Mathematical and Computational Applications

Open AccessFeature PaperEditor’s ChoiceArticle

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

by

Dmytro Sytnyk

and

Roderick Melnik

Math. Comput. Appl. 2021, 26(4), 73; https://doi.org/10.3390/mca26040073 - 23 Oct 2021

Cited by 5 | Viewed by 3017

Abstract

Nonlocal models are ubiquitous in all branches of science and engineering, with a rapidly expanding range of mathematical and computational applications due to the ability of such models to capture effects and phenomena that traditional models cannot. While spatial nonlocalities have received considerable [...] Read more.

Nonlocal models are ubiquitous in all branches of science and engineering, with a rapidly expanding range of mathematical and computational applications due to the ability of such models to capture effects and phenomena that traditional models cannot. While spatial nonlocalities have received considerable attention in the research community, the same cannot be said about nonlocality in time, in particular when nonlocal initial conditions are present. This paper aims at filling this gap, providing an overview of the current status of nonlocal models and focusing on the mathematical treatment of such models when nonlocal initial conditions are at the heart of the problem. Specifically, our representative example is given for a nonlocal-in-time problem for the abstract Schrödinger equation. By exploiting the linear nature of nonlocal conditions, we derive an exact representation of the solution operator under assumptions that the spectrum of Hamiltonian is contained in the horizontal strip of the complex plane. The derived representation permits us to establish the necessary and sufficient conditions for the problem’s well-posedness and the existence of its solution under different regularities. Furthermore, we present new sufficient conditions for the existence of the solution that extend the existing results in this field to the case when some nonlocal parameters are unbounded. Two further examples demonstrate the developed methodology and highlight the importance of its computer algebra component in the reduction procedures and parameter estimations for nonlocal models. Finally, a connection of the considered models and developed analysis is discussed in the context of other reduction techniques, concentrating on the most promising from the viewpoint of data-driven modelling environments, and providing directions for further generalizations. Full article

► Show Figures

Figure 1

Open AccessEditor’s ChoiceArticle

Modelling Forest Fires Using Complex Networks

by

,

,

and

Math. Comput. Appl. 2021, 26(4), 68; https://doi.org/10.3390/mca26040068 - 28 Sep 2021

Cited by 2 | Viewed by 2218

Abstract

Forest fires have been a major threat to the environment throughout history. In order to mitigate its consequences, we present, in a first of a series of works, a mathematical model with the purpose of predicting fire spreading in a given land portion [...] Read more.

Forest fires have been a major threat to the environment throughout history. In order to mitigate its consequences, we present, in a first of a series of works, a mathematical model with the purpose of predicting fire spreading in a given land portion divided into patches, considering the area and the rate of spread of each patch as inputs. The rate of spread can be estimated from previous knowledge on fuel availability, weather and terrain conditions. We compute the time duration of the spreading process in a land patch in order to construct and parametrize a landscape network, using cellular automata simulations. We use the multilayer network model to propose a network of networks at the landscape scale, where the nodes are the local patches, each with their own spreading dynamics. We compute some respective network measures and aim, in further work, for the establishment of a fire-break structure according to increasing accuracy simulation results. Full article

(This article belongs to the Special Issue Numerical and Symbolic Computation: Developments and Applications 2021)

► Show Figures

Figure 1

Open AccessFeature PaperEditor’s ChoiceReview

Review of Multi-Physics Modeling on the Active Magnetic Regenerative Refrigeration

by

,

,

and

Math. Comput. Appl. 2021, 26(2), 47; https://doi.org/10.3390/mca26020047 - 15 Jun 2021

Cited by 6 | Viewed by 2643

Abstract

Compared to conventional vapor-compression refrigeration systems, magnetic refrigeration is a promising and potential alternative technology. The magnetocaloric effect (MCE) is used to produce heat and cold sources through a magnetocaloric material (MCM). The material is submitted to a magnetic field with active magnetic [...] Read more.

Compared to conventional vapor-compression refrigeration systems, magnetic refrigeration is a promising and potential alternative technology. The magnetocaloric effect (MCE) is used to produce heat and cold sources through a magnetocaloric material (MCM). The material is submitted to a magnetic field with active magnetic regenerative refrigeration (AMRR) cycles. Initially, this effect was widely used for cryogenic applications to achieve very low temperatures. However, this technology must be improved to replace vapor-compression devices operating around room temperature. Therefore, over the last 30 years, a lot of studies have been done to obtain more efficient devices. Thus, the modeling is a crucial step to perform a preliminary study and optimization. In this paper, after a large introduction on MCE research, a state-of-the-art of multi-physics modeling on the AMRR cycle modeling is made. To end this paper, a suggestion of innovative and advanced modeling solutions to study magnetocaloric regenerator is described. Full article

(This article belongs to the Special Issue Characterization of Magnetocaloric Devices and Materials through Mathematical Models)

► Show Figures

Figure 1

Open AccessFeature PaperEditor’s ChoiceArticle

Derivative-Free Multiobjective Trust Region Descent Method Using Radial Basis Function Surrogate Models

by

Manuel Berkemeier

and

Sebastian Peitz

Math. Comput. Appl. 2021, 26(2), 31; https://doi.org/10.3390/mca26020031 - 15 Apr 2021

Cited by 3 | Viewed by 2457

Abstract

We present a local trust region descent algorithm for unconstrained and convexly constrained multiobjective optimization problems. It is targeted at heterogeneous and expensive problems, i.e., problems that have at least one objective function that is computationally expensive. Convergence to a Pareto critical point [...] Read more.

We present a local trust region descent algorithm for unconstrained and convexly constrained multiobjective optimization problems. It is targeted at heterogeneous and expensive problems, i.e., problems that have at least one objective function that is computationally expensive. Convergence to a Pareto critical point is proven. The method is derivative-free in the sense that derivative information need not be available for the expensive objectives. Instead, a multiobjective trust region approach is used that works similarly to its well-known scalar counterparts and complements multiobjective line-search algorithms. Local surrogate models constructed from evaluation data of the true objective functions are employed to compute possible descent directions. In contrast to existing multiobjective trust region algorithms, these surrogates are not polynomial but carefully constructed radial basis function networks. This has the important advantage that the number of data points needed per iteration scales linearly with the decision space dimension. The local models qualify as fully linear and the corresponding general scalar framework is adapted for problems with multiple objectives. Full article

(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)

► Show Figures

Figure 1

Open AccessFeature PaperEditor’s ChoiceArticle

ROM-Based Inexact Subdivision Methods for PDE-Constrained Multiobjective Optimization

by

,

,

and

Math. Comput. Appl. 2021, 26(2), 32; https://doi.org/10.3390/mca26020032 - 15 Apr 2021

Cited by 1 | Viewed by 1703

Abstract

The goal in multiobjective optimization is to determine the so-called Pareto set. Our optimization problem is governed by a parameter-dependent semi-linear elliptic partial differential equation (PDE). To solve it, we use a gradient-based set-oriented numerical method. The numerical solution of the PDE by [...] Read more.

The goal in multiobjective optimization is to determine the so-called Pareto set. Our optimization problem is governed by a parameter-dependent semi-linear elliptic partial differential equation (PDE). To solve it, we use a gradient-based set-oriented numerical method. The numerical solution of the PDE by standard discretization methods usually leads to high computational effort. To overcome this difficulty, reduced-order modeling (ROM) is developed utilizing the reduced basis method. These model simplifications cause inexactness in the gradients. For that reason, an additional descent condition is proposed. Applying a modified subdivision algorithm, numerical experiments illustrate the efficiency of our solution approach. Full article

(This article belongs to the Special Issue Set Oriented Numerics 2022)

► Show Figures

Figure 1

Open AccessFeature PaperEditor’s ChoiceArticle

Data Augmentation and Feature Selection for Automatic Model Recommendation in Computational Physics

by

,

,

and

Math. Comput. Appl. 2021, 26(1), 17; https://doi.org/10.3390/mca26010017 - 16 Feb 2021

Cited by 3 | Viewed by 1987

Abstract

Classification algorithms have recently found applications in computational physics for the selection of numerical methods or models adapted to the environment and the state of the physical system. For such classification tasks, labeled training data come from numerical simulations and generally correspond to [...] Read more.

Classification algorithms have recently found applications in computational physics for the selection of numerical methods or models adapted to the environment and the state of the physical system. For such classification tasks, labeled training data come from numerical simulations and generally correspond to physical fields discretized on a mesh. Three challenging difficulties arise: the lack of training data, their high dimensionality, and the non-applicability of common data augmentation techniques to physics data. This article introduces two algorithms to address these issues: one for dimensionality reduction via feature selection, and one for data augmentation. These algorithms are combined with a wide variety of classifiers for their evaluation. When combined with a stacking ensemble made of six multilayer perceptrons and a ridge logistic regression, they enable reaching an accuracy of

90 %

on our classification problem for nonlinear structural mechanics. Full article

► Show Figures

Figure 1

Open AccessEditor’s ChoiceArticle

Mathematical Model and Numerical Simulation for Electric Field Induced Cancer Cell Migration

by

Antonino Amoddeo

Math. Comput. Appl. 2021, 26(1), 4; https://doi.org/10.3390/mca26010004 - 31 Dec 2020

Viewed by 1985

Abstract

A mathematical model describing the interaction of cancer cells with the urokinase plasminogen activation system is represented by a system of partial differential equations, in which cancer cell dynamics accounts for diffusion, chemotaxis, and haptotaxis contributions. The mutual relations between nerve fibers and [...] Read more.

A mathematical model describing the interaction of cancer cells with the urokinase plasminogen activation system is represented by a system of partial differential equations, in which cancer cell dynamics accounts for diffusion, chemotaxis, and haptotaxis contributions. The mutual relations between nerve fibers and tumors have been recently investigated, in particular, the role of nerves in the development of tumors, as well neurogenesis induced by cancer cells. Such mechanisms are mediated by neurotransmitters released by neurons as a consequence of electrical stimuli flowing along the nerves, and therefore electric fields can be present inside biological tissues, in particular, inside tumors. Considering cancer cells as negatively charged particles immersed in the correct biological environment and subjected to an external electric field, the effect of the latter on cancer cell dynamics is still unknown. Here, we implement a mathematical model that accounts for the interaction of cancer cells with the urokinase plasminogen activation system subjected to a uniform applied electric field, simulating the first stage of cancer cell dynamics in a three-dimensional axial symmetric domain. The obtained numerical results predict that cancer cells can be moved along a preferred direction by an applied electric field, suggesting new and interesting strategies in cancer therapy. Full article

(This article belongs to the Section Natural Sciences)

► Show Figures

Figure 1

Open AccessFeature PaperEditor’s ChoiceReview

Surrogate Modeling Approaches for Multiobjective Optimization: Methods, Taxonomy, and Results

by

Kalyanmoy Deb

,

Proteek Chandan Roy

and

Rayan Hussein

Math. Comput. Appl. 2021, 26(1), 5; https://doi.org/10.3390/mca26010005 - 31 Dec 2020

Cited by 16 | Viewed by 3197

Abstract

Most practical optimization problems are comprised of multiple conflicting objectives and constraints which involve time-consuming simulations. Construction of metamodels of objectives and constraints from a few high-fidelity solutions and a subsequent optimization of metamodels to find in-fill solutions in an iterative manner remain [...] Read more.

Most practical optimization problems are comprised of multiple conflicting objectives and constraints which involve time-consuming simulations. Construction of metamodels of objectives and constraints from a few high-fidelity solutions and a subsequent optimization of metamodels to find in-fill solutions in an iterative manner remain a common metamodeling based optimization strategy. The authors have previously proposed a taxonomy of 10 different metamodeling frameworks for multiobjective optimization problems, each of which constructs metamodels of objectives and constraints independently or in an aggregated manner. Of the 10 frameworks, five follow a generative approach in which a single Pareto-optimal solution is found at a time and other five frameworks were proposed to find multiple Pareto-optimal solutions simultaneously. Of the 10 frameworks, two frameworks (M3-2 and M4-2) are detailed here for the first time involving multimodal optimization methods. In this paper, we also propose an adaptive switching based metamodeling (ASM) approach by switching among all 10 frameworks in successive epochs using a statistical comparison of metamodeling accuracy of all 10 frameworks. On 18 problems from three to five objectives, the ASM approach performs better than the individual frameworks alone. Finally, the ASM approach is compared with three other recently proposed multiobjective metamodeling methods and superior performance of the ASM approach is observed. With growing interest in metamodeling approaches for multiobjective optimization, this paper evaluates existing strategies and proposes a viable adaptive strategy by portraying importance of using an ensemble of metamodeling frameworks for a more reliable multiobjective optimization for a limited budget of solution evaluations. Full article

(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)

► Show Figures

Figure 1

Open AccessFeature PaperEditor’s ChoiceArticle

A Localized Collocation Solver Based on T-Complete Functions for Anti-Plane Transverse Elastic Wave Propagation Analysis in 2D Phononic Crystals

by

,

,

and

Math. Comput. Appl. 2021, 26(1), 2; https://doi.org/10.3390/mca26010002 - 30 Dec 2020

Cited by 2 | Viewed by 1719

Abstract

In this paper, we introduce a novel localized collocation solver for two-dimensional (2D) phononic crystal analysis. In the proposed collocation solver, the displacement at each node is expressed as a linear combination of T-complete functions in each stencil support and the sparse linear [...] Read more.

In this paper, we introduce a novel localized collocation solver for two-dimensional (2D) phononic crystal analysis. In the proposed collocation solver, the displacement at each node is expressed as a linear combination of T-complete functions in each stencil support and the sparse linear system is obtained by satisfying the considered governing equation at interior nodes and boundary conditions at boundary nodes. As compared with finite element method (FEM) results and the analytical solutions, the efficiency and accuracy of the proposed localized collocation solver are verified under a benchmark example. Then, the proposed method is applied to 2D phononic crystals with various lattice forms and scatterer shapes, where the related band structures, transmission spectra, and displacement amplitude distributions are calculated as compared with the FEM. Full article

(This article belongs to the Section Engineering)

► Show Figures

Figure 1

Open AccessFeature PaperEditor’s ChoiceArticle

A Transformational Modified Markov Process for Chord-Based Algorithmic Composition

by

,

,

and

Math. Comput. Appl. 2020, 25(3), 43; https://doi.org/10.3390/mca25030043 - 10 Jul 2020

Cited by 1 | Viewed by 2072

Abstract

The goal of this research is to maximize chord-based composition possibilities given a relatively small amount of information. A transformational approach, based in group theory, was chosen, focusing on chord intervals as the components of a modified Markov process. The Markov process was [...] Read more.

The goal of this research is to maximize chord-based composition possibilities given a relatively small amount of information. A transformational approach, based in group theory, was chosen, focusing on chord intervals as the components of a modified Markov process. The Markov process was modified to balance between average harmony, representing familiarity, and entropy, representing novelty. Uniform triadic transformations are suggested as a further extension of the transformational approach, improving the quality of tonality. The composition algorithms are demonstrated given a short chord progression and also given a larger database of albums by the Beatles. Results demonstrate capabilities and limitations of the algorithms. Full article

(This article belongs to the Section Natural Sciences)

► Show Figures

Figure 1

Open AccessFeature PaperEditor’s ChoiceArticle

Parallel Matrix-Free Higher-Order Finite Element Solvers for Phase-Field Fracture Problems

by

Daniel Jodlbauer

,

Ulrich Langer

and

Thomas Wick

Math. Comput. Appl. 2020, 25(3), 40; https://doi.org/10.3390/mca25030040 - 07 Jul 2020

Cited by 13 | Viewed by 2405

Abstract

Phase-field fracture models lead to variational problems that can be written as a coupled variational equality and inequality system. Numerically, such problems can be treated with Galerkin finite elements and primal-dual active set methods. Specifically, low-order and high-order finite elements may be employed, [...] Read more.

Phase-field fracture models lead to variational problems that can be written as a coupled variational equality and inequality system. Numerically, such problems can be treated with Galerkin finite elements and primal-dual active set methods. Specifically, low-order and high-order finite elements may be employed, where, for the latter, only few studies exist to date. The most time-consuming part in the discrete version of the primal-dual active set (semi-smooth Newton) algorithm consists in the solutions of changing linear systems arising at each semi-smooth Newton step. We propose a new parallel matrix-free monolithic multigrid preconditioner for these systems. We provide two numerical tests, and discuss the performance of the parallel solver proposed in the paper. Furthermore, we compare our new preconditioner with a block-AMG preconditioner available in the literature. Full article

(This article belongs to the Special Issue High-Performance Computing 2020)

► Show Figures

Figure 1

Open AccessFeature PaperEditor’s ChoiceArticle

Numerical Approach to a Nonlocal Advection-Reaction-Diffusion Model of Cartilage Pattern Formation

by

Tilmann Glimm

and

Jianying Zhang

Math. Comput. Appl. 2020, 25(2), 36; https://doi.org/10.3390/mca25020036 - 19 Jun 2020

Cited by 1 | Viewed by 2293

Abstract

We propose a numerical approach that combines a radial basis function (RBF) meshless approximation with a finite difference discretization to solve a nonlinear system of integro-differential equations. The equations are of advection-reaction-diffusion type modeling the formation of pre-cartilage condensations in embryonic chicken limbs. [...] Read more.

We propose a numerical approach that combines a radial basis function (RBF) meshless approximation with a finite difference discretization to solve a nonlinear system of integro-differential equations. The equations are of advection-reaction-diffusion type modeling the formation of pre-cartilage condensations in embryonic chicken limbs. The computational domain is four dimensional in the sense that the cell density depends continuously on two spatial variables as well as two structure variables, namely membrane-bound counterreceptor densities. The biologically proper Dirichlet boundary conditions imposed in the semi-infinite structure variable region is in favor of a meshless method with Gaussian basis functions. Coupled with WENO5 finite difference spatial discretization and the method of integrating factors, the time integration via method of lines achieves optimal complexity. In addition, the proposed scheme can be extended to similar models with more general boundary conditions. Numerical results are provided to showcase the validity of the scheme. Full article

(This article belongs to the Section Natural Sciences)

► Show Figures

Figure 1

Open AccessEditor’s ChoiceArticle

Mechanism of Coup and Contrecoup Injuries Induced by a Knock-Out Punch

by

,

,

and

Math. Comput. Appl. 2020, 25(2), 22; https://doi.org/10.3390/mca25020022 - 15 Apr 2020

Cited by 15 | Viewed by 7833

Abstract

Primary Objective: The interaction of cerebrospinal fluid with the brain parenchyma in an impact scenario is studied. Research Design: A computational fluid-structure interaction model is used to simulate the interaction of cerebrospinal fluid with a comprehensive brain model. Methods and Procedures: The method [...] Read more.

Primary Objective: The interaction of cerebrospinal fluid with the brain parenchyma in an impact scenario is studied. Research Design: A computational fluid-structure interaction model is used to simulate the interaction of cerebrospinal fluid with a comprehensive brain model. Methods and Procedures: The method of smoothed particle hydrodynamics is used to simulate the fluid flow, induced by the impact, simultaneously with finite element analysis to solve the large deformations in the brain model. Main Outcomes and Results: Mechanism of injury resulting in concussion is demonstrated. The locations with the highest stress values on the brain parenchyma are shown. Conclusions: Our simulations found that the damage to the brain resulting from the contrecoup injury is more severe than that resulting from the coup injury. Additionally, we show that the contrecoup injury does not always appear on the side opposite from where impact occurs. Full article

(This article belongs to the Special Issue Numerical Modelling and Simulation Applied to Head Trauma)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Editor’s Choice Articles

Further Information

Guidelines

MDPI Initiatives

Follow MDPI