Implicit Approximation Research Articles

Generalized reproducing kernel peridynamics: unification of local and non-local meshfree methods, non-local derivative operations, and an arbitrary-order state-based peridynamic formulation

State-based peridynamics is a non-local reformulation of solid mechanics that replaces the force density of the divergence of stress with an integral of the action of force states on bonds local to a given position, which precludes differentiation with the aim to model strong discontinuities effortlessly. A popular implementation is a meshfree formulation where the integral is discretized by quadrature points, which results in a series of unknowns at the points under the strong-form collocation framework. In this work, the meshfree discretization of state-based peridynamics under the correspondence principle is examined and compared to traditional meshfree methods based on the classical local formulation of solid mechanics. It is first shown that the way in which the peridynamic formulation approximates differentiation can be unified with the implicit gradient approximation, and this is termed the reproducing kernel peridynamic approximation. This allows the construction of non-local deformation gradients with arbitrary-order accuracy, as well as non-local approximations of higher-order derivatives. A high-order accurate non-local divergence of stress is then proposed to replace the force density in the original state-based peridynamics, in order to obtain global arbitrary-order accuracy in the numerical solution. These two operators used in conjunction with one another is termed the reproducing kernel peridynamic method. The strong-form collocation version of the method is tested against benchmark solutions to examine and verify the high-order accuracy and convergence properties of the method. The method is shown to exhibit superconvergent behavior in the nodal collocation setting.

A Class of Weighted Point Schemes for the Grünwald Implicit Finite Difference Solution of Time-Fractional Parabolic Equations Using KSOR method

In this study, system of Grünwald implicit approximation equations has been developed through the discretization of one-dimensional linear time-fractional parabolic equations using the Grünwald fractional derivative operator and second-order implicit finite difference scheme. The aim of this paper is to examine the effectiveness of Kaudd Successive Over-Relaxation (KSOR) iterative method, which is one of the weighted point iterative schemes for solving the proposed time-fractional parabolic equations by considering the Grünwald implicit approximation equation. To investigate the effectiveness of the proposed iterative method, numerical experiments and comparison are made in terms of number of iterations, execution time, and maximum absolute error. Based on numerical results, the accuracy of Grünwald implicit solution obtained by proposed iterative method is in excellent agreement, and it can be concluded that the proposed KSOR iterative method requires less number of iterations and execution time as compared to the existing point iterative method.

Implicit untangling

We propose a robust method for untangling an arbitrary number of cloth layers, possibly exhibiting deep interpenetrations, to a collision-free state, ready for animation. Our method relies on an intermediate, implicit representation to solve the problem: the user selects a few garments stored in a library together with their implicit approximations, and places them over a mannequin while specifying the desired order between layers. The intersecting implicit surfaces are then combined using a new family of N-ary composition operators, specially designed for untangling layers. Garment meshes are finally projected to the deformed implicit surfaces in linear time, while best preserving triangles and avoiding loss of details. Each of the untangling operators computes the target surface for a given garment in a single step, while accounting for the order between cloth layers and their individual thicknesses. As a group, they guarantee an intersection-free output configuration. Moreover, a weight can be associated with each layer to tune their relative influence during untangling, such as leather being less deformed than cloth. Results for each layer then reflect the combined effect of the other layers, enabling us to output a plausible configuration in contact regions. As our results show, our method can be used to generate plausible, new static shapes of garments when underwear has been added, as well as collision-free configurations enabling a user to safely launch animations of arbitrarily complex layered clothing.

Approximation Schemes for Mixed Optimal Stopping and Control Problems with Nonlinear Expectations and Jumps

We propose a class of numerical schemes for mixed optimal stopping and control of processes with infinite activity jumps and where the objective is evaluated by a nonlinear expectation. Exploiting an approximation by switching systems, piecewise constant policy timestepping reduces the problem to nonlocal semi-linear equations with different control parameters, uncoupled over individual time steps, which we solve by fully implicit monotone approximations to the controlled diffusion and the nonlocal term, and specifically the Lax–Friedrichs scheme for the nonlinearity in the gradient. We establish a comparison principle for the switching system and demonstrate the convergence of the schemes, which subsequently gives a constructive proof for the existence of a solution to the switching system. Numerical experiments are presented for a recursive utility maximization problem to demonstrate the effectiveness of the new schemes.

Numerical solution of a parabolic optimal control problem arising in economics and management

We investigate a parabolic optimal control problem that serves as a mathematical model for a class of problems of economics and management. The problem is to minimize the objective functional that is not convex and not coercive, and the state equation is a two-dimensional linear parabolic diffusion-advection equation controlled by the coefficients of the advective part. The main property of a solution of state equation is its non-negativity for a non-negative initial data. We prove that implicit (backward Euler) and fractional steps (operator splitting) approximations of the state equation have strictly positive solutions for positive time and use this fact to prove the existence of a solution for a discrete optimal control problem without imposing any additional constraints on the control function and mesh parameters. We derive first order optimality conditions and apply gradient-type iterative methods for the constructed discrete optimal control problem. Numerical tests confirm theoretically valid results and demonstrate the effectiveness of the proposed solution methods.

Convergence of a hybrid viscosity approximation method for finding zeros of m-accretive operators

For finding a zero of an m-accretive operator in a real Banach space, we consider an implicit hybrid viscosity approximation method and show that its explicit variant converges strongly to a zero under weaker assumptions than the ones used recently by Ceng et al. (Numer. Func. Anal. Opt. 35(2), 142–165, 2012). First, we examine the case of spaces which are uniformly smooth or reflexive and strictly convex with a uniformly Gâteaux differentiable norm. Afterwards, we improve our convergence results when the space is uniformly convex. Furthermore, we show that the strong convergence of some modified Halpern and Krasnosel’skii–Mann type methods can also be deduced from our results. Finally, a numerical example is also given to illustrate the convergence analysis of the considered method.

Use of Interaction Energies in QM/MM Free Energy Simulations.

The use of the most accurate (i.e., QM or QM/MM) levels of theory for free energy simulations (FES) is typically not possible. Primarily, this is because the computational cost associated with the extensive configurational sampling needed for converging FES is prohibitive. To ensure the feasibility of QM-based FES, the "indirect" approach is generally taken, necessitating a free energy calculation between the MM and QM/MM potential energy surfaces. Ideally, this step is performed with standard free energy perturbation (Zwanzig's equation) as it only requires simulations be carried out at the low level of theory; however, work from several groups over the past few years has conclusively shown that Zwanzig's equation is ill-suited to this task. As such, many approximations have arisen to mitigate difficulties with Zwanzig's equation. One particularly popular notion is that the convergence of Zwanzig's equation can be improved by using interaction energy differences instead of total energy differences. Although problematic numerical fluctuations (a major problem when using Zwanzig's equation) are indeed reduced, our results and analysis demonstrate that this "interaction energy approximation" (IEA) is theoretically incorrect, and the implicit approximation invoked is spurious at best. Herein, we demonstrate this via solvation free energy calculations using IEA from two different low levels of theory to the same target high level. Results from this proof-of-concept consistently yield the wrong results, deviating by ∼1.5 kcal/mol from the rigorously obtained value.

Explicit-Implicit schemes for solving the problems of the dynamics of isotropic and anisotropic elastoviscoplastic media

A method is developed for the numerical solution of dynamic elastic-visco-plasticity equations for isotropic and anisotropic cases. Anisotropic elastic-visco-plastic systems of equations often describe models of deformable solid media with a discrete set of slip planes (in layered and block media) with nonlinear slip conditions at contact boundaries. For a stable numerical solution of such systems of differential equations, an explicitly implicit method is proposed. An explicit approximation is used for the equations of motion while an implicit approximation is applied to the constitutive relations containing a small parameter in the denominator of the free term. Examples of numerical solutions are presented for dynamic problems on the reflection of waves from obstacles in the inelastic layered and block media with the formation of slip and delamination zones at the boundaries of structural elements.

An inverse problem for determination of the right part of parabolic equation by conjugate gradient method

In the paper for multidimensional parabolic equation we consider a problem of the right-hand side identification which is a product of two functions, one of which depends on time and the other one depends on space variables. For numerical solution of the posed inverse initial-boundary value problem, the conjugate gradient method is used in combination with the method of finite differences with a purely implicit time approximation. The authors discuss the results of the computational experiment for model problems with quasi-real solutions

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L∞‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.

Generalized implicit viscosity approximation method for multivalued mappings in CAT(0) spaces

Abstract We prove strong convergence of the sequence generated by implicit viscosity approximation method involving a multivalued nonexpansive mapping in framework of CAT(0) space. Under certain appropriate conditions on parameters, we show that such a sequence converges strongly to a fixed point of the mapping which solves a variational inequality. We also present some convergence results for the implicit viscosity approximation method in complete ℝ-trees without the endpoint condition.

Economic explicit-implicit schemes for solving multidimensional diffusion-convection problems

This work concerns the development of an efficient parallel algorithm for numerical solution of nonstationary diffusion-convection problems by means of a multiprocessor computer system with distributed memory. Economically explicit-implicit difference schemes and the method of splitting into physical processes are used as a basis. The original problem is replaced by a sequence of one-dimensional and two-dimensional difference problems using complex schemes that approximate the original problem in the general sense. Explicit-implicit difference schemes involve explicit approximation in horizontal directions and implicit approximation with weights in a vertical direction and require less time for solving diffusion-convection problems compared to explicit schemes while maintaining acceptable accuracy of solutions. The algorithm is proposed to find the optimal weight value and it yields the lowest approximation error in the solution of the diffusion-convection problem in the vertical direction for given time grid steps. The three-dimensional model problem of transport of suspensions in the water environment is considered. The model takes into account the following processes: advective transport caused by the movement of the water medium, microturbulent diffusion and gravitational deposition of suspended particles, as well as changes in the geometry of the bottom caused by the deposition of suspended particles or the rise of sediment particles. Application of the parallel algorithm developed for numerical modeling of suspension transport makes it possible to significantly improve the real-time forecast accuracy and the validity of the accepted engineering decisions used in designing the objects of coastal infrastructure.

Continual models of layered and block media with slippage and delamination

Continual models of solid media with a discrete set of slip planes (layered, block media) and with nonlinear viscous-plastic type slip conditions at the contact boundaries of structural elements are constructed. The resulting system of equations contains a small viscosity parameter in the denominator of nonlinear free terms. For a stable numerical solution of a system of differential equations, an explicit-implicit method is proposed with an explicit approximation of the equations of motion and an implicit approximation of the constitutive relations containing a small parameter. Examples of numerical solutions of dynamic problems on the scattering of waves on layered cluster in an elastic medium, on the formation of slip and delamination zones at the boundaries of structural elements are given.

System of Variational Inclusions and Fixed Points of Pseudocontractive Mappings in Banach Spaces

The purpose of this paper is to solve the general system of variational inclusions (GSVI) with hierarchical variational inequality (HVI) constraint, for an infinite family of continuous pseudocontractive mappings in Banach spaces. By utilizing the equivalence between the GSVI and the fixed point problem, we construct an implicit multiple-viscosity approximation method for solving the GSVI. Under very mild conditions, we prove the strong convergence of the proposed method to a solution of the GSVI with the HVI constraint, for infinitely many pseudocontractions.

Implicit numerical solutions to neutral-type stochastic systems with superlinearly growing coefficients

In this paper, our main aim is to investigate the stability and strong convergence of an implicit numerical approximations for neutral-type stochastic differential equations with superlinearly growing coefficients. After providing moment boundedness and exponential stability for the exact solutions, we show that the backward Euler–Maruyama numerical method preserves stability and boundedness of moments, and the numerical approximations converge strongly to the true solutions for sufficiently small step size.

Application of EGSOR Iteration with Nonlocal Arithmetic Discretization Scheme for Solving Burger’s Equation

The purpose of this paper is to investigate the performance of 2-Point Explicit Group Successive Over Relaxation (EGSOR) method with nonlocal arithmetic discretization scheme for solving Burger’s equations. To do this matter, the proposed problems need to be discretized by using second-order implicit scheme to derive the corresponding nonlinear second order implicit finite difference approximate equation. Then, the nonlocal arithmetic discretization scheme is used to transform the corresponding nonlinear implicit approximation equation into a system of linear equations. Furthermore, numerical results of four proposed problems are also included in order to verify the effectiveness of the EGSOR method compared to Gauss-Seidel (GS) and SOR method. Based on numerical results, it can be concluded that the performance of EGSOR method is better than GS and SOR in terms of number of iterations and computational time.

Adaptive cluster approximation for reduced density-matrix functional theory

A method, called the adaptive cluster approximation (ACA), for single-impurity Anderson models is proposed. It is based on reduced density-matrix functional theory, where the one-particle reduced density matrix is used as the basic variable. The adaptive cluster approximation introduces a unitary transformation of the bath states such that the effect of the bath is concentrated to a small cluster around the impurity. For this small effective system one can then either calculate the reduced density-matrix functional numerically exact from Levy's constrained-search formalism or approximate it by an implicit approximation of the reduced density-matrix functional. The method is evaluated for single-impurity Anderson models with finite baths. The method converges rapidly to the exact result with the size of the effective bath.

The analytical solution and numerical solutions for a two-dimensional multi-term time fractional diffusion and diffusion-wave equation

In this paper we consider the analytical and numerical solutions for a two-dimensional multi-term time-fractional diffusion and diffusion-wave equation. We derive the analytical solution for the equation using the method of separation of variables and properties of the multivariate Mittag-Leffler function. An implicit difference approximation is constructed. Stability and convergence analysis of the numerical scheme are proved by the energy method. Numerical examples are constructed to evaluate the working of the numerical scheme as compared to theoretical analysis. • A new two-dimensional multi-term time-fractional diffusion and diffusion. • Wave equation (2D-MT-TFD-DWE) is considered. • The analytical solution for the 2D-MT-TFD-DWE is derived. • A novel implicit difference method (IDM) for 2D-MT-TFD-DWE is proposed. • The stability and convergence of the IDM are proved by the energy method. • Numerical examples are given.

On a Double Porosity Model of Fractured-Porous Reservoirs Based on a Hybrid Flow Function

In this paper, a model of double porosity for a fractured porous medium using a combination of classical and gradient functions of mass transfer between the cracks and porous blocks in a weakly compressible single-phase fluid flow is considered. As compared to the wellknown models, the model with such a mass transfer function allows one to take into account the anisotropic properties of filtration in a more general form. The results of numerical tests for two- and three-dimensional model problems are presented. The computational algorithm is based on a finite element approximation with respect to space and a completely implicit approximation with respect to time.

A multi-resolution collocation procedure for time-dependent inverse heat problems

In this paper, a Haar wavelet collocation method (HWCM) is developed for PDEs related to the framework of so-called inverse problem. These include PDEs with unknown time dependent heat source and unknown solution in interior of the domain. To this end, a transformation is used to eliminate the unknown heat source to obtain a PDE without a heat source. After elimination of unknown non-homogeneous term, an implicit finite-difference approximations is used to approximate the time derivative and Haar wavelets are used for approximation of the space derivatives. Several numerical experiments related to one- and two-dimensional heat sources are included to validate small condition number of coefficient matrix, accuracy and simple applicability of the proposed approach.