Scalar Problem Research Articles

Riemann–Hilbert problem and soliton solutions with their asymptotic analysis for the focusing nonlocal Hirota equation with step-like initial data

The Riemann–Hilbert (RH) problem is developed to study the Cauchy problem of focusing space symmetric nonlocal Hirota (fsNH) equation with step-like initial data. Firstly, we analyze the characteristics of the Jost solutions and spectral functions, including analyticity, symmetry, and asymptotics as k→∞ and k→0. Then, we discuss three cases of zero point distributions of the spectral functions, and derive the expressions of spectral functions by constructing the special analytic functions and establishing the scalar RH problem. Furthermore, the constraint relations among the zeros of the spectral functions are analyzed. Next, the corresponding RH problem equipped with residual and singularity conditions is successfully derived. These singularity conditions are then transformed into a general residue condition, which reduces the problem of solving the RH problem with reflection-less potential to solving a system of algebraic equations. Ultimately, the corresponding soliton solutions of the fsNH equation and their asymptotics are derived from the solution of the RH problem.

A novel direct interpolation boundary element method formulation for solving diffusive–advective problems

The capability of dealing with the advective transport term constitutes a challenging issue for the performance of the majority of the numerical techniques, which significantly lose their precision with the increasing in the relative magnitude of this term. Recently, a new boundary element technique called direct interpolation (DIBEM) has emerged, mainly characterized by the approximation of the entire kernel of the remaining domain integrals. The DIBEM model has been successfully applied to several scalar field problems and recently applied to certain diffusive–advective problems with superior accuracy compared to dual reciprocity formulation (DRBEM). Using DIBEM, a broader range of precise responses for flows with higher Peclet numbers can be reached beyond a lower computational cost due to the simplicity of its matrix operations. These advantages are due DIBEM concept that employs a simple interpolation procedure to approximate the kernel of the advective domain integral. However, it was noticed that in some physical scenarios with spatial variation in the velocity field, the accuracy of DIBEM did not have the same quality observed in other applications. Therefore, this work presents a new DIBEM model so that the integral equations include the explicit calculation of spatial derivatives and simultaneously solve the variation of the divergent velocity field if this is not zero. Conversely, reactive and source terms were also included to expand numerical comparisons. Thus, in the proposed examples, the results of two DIBEM models, the standard (DIBEM-S) and the new, so-called alternative model (DIBEM-A), are compared with DRBEM and also with available analytical solutions.

Vector optimization problems in near vector space

ABSTRACT The conventional vector optimization problems intend to optimize the objective functions, which are defined on a vector space into another vector space. In this paper, we are going to study the vector optimization problems by minimizing the objective functions, which are defined on a vector space into another so-called near vector space. The concept of near vector space is axiomatically defined, and it does not necessarily contain the additive inverse element. A so-called pointed-like convex cone is introduced to propose the concept of minimizer of this new type vector optimization problem. Its corresponding scalar optimization problem is also formulated. Finally, we show that the optimal solution of the corresponding scalar optimization problem is also a minimizer of the original vector optimization problem based on the near vector space.

Optimal impulse control problems with time delays: An illustrative example

For impulse control systems described by a measure driven differential equation, depending linearly on the measure, it is customary to interpret the state trajectory corresponding to an impulse control, specified by a measure, as the limit of state trajectories associated with some sequence of conventional controls approximating the measure. It is known that, when the measure is vector valued, it is possible that different choices of approximating sequences for the measure give rise to different limiting state trajectories. If the measure is scalar valued, however, there is a unique limiting trajectory. Now consider impulse control systems, in which the right side of the measure driven differential equation depends on both the current and delayed states. In recent work by the authors it has been shown that, for such impulse control systems with time delay, the state trajectory corresponding to a given measure may be non-unique, even when the measure is scalar valued. It was also shown that each limiting state trajectory can be identified with the unique state trajectory associated with some measure together with a family of ‘attached controls’. (The attached controls capture the nature of the measure approximation.) The authors also derived a maximum principle governing minimizers for a general class of impulse optimal control problems with time delay, in which the domain of the optimization problem comprises measures coupled with a family of ‘attached controls’. The purpose of this paper is both to illustrate, by means of an example, this newly discovered non-uniqueness phenomenon and to provide the first application of the new maximum principle, to investigate minimizers for scalar input impulse optimal control problems with time delay, in circumstances when limiting state trajectories associated with a given measure control are not unique. The example is an optimal control problem, for which the underlying control system is a forced harmonic oscillator, with scalar impulse control, in which the control gain is a nonlinear function of the current and delayed states.

Discovering artificial viscosity models for discontinuous Galerkin approximation of conservation laws using physics-informed machine learning

Finite element-based high-order solvers of conservation laws offer large accuracy but face challenges near discontinuities due to the Gibbs phenomenon. Artificial viscosity is a popular and effective solution to this problem based on physical insight. In this work, we present a physics-informed machine learning algorithm to automate the discovery of artificial viscosity models. We refer to the proposed approach as an “hybrid” approach which stands at the edge between supervised and unsupervised learning. More precisely, the proposed “hybrid” paradigm is not supervised in the classical sense as it does not utilize labeled data in the traditional way but relies on the intrinsic properties of the reference solution. The algorithm is inspired by reinforcement learning and trains a neural network acting cell-by-cell (the viscosity model) by minimizing a loss defined as the difference with respect to a reference solution thanks to automatic differentiation. This enables a dataset-free training procedure. We prove that the algorithm is effective by integrating it into a state-of-the-art Runge-Kutta discontinuous Galerkin solver. We showcase several numerical tests on scalar and vectorial problems, such as Burgers' and Euler's equations in one and two dimensions. Results demonstrate that the proposed approach trains a model that is able to outperform classical viscosity models. Moreover, we show that the learnt artificial viscosity model is able to generalize across different problems and parameters.

Lagrange Duality and Saddle-Point Optimality Conditions for Nonsmooth Interval-Valued Multiobjective Semi-Infinite Programming Problems with Vanishing Constraints

This article deals with a class of nonsmooth interval-valued multiobjective semi-infinite programming problems with vanishing constraints (NIMSIPVC). We introduce the VC-Abadie constraint qualification (VC-ACQ) for NIMSIPVC and employ it to establish Karush–Kuhn–Tucker (KKT)-type necessary optimality conditions for the considered problem. Regarding NIMSIPVC, we formulate interval-valued weak vector, as well as interval-valued vector Lagrange-type dual and scalarized Lagrange-type dual problems. Subsequently, we establish the weak, strong, and converse duality results relating to the primal problem, NIMSIPVC, and the corresponding dual problems. Moreover, we introduce the notion of saddle points for the interval-valued vector Lagrangian and scalarized Lagrangian of NIMSIPVC. Furthermore, we derive the saddle-point optimality criteria for NIMSIPVC by establishing the relationships between the solutions of NIMSIPVC and the saddle points of the corresponding Lagrangians of NIMSIPVC, under convexity assumptions. Non-trivial illustrative examples are provided to demonstrate the validity of the established results. The results presented in this paper extend the corresponding results derived in the existing literature from smooth to nonsmooth optimization problems, and we further generalize them for interval-valued multiobjective semi-infinite programming problems with vanishing constraints.

A branch and cut algorithm to optimize a weighted sum-of-ratios in multiobjective mixed-integer fractional programming

AbstractMultiobjective linear fractional programming is useful to model multiobjective problems where all or some of the objective functions are a ratio or proportion of one linear/affine function to another linear/affine function. In practice, many of such problems include integer variables. If the weighted-sum scalarization is used to compute efficient solutions to the multiobjective problem, then the scalar problem to be solved for each weight vector turns out to be a weighted sum-of-ratios. There are several algorithms reported in the literature to optimize weighted sum-of-ratios, but almost all of them cannot deal with integer variables. In this paper we propose a Branch & Cut algorithm to optimize weighted-sums of the objective functions in multiobjective mixed integer fractional programming (MOMIFP). Several theoretical properties that support the algorithm are presented and proved. Computational experiments with randomly generated general problems are presented and discussed, which show that the algorithm is able to deal with practical MOMIFP problems.

A new local and explicit kinetic method for linear and non-linear convection-diffusion problems with finite kinetic speeds: I. One-dimensional case

We propose a numerical approach, of the BGK kinetic type, that is able to approximate with a given, but arbitrary, order of accuracy the solution of linear and non-linear convection-diffusion type problems: scalar advection-diffusion, non-linear scalar problems of this type and the compressible Navier-Stokes equations. Our kinetic model can use finite advection speeds that are independent of the relaxation parameter, and the time step does not suffer from a parabolic constraint. Having finite speeds is in contrast with many of the previous works about this kind of approach, and we explain why this is possible: paraphrasing more or less [1], the convection-diffusion like PDE is not a limit of the BGK equation, but a correction of the same PDE without the parabolic term at the second order in the relaxation parameter that is interpreted as Knudsen number. We then show that introducing a relaxation matrix instead of the well-known BGK relaxation makes it possible to target a desired convection-diffusion system.Several numerical examples, ranging from a simple pure diffusion model to the compressible Navier-Stokes equations illustrate our approach.

Asymptotics of solutions to systems of (pseudo)differential equations with localized right-hand sides

Abstract The article describes a method for constructing semiclassical asymptotic solutions of multidimensional (pseudo)differential partial differential equations with localized right-hand sides. The method is based on the asymptotic reduction of vector problems (under certain additional conditions) to scalar (pseudo)differential problems, for which efficient asymptotic formulas have recently been constructed. The method is based on the Feynman–Maslov operator calculus. Even if the original problems are stated for differential equations, the operators in reduced problems, as a rule, turn out to be pseudodifferential. Further, nontrivial problems of calculating operator symbols in the reduced scalar problems arise in the reduction process. We discuss approaches to the transformation of these symbols so as to enable their effective use when solving real problems. The method is illustrated by several physical examples.

On the Selection of Weights for Difference Schemes to Approximate Systems of Differential Equations

We consider the problem of determining the weights of difference schemes whose form is specified by a particular symbolic expression. The order of approximation of the differential equation is equal to a given number. To solve it, it was propose to proceed from considering systems of differential equations of a general form to one scalar equation. This method provides us with some values for the weights, which we propose to test using Richardson’s method. The method was shown to work in the case of low-order schemes. However, when transitioning from the scalar problem to the vector and nonlinear problems, the reduction of the order of the scheme, whose weights are selected for the scalar problem, occurs in different families of schemes. This was first discovered when studying the Shanks scheme, which belongs to the family of explicit Runge–Kutta schemes. This does not deteriorate the proposed strategy itself concerning the simplification of the weight-determination problem, which should include a clause on mandatory testing of the order using the Richardson method.

A Learned-SVD Approach to the Electromagnetic Inverse Source Problem.

We propose an artificial intelligence approach based on deep neural networks to tackle a canonical 2D scalar inverse source problem. The learned singular value decomposition (L-SVD) based on hybrid autoencoding is considered. We compare the reconstruction performance of L-SVD to the Truncated SVD (TSVD) regularized inversion, which is a canonical regularization scheme, to solve an ill-posed linear inverse problem. Numerical tests referring to far-field acquisitions show that L-SVD provides, with proper training on a well-organized dataset, superior performance in terms of reconstruction errors as compared to TSVD, allowing for the retrieval of faster spatial variations of the source. Indeed, L-SVD accommodates a priori information on the set of relevant unknown current distributions. Different from TSVD, which performs linear processing on a linear problem, L-SVD operates non-linearly on the data. A numerical analysis also underlines how the performance of the L-SVD degrades when the unknown source does not match the training dataset.

An unconditionally-stable well-posed relativistic particle pusher

Particle pushers widely used in Particle-in-Cell(PIC) simulations are commonly required to be unconditionally stable and meet the basic physical laws. In this work, basing on central difference, we propose an unconditionally stable and well posed particle pusher. By mathematical deduction, the high-dimensional non-linear problem for solving tensor-form relativistic Lorentz force law equation is transformed to a quartic scalar problem and a following linear problem. Some practical suggestions for programming and some numerical results are also given.

Multi-objective optimization for perishable product dispatch in a FEFO system for a food bank single warehouse

One of the main challenges of food bank warehouses in developing countries is to determine how to allocate perishable products to beneficiary agencies with different expiry dates while ensuring food safety, meeting nutritional requirements, and minimizing the shortage. The contribution of this research is to introduce a new multi-objective, multi-product, and multi-period perishable food allocation problem based on a single warehouse management system for a First Expired-First Out (FEFO) policy. Moreover, it incorporates the temporal aspect, guaranteeing the dispatch of only those perishable products that meet the prescribed minimum quality standards. A weighted sum approach converts the multi-objective problem of minimizing a vector of objective functions into a scalar problem by constructing a weighted sum of all the objectives. The problem can then be solved using a standard constrained optimization procedure. The proposed mixed integer linear model is solved by using the CPLEX solver. The solution obtained from the multi-objective problem allows us to identify days and products experiencing shortages. In such cases, when there is insufficient available inventory, the total quantity of product to be dispatched is redistributed among beneficiaries according to a pre-established prioritization. These redistributions are formulated as integer programming problems using a score-based criterion and solved by an exact method based on dynamic programming. Computational results demonstrate the applicability of the novel model for perishable items to a real-world study case.

Parallel ensemble Kalman method with total variation regularization for large-scale field inversion

Field inversion is often encountered in data-driven computational modeling to infer latent spatial–varying parameters from available observations. The ensemble Kalman method is emerging as a useful tool for solving field inversion problems due to its derivative-free merits. However, the method is computationally prohibitive for large-scale field inversion with high-dimensional observation data, which necessitates developing a practical efficient implementation strategy. In this work, we propose a parallel implementation of the ensemble Kalman method with total variation regularization for large-scale field inversion problems. It is achieved by partitioning the computational domain into non-overlapping subdomains and performing local ensemble Kalman updates at each subdomain parallelly. In doing so, the computational complexity of the ensemble-based inversion method is significantly reduced to the level of local subdomains. Further, the total variation regularization is employed to smoothen the physical field over the entire domain, which can reduce the inference discrepancy caused by missing covariances near subdomain interfaces. The capability of the proposed method is demonstrated in three field inversion problems with increasing complexity, i.e., the diffusion problem, the scalar transport problem and the Reynolds averaged Navier-Stokes closure problem. The numerical results show that the proposed method can significantly improve computational efficiency with satisfactory inference accuracy.

Intralinguistic Motivation for Pluralism About Truth

Critics of the scope problem that motivates pluralism about truth have argued that it is a pseudo-problem. If the criticism is correct, then truth pluralism is left unmotivated and potentially bankrupt. In this paper, I argue that closely related to the scope problem is another problem, which I call “the scalar problem.” If the property of truth is sensitive to how an agent expresses the truth predicate within a single linguistic discourse and different agents or groups of agents express truth differently within that discourse, then there are different ways of being true within the same linguistic discourse. Given this possibility, even if the scope problem fails, truth pluralism remains fully motivated. Keywords: alethic pluralism, truth, Quine-Sainsbury problem, experimental philosophy

Using Genetic Algorithms and Core Values of Cooperative Games to Solve Fuzzy Multiobjective Optimization Problems

A new methodology for solving the fuzzy multiobjective optimization problems is proposed in this paper by considering the fusion of cooperative game theory and genetic algorithm. The original fuzzy multiobjective optimization problem needs to be transformed into a scalar optimization problem, which is a conventional optimization problem. Usually, the assignments of suitable coefficients to the corresponding scalar optimization problem are subjectively determined by the decision makers. However, these assignments may cause some biases by their subjectivity. Therefore, this paper proposes a mechanical procedure to avoid this subjective biases. We are going to formulate a cooperative game using the α-level functions of the multiple fuzzy objective functions. Under this setting, the suitable coefficients can be determined mechanically by involving the core values of the cooperative game, which is formulated using the multiple fuzzy objective functions. We shall prove that the optimal solutions of the transformed scalar optimization problem are indeed the nondominated solutions of fuzzy multiobjective optimization problem. Since the core-nondominated solutions will depend on the coefficients that are determined by the core values of cooperative game, there will be a lot of core-nondominated solutions that will also depend on the corresponding coefficients. In order to obtain the best core-nondominated solution, we shall invoke the genetic algorithms by evolving the coefficients.

Correction: A Prescribed Scalar and Boundary Mean Curvature Problem and the Yamabe Classification on Asymptotically Euclidean Manifolds with Inner Boundary

Correction: A Prescribed Scalar and Boundary Mean Curvature Problem and the Yamabe Classification on Asymptotically Euclidean Manifolds with Inner Boundary

Continuously bounds-preserving discontinuous Galerkin methods for hyperbolic conservation laws

For finite element approximations of transport phenomena, it is often necessary to apply a form of limiting to ensure that the discrete solution remains well-behaved and satisfies physical constraints. However, these limiting procedures are typically performed at discrete nodal locations, which is not sufficient to ensure the robustness of the scheme when the solution must be evaluated at arbitrary locations (e.g., for adaptive mesh refinement, remapping in arbitrary Lagrangian–Eulerian solvers, overset meshes, etc.). In this work, a novel limiting approach for discontinuous Galerkin methods is presented which ensures that the solution is continuously bounds-preserving (i.e., across the entire solution polynomial) for any arbitrary choice of basis, approximation order, and mesh element type. Through a modified formulation for the constraint functionals, the proposed approach requires only the solution of a single spatial scalar minimization problem per element for which a highly efficient numerical optimization procedure is presented. The efficacy of this approach is shown in numerical experiments by enforcing continuous constraints in high-order unstructured discontinuous Galerkin discretizations of hyperbolic conservation laws, ranging from scalar transport with maximum principle preserving constraints to compressible gas dynamics with positivity-preserving constraints.

A conservative discontinuous-Galerkin-in-time (DGiT) multirate time integration framework for interface-coupled problems with applications to solid–solid interaction and air–sea models

In this paper we extend the DGiT multirate framework, developed in Connors and Sockwell (2022) for scalar transmission problems, to a solid–solid interaction (SSI) problem involving two coupled elastic solids and a coupled air–sea model with the rotating, thermal shallow water equations. In so doing we aim to demonstrate the broad applicability of the mathematical theory and governing principles established in Connors and Sockwell (2022) to coupled problems characterized by subproblems evolving at different temporal scales. Multirate time integration algorithms employing different time steps, optimized for the dynamics of each subproblem, can significantly improve simulation efficiency for such coupled problems. However, development of multirate algorithms is a highly non-trivial task due to the coupling, which can impact accuracy, stability or other desired properties such as preservation of system invariants. DGiT provides a general template for multirate time integration that can achieve these properties. To elucidate the manner in which DGiT accomplishes this task, we fully detail each step in the application of the framework to the SSI and air–sea coupled problems. Numerical examples illustrate key properties of the resulting multirate schemes for both problems.

WELL-POSEDNESS OF A NONLINEAR INTERFACE PROBLEM DESCRIBED BY HEMIVARIATIONAL INEQUALITIES VIA BOUNDARY INTEGRAL OPERATORS

This paper is devoted to the well-posedness of a novel nonlinear interface problem on an unbounded domain with nonmonotone set-valued transmission conditions. This interface problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models nonmonotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality (HVI), which however lives on the unbounded domain, and thus cannot be analyzed in a reflexive Banach space setting. Boundary integral methods lead to another HVI that is amenable to functional analytic methods using standard Sobolev spaces on the interior domain and Sobolev spaces of fractional order on the coupling boundary. Broadening the scope of the paper, we consider extended real-valued HVIs augmented by convex extended real-valued functions. Under a smallness hypothesis, we provide existence and uniqueness results and, moreover, establish a stability result for extended real-valued HVIs with respect to the extended real-valued function as a parameter. Based on the latter general stability result, we provide various stability results for the interface problem, as well as the stability of a related bilateral obstacle interface problem with respect to the obstacles.