Due to the shortcomings such as the premature convergence and the bad local optimal searching capability in traditional intelligence methods for pattern synthesis, a new type of wolf pack algorithm named Levy-Cultural Wolf Pack Algorithm（LCWPA） was designed on the basis of the Cultural Wolf Pack Algorithm（CWPA）, which obeys the selective Levy flight. Because of the good overall management ability provided by the cultural algorithm in optimization process and the characteristics of excellent population diversity brought by Levy flight, the search efficiency of the new algorithm was greatly improved. When the algorithm was applied in the pattern synthesis of array antenna, the simulation results showed its high performance with multi-null and low side-lobe restrictions. In addition, the algorithm was superior to the Quantum Particle Swarm Optimization（QPSO）, Particle Swarm Optimization（PSO）, and Genetic Algorithm（GA） in optimization accuracy and operation speed, and is of very good generalization.

The homotopy continuation method has been widely used to compute multiple solutions of nonlinear differential equations, but the computational cost grows exponentially based on the traditional finite difference and finite element discretizations. In this work, we presented a new method by constructing a spectral approximation space adaptively based on a greedy algorithm for nonlinear differential equations. Then multiple solutions were computed by the homotopy continuation method on this low-dimensional approximation space. Various numerical examples were given to illustrate the feasibility and the efficiency of this new approach.

This paper investigates lowest-order weak Galerkin finite element methods for solving linear elasticity problems on rectangular and brick meshes. Specifically, constant vectors are used in element interiors and on element interfaces respectively for approximating displacement. For these constant basis functions, their discrete weak gradients are calculated in the local Raviart–Thomas spaces \( RT_{[0]}^d \) (\( d=2 \) or 3), whereas their discrete weak divergences are calculated as elementwise constants. Discrete weak strains are calculated accordingly. Then these quantities are used to develop finite element schemes in both strain-div and grad-div formulations, on both rectangular and brick meshes. A theoretical analysis supported by numerical experiments in both 2-dim and 3-dim reveal that the methods are locking-free and have optimal 1st order convergence in displacement, stress, and dilation (divergence of displacement), when the exact solution has full regularity. The methods can also capture low-regularity solutions very well. Strategies for efficient implementation including Schur complement are presented. Extension to quadrilateral and hexahedral meshes, in both theoretical analysis and numerical experiments, is also examined.

The feedback between planetary warming and soil carbon loss has been the focus of considerable scientific attention in recent decades, due to its potential to accelerate anthropogenic climate change. The soil carbon temperature sensitivity is traditionally estimated from short-term respiration measurements – either from laboratory incubations that are artificially manipulated or from field measurements that cannot distinguish between plant and microbial respiration. To address these limitations of previous approaches, we developed a new method to estimate soil temperature sensitivity (Q10) of soil carbon directly from warming-induced changes in soil carbon stocks measured in 36 field experiments across the world. Variations in warming magnitude and control organic carbon percentage explained much of field-warmed organic carbon percentage (R2 = 0.96), revealing Q10 across sites of 2.2 [1.6, 2.7] 95 % confidence interval (CI). When these field-derived Q10 values were extrapolated over the 21st century using a post hoc correction of 20 Coupled Model Intercomparison Project Phase 5 (CMIP5) Earth system model outputs, the multi-model mean soil carbon stock changes shifted from the previous value of 88 ± 153 Pg carbon (weighted mean ± 1 SD) to 19 ± 155 Pg carbon with a Q10-driven 95 % CI of 248 ± 191 to −95 ± 209 Pg carbon. On average, incorporating the field-derived Q10 values into Earth system model simulations led to reductions in the projected amount of carbon sequestered in the soil over the 21st century. However, the considerable parameter uncertainty led to extremely high variability in soil carbon stock projections within each model; intra-model uncertainty driven by the field-derived Q10 was as great as that between model variation. This study demonstrates that data integration should capture the variation of the system, as well as mean trends.

In this work, we develop an hp-adaptivity strategy for the minimum action method (MAM) using a posteriori error estimate. MAM plays an important role in minimizing the Freidlin-Wentzell action functional, which is the central object of the Freidlin-Wentzell theory of large deviations for noise-induced transitions in stochastic dynamical systems. Because of the demanding computation cost, especially in spatially extended systems, numerical efficiency is a critical issue for MAM. Difficulties come from both temporal and spatial discretizations. One severe hurdle for the application of MAM to large scale systems is the global reparametrization in time direction, which is needed inmost versions of MAM to achieve accuracy. We recently introduced a new version of MAM in [22], called tMAM, where we used some simple heuristic criteria to demonstrate that tMAM can be effectively coupled with h-adaptivity, i.e., the global reparametrization can be removed. The target of this paper is to integrate hp-adaptivity into tMAM using a posteriori error estimation techniques, which provides a general adaptive MAM more suitable for parallel computing. More specifically, we use the zero-Hamiltonian constraint to define an indicator to measure the error induced by linear time scaling, and the derivative recovery technique to construct an error indicator and a regularity indicator for the transition paths approximated by finite elements. Strategies for hp-adaptivity have been developed. Numerical results are presented.

In this paper, we first present the THex algorithm that refines a tetrahedral mesh into a hexahedral mesh. Strategies for efficient implementation of the THex algorithm are discussed. Then we present the lowest order weak Galerkin (WG) (Q0,Q0; RT[0]) finite element method for solving the Darcy equation on general hexahedral meshes. This simple solver uses constant pressure unknowns inside hexahedra and on faces but specifies the discrete weak gradients of these basis functions in local Raviart-Thomas RT[0] spaces. Implementation of this solver is straightforward. The solver is locally mass-conservative, and produces continuous normal fluxes, regardless of hexahedral mesh quality. When the mesh is asymptotically parallelopiped, this Darcy solver exhibits optimal order convergence in pressure, velocity, and flux, as demonstrated by numerical results.

In this paper, we study fast iterative solvers for the solution of fourth order parabolic equations discretized by mixed finite element methods. We propose to use consistent mass matrix in the discretization and use lumped mass matrix to construct efficient preconditioners. We provide eigenvalue analysis for the preconditioned system and estimate the convergence rate of the preconditioned GMRes method. Furthermore, we show that these preconditioners only need to be solved inexactly by optimal multigrid algorithms. Our numerical examples indicate that the proposed preconditioners are very efficient and robust with respect to both discretization parameters and diffusion coefficients. We also investigate the performance of multigrid algorithms with either collective smoothers or distributive smoothers when solving the preconditioner systems.

In this paper we study the a priori error estimates of finite element method for the system of time-dependent Poisson–Nernst–Planck equations, and for the first time, we obtain its optimal error estimates in L∞(H1) and L2(H1) norms, and suboptimal error estimates in L∞(L2) norm, with linear element, and optimal error estimates in L∞(L2) norm with quadratic or higher-order element, for both semi- and fully discrete finite element approximations. Numerical experiments are also given to validate the theoretical results.

Biomolecules exhibit conformational fluctuations near equilibrium states, inducing uncertainty in various biological properties in a dynamic way. We have developed a general method to quantify the uncertainty of target properties induced by conformational fluctuations. Using a generalized polynomial chaos (gPC) expansion, we construct a surrogate model of the target property with respect to varying conformational states. To alleviate the high-dimensionality of the corresponding stochastic space, we propose a method to increase the sparsity of the gPC expansion by defining a set of conformational "active space" random variables. With the increased sparsity, we employ the compressive sensing method to accurately construct the surrogate model. We demonstrate the performance of the surrogate model by evaluating fluctuation-induced uncertainty in solvent-accessible surface area for the bovine trypsin inhibitor protein system and show that the new approach offers more accurate statistical information than standard Monte Carlo approaches. Furthermore, the constructed surrogate model also enables us to directly evaluate the target property under various conformational states, yielding a more accurate response surface than standard sparse grid collocation methods. In particular, the new method provides higher accuracy in high-dimensional systems, such as biomolecules, where sparse grid performance is limited by the accuracy of the computed quantity of interest. Our new framework is generalizable and can be used to investigate the uncertainty of a wide variety of target properties in biomolecular systems.

This report describes our initial research to quantify uncertainties in the identification and characterization of possible attack states in a network. As a result, we should be able to estimate the current state in which the network is operating, based on a wide variety of network data, and attach a defensible measure of confidence to these state estimates. The output of this research will be new uncertainty quantification (UQ) methods to help develop a process for model development and apply UQ to characterize attacks/adversaries, create an understanding of the degree to which methods scale to "big" data, and offer methods for addressing model approaches with regard to validation and accuracy.

Adaptive mesh refinement and the Borgers algorithm are combined to generate a body-fitted mesh which can resolve the interface with fine geometric details. Standard linear finite element method based on such body-fitted meshes is applied to the elliptic interface problem and proven to be superclose to the linear interpolation of the exact solution. Based on this superconvergence result, a maximal norm error estimate of order one and half is obtained without using the discrete maximum principle. The data structure and meshing algorithms, including local refinement and coarsening, are very simple. In particular, no tree structure is needed. An efficient solver for solving the resulting linear algebraic systems is also developed and shown be robust with respect to both the problem size and the jump of the diffusion coefficients.

We have developed efficient numerical algorithms for solving 3D steady-state Poisson-Nernst-Planck (PNP) equations with excess chemical potentials described by the classical density functional theory (cDFT). The coupled PNP equations are discretized by a finite difference scheme and solved iteratively using the Gummel method with relaxation. The Nernst-Planck equations are transformed into Laplace equations through the Slotboom transformation. Then, the algebraic multigrid method is applied to efficiently solve the Poisson equation and the transformed Nernst-Planck equations. A novel strategy for calculating excess chemical potentials through fast Fourier transforms is proposed, which reduces computational complexity from O(N-2) to O(N log N), where N is the number of grid points. Integrals involving the Dirac delta function are evaluated directly by coordinate transformation, which yields more accurate results compared to applying numerical quadrature to an approximated delta function. Numerical results for ion and electron transport in solid electrolyte for lithium-ion (Li-ion) batteries are shown to be in good agreement with the experimental data and the results from previous studies.

In this paper we present a nonconforming finite element method for solving fourth order curl equations in three dimensions arising from magnetohydrodynamics models. We show that the method has an optimal error estimate for a model problem involving both curl^2 and curl^4 operators. The element has a very small number of degrees of freedom and it imposes the inter-element continuity along the tangential direction which is appropriate for the approximation of magnetic fields. We also provide explicit formulae of basis functions for this element.

In this paper, we develop a postprocessing derivative recovery scheme for the finite element solution $u_h$ on general unstructured but shape regular triangulations. In the case of continuous piecewise polynomials of degree $p\geq 1$, by applying the global $L^2$ projection ($Q_h$) and a smoothing operator ($S_h$), the recovered $p$th derivatives ($S_h^m Q_h\partial^p u_h$) superconverge to the exact derivatives ($\partial^p u$). Based on this technique we are able to derive a local error indicator depending only on the geometry of corresponding element and the $(p+1)$st derivatives approximated by $\partial S_h^m Q_h\partial^p u_h$. We provide several numerical examples illustrating the effectiveness of our schemes. We also observe that higher order elements are likely to require more conservative refinement strategies to create meshes corresponding to optimal orders of convergence.