain problems

A new gradient-enhanced strain-tensor formulation for finite-strain problems is introduced, based on the Raviart-Thomas face-interpolation scheme and the Hellinger-Reissner variational principle. The screened-Poisson equation is employed to relate the kinematic Green-Lagrange strain with the mixed strain. The strain vector obtained from the face normals is now a (vector) degree-of-freedom at each face. In contrast with variational multiscale methods, there are no parameters to fit and stability in compression is verified. When compared with smoothed finite-elements, the formulation is straightforward and sparsity pattern of the classical system retained, albeit with high computational cost. In contrast with traditional gradient-enhanced formulations, a theoretically sound mixed formulation underlies the algorithm. High accuracy is obtained for four-node tetrahedra with incompressibility and bending benchmarks being solved. Traditional finite-strain benchmarks and a quasi-brittle damage numerical test are performed, with very competitive results.

Electrostatic pair-potentials based on the Poisson equation

Electrostatic pair-potentials within molecular simulations are often based on empirical data, cancellation of derivatives or moments, or statistical distributions of image-particles. In this work we start with the fundamental Poisson equation and show that no truncated Coulomb pair-potential, unsurprisingly, can solve the Poisson equation. For any such pair-potential the Poisson equation gives two incompatible constraints, yet we find a single unique expression which, pending two physically connected smoothness parameters, can obey either one of these. This expression has a general form which covers several recently published pair-potentials. For sufficiently large degree of smoothness we find that the solution implies a Gaussian distribution of the charge, a feature which is frequently assumed in pair-potential theory. We end up by recommending a single pair-potential based both on theoretical arguments and empirical evaluations of non-thermal lattice- and thermal water-systems. The same derivations have also been made for the screened Poisson equation, i.e. for Yukawa potentials, with a similar solution.

Cubic B20 helimagnets with quenched disorder in magnetic field

We theoretically address the problem of cubic B20 helimagnets with a small concentration $c\ensuremath{\ll}1$ of defect bonds in external magnetic field $\mathbf{H}$, which is relevant to mixed B20 compounds at small dopant concentrations. We assume that the Dzyaloshinskii-Moriya interaction and the exchange coupling constant are changed on imperfect bonds which leads to distortion of the conical spiral ordering. In a one-impurity problem we find that the distortion of the spiral pitch is long ranged and it is governed by the Poisson equation for an electric dipole. The variation of the cone angle is described by the screened Poisson equation for two electric charges with the screening length being of the order of the spiral period. We calculate corrections to the spiral vector and to the cone angle at finite $c$. The correction to the spiral vector is shown to be independent of $H$. We demonstrate that diffuse neutron scattering caused by disorder appears in the elastic cross section as power-law decaying tails centered at magnetic Bragg peaks.

Impact of Dehazing on Underwater Marker Detection for Augmented Reality.

Underwater augmented reality is a very challenging task and amongst several issues, one of the most crucial aspects involves real-time tracking. Particles present in water combined with the uneven absorption of light decrease the visibility in the underwater environment. Dehazing methods are used in many areas to improve the quality of digital image data that is degraded by the influence of the environment. This paper describes the visibility conditions affecting underwater scenes and shows existing dehazing techniques that successfully improve the quality of underwater images. Four underwater dehazing methods are selected for evaluation of their capability of improving the success of square marker detection in underwater videos. Two reviewed methods represent approaches of image restoration: Multi-Scale Fusion, and Bright Channel Prior. Another two methods evaluated, the Automatic Color Enhancement and the Screened Poisson Equation, are methods of image enhancement. The evaluation uses diverse test data set to evaluate different environmental conditions. Results of the evaluation show an increased number of successful marker detections in videos pre-processed by dehazing algorithms and evaluate the performance of each compared method. The Screened Poisson method performs slightly better to other methods across various tested environments, while Bright Channel Prior and Automatic Color Enhancement shows similarly positive results.

A Gradient-Domain Based Geometry Processing Framework for Point Clouds

The use of point clouds is becoming increasingly popular. We present a general framework for performing geometry filtering on point-based surface through applying the meshless local Petrol-Galelkin (MLPG) to obtain the solution of a screened Poisson equation. The enhancement or smoothing of surfaces is controlled by a gradient scale parameter. Anisotropic filtering is supported by the adapted Riemannian metric. Contrary to the other approaches of partial differential equation for point-based surface, the proposed approach neither needs to construct local or global triangular meshes, nor needs global parameterization. It is only based on the local tangent space and local interpolated surfaces. Experiments demonstrate the efficiency of our approach.

Charge Transfer Excitations with Range Separated Functionals Using Improved Virtual Orbitals.

We present an implementation of range separated functionals utilizing the Slater function on grids in real space in the projector augmented waves method. The screened Poisson equation is solved to evaluate the necessary screened exchange integrals on Cartesian grids. The implementation is verified against existing literature and applied to the description of charge transfer excitations. We find very slow convergence for calculations within linear response time-dependent density functional theory and unoccupied orbitals of the canonical Fock operator. Convergence can be severely improved by using Huzinaga's virtual orbitals instead. This combination furthermore enables an accurate determination of long-range charge transfer excitations by means of ground-state calculations.

Combinatorial optimal control of semilinear elliptic PDEs

Optimal control problems (OCPs) containing both integrality and partial differential equation (PDE) constraints are very challenging in practice. The most wide-spread solution approach is to first discretize the problem, which results in huge and typically nonconvex mixed-integer optimization problems that can be solved to proven optimality only in very small dimensions. In this paper, we propose a novel outer approximation approach to efficiently solve such OCPs in the case of certain semilinear elliptic PDEs with static integer controls over arbitrary combinatorial structures, where we assume the nonlinear part of the PDE to be non-decreasing and convex. The basic idea is to decompose the OCP into an integer linear programming (ILP) master problem and a subproblem for calculating linear cutting planes. These cutting planes rely on the pointwise concavity or submodularity of the PDE solution with respect to the control variables. The decomposition allows us to use standard solution techniques for ILPs as well as for PDEs. We further benefit from reoptimization strategies for the PDE solution due to the iterative structure of the algorithm. Experimental results show that the new approach is capable of solving the combinatorial OCP of a semilinear Poisson equation with up to 180 binary controls to global optimality within a 5 h time limit. In the case of the screened Poisson equation, which yields semi-infinite integer linear programs, problems with as many as 1400 binary controls are solved.

Effective 2D and 3D crack propagation with local mesh refinement and the screened Poisson equation

In this paper, we propose a simple 2D and 3D crack evolution algorithm which avoids the variable/DOF mapping within mesh adaptation algorithms. To this end, a new area/volume minimization algorithm for damaged elements is introduced with the goal of improving the crack path representation. In addition, the new algorithm consists of: (i) mesh-creation stage where a damage model is employed to drive the remeshing procedure (ii) a subsequent analysis stage with a localization limiter in the form of a modified screened Poisson equation. This is exempt of crack path calculations. In the second stage, the crack naturally occurs within the refined region. A staggered algorithm for equilibrium and screened Poisson equations is used in this second stage. Element subdivision is based on edge split operations in 2D and 3D using the damage variable. Both 2D and 3D operations are described in detail. With the objective of assessing the robustness and accuracy of the algorithm, we test its capabilities by means of four quasi-brittle benchmark applications.

Effect of potential screening on the H2 autoionizing states

We study the behavior of autoionizing states of the hydrogen molecule subject to screened Coulomb interactions using an ab initio Feshbach configuration interaction method. Special attention is given to the algorithms developed for the evaluation of (i) screened molecular orbitals expressed in terms of one-center expansions using B-spline polynomial basis functions and (ii) screened two-electron integrals between configurations expressed in terms of such molecular orbitals, by solving the screened Poisson equation. As an illustration of the method we focus on the lowest Feshbach resonance of the ${\mathcal{Q}}_{1}{\phantom{\rule{4pt}{0ex}}}^{1}{\mathrm{\ensuremath{\Sigma}}}_{g}^{+}$ series of doubly excited states of ${\mathrm{H}}_{2}$, which lies between the first ${\mathrm{H}}_{2}{}^{+}$ ($1s{\ensuremath{\sigma}}_{g}$) and the second ${\mathrm{H}}_{2}{}^{+}$ ($2p{\ensuremath{\sigma}}_{u}$) ionization thresholds. We show that Coulomb screening in the electron-proton interaction and between electrons may significantly alter the resonance position and autoionizing decay as a function of internuclear distance. In general, screening increases the resonance lifetime. However, when electron-proton screening dominates over electron-electron screening, we find that the ${\mathcal{Q}}_{1}$ resonance acquires a pronounced shape-resonance character at internuclear distances where the resonance approaches the lower ionization threshold, thus leading to a pronounced decrease of its lifetime.

A novel two-stage discrete crack method based on the screened Poisson equation and local mesh refinement

We propose an alternative crack propagation algorithm which effectively circumvents the variable transfer procedure adopted with classical mesh adaptation algorithms. The present alternative consists of two stages: a mesh-creation stage where a local damage model is employed with the objective of defining a crack-conforming mesh and a subsequent analysis stage with a localization limiter in the form of a modified screened Poisson equation which is exempt of crack path calculations. In the second stage, the crack naturally occurs within the refined region. A staggered scheme for standard equilibrium and screened Poisson equations is used in this second stage. Element subdivision is based on edge split operations using a constitutive quantity (damage). To assess the robustness and accuracy of this algorithm, we use five quasi-brittle benchmarks, all successfully solved.

Normals and texture fusion for enhancing orthogonal projections of 3D models

Abstract Orthogonal projections of objects play an important role in the process of making archaeological illustrations. We present a method to generate the detailed orthogonal projection of a 3D model by fusing normal and texture information in gradient domain. We first render the model into a texture image from a perpendicular view. A normal map is then obtained from the same view with pseudocolors converted from vertex normals. Finally, we make a non-photorealistic projection image that combines the texture image and the normal map by solving the 2D screened Poisson equation. The non-photorealistic projection is both geometry-aware and texture-aware and enhances the subtle details that are hard to see in the texture image or the normal map alone. It is more convenient for archaeologists to make line drawings by using tools such as Adobe Illustrator ® to trace over the fused images.