Solution Map Research Articles

Raman Mapping of Biological Systems Interacting with a Disordered Nanostructured Surface: A Simple and Powerful Approach to the Label-Free Analysis of Single DNA Bases.

This article demonstrates the possibility to use a novel powerful approach based on Raman mapping of analyte solutions drop casted on a disordered array of Ag covered silicon nanowires (Ag/SiNWs), to identify the characteristic spectral signal of the four DNA bases, adenine (A), thymine (T), cytosine (C), and guanine (G), at concentration as low as 10 ng/µL, and to study their specific way of interacting with the nanostructured substrate. The results show a distinctive and amplified interaction of guanine, the base that is most susceptible to oxidation, with the nanostructured surface. Our findings explain the recently revealed diverse behaviour of cancer and normal DNA deposited on the same Ag/SiNWs, which is ascribed to mechanical deformation and base lesions present on the oxidised DNA molecule backbone and causes detectable variation in the Raman signal, usable for diagnostic purposes. The notable bio-analytical capability of the presented platform, and its sensitivity to the molecule mechanical conformation at the single-base level, thus provides a new reliable, rapid, label-free DNA diagnostic methodology alternative to more sophisticated and expensive sequencing ones.

A solution in search of problems: a cognitive tool for solution mapping to promote divergent thinking

Engineering design processes are often defined as beginning with a problem and diverging to generate possible solutions; however, design processes can start with a newly developed technological solution, followed by a divergent search for potential problem applications it can solve, termed ‘solution mapping'. Building on previous research where engineering practitioners described their successful strategies for solution mapping, we created a tool to support solution mapping and tested its impact with engineering students. In a single session, graduate and advanced undergraduate engineering students were presented with a novel technology and worked to identify potential problem applications for it. Comparing students using the Solution Mapping Design Tool to two control groups, more diverse problem applications were produced when using the tool. Considering diverse options is an important feature of design processes shown to promote creativity and innovation. With this successful proof of concept, future work on solution mapping will identify how to support engineers seeking problem applications by making use of new technologies.

A novel approach to Hölder continuity of a class of parametric variational–hemivariational inequalities

In this paper, we first recall a class of parametric variational–hemivariational inequalities (PVHIs) introduced in Jiang et al. (2020). Then, based on the properties of the Clarke generalized gradient, we establish the Hölder continuity of the solution mapping for PVHIs in terms of regularized gap functions under some assumptions imposed on the data of PVHIs. Finally, an example is given to illustrate our main results.

Well-posedness and stability in set optimization with applications

This paper is about the well-posedness and stability in set optimization with applications to vector-valued games involving uncertainty. Some characterizations for several types of well-posedness with their relations in set optimization are given under mild conditions. Some sufficient and necessary conditions for these types of well-posedness in set optimization are also given by using the local C-Lipschitz continuity. Moreover, the upper semi-continuity, lower semi-continuity and compactness of minimal solution mappings are studied for parametric set optimization involving the cone Lipschitz continuous set-valued mapping. Finally, some obtained results are applied to derive the well-posedness for vector-valued games with uncertainty.

Efficient incrementing heuristics for generalized p-location problems

This paper studies effective solving techniques for large p-location problems, in which generalized disutility is applied to incorporate stochastic behavior of real service system. The concept of generalized disutility follows the idea that the nearest located service center may be temporarily unavailable at the moment of currently arisen demand and the demand is serviced from a more distant service center with given probability. The more complex approach to the real p-location asks for enormous computational time of exact solving tools and thus evokes the question whether the faster heuristics can compete with the exact approaches in accuracy of results obtained. To answer the question, two heuristics were proposed for the studied problem. The first of them is a common swap local search with the best-admissible strategy and the second is based on the path-relinking method. The both heuristics make use of so called uniformly deployed sets used for preliminary mapping of the feasible solutions to accelerate the performance of the heuristics. As uniformly deployed set construction can be done in two modes, the paper is also focused on finding the most effective combination of the mode and heuristic. A real case study was performed with seven road networks with number of nodes varying from 250 to 660. In addition, ten different families of uniformly deployed sets were generated for each network and mode. The results of the case study confirm that the suggested approaches are able to produce the solutions of the same quality as the exact approaches. Furthermore, the study indicates drawbacks, which can be met in connection with usage of specific combinations of heuristic and the mode of uniformly deployed set construction.

On the Quantitative Solution Stability of Parameterized Set-Valued Inclusions

The subject of the present paper are stability properties of the solution set to set-valued inclusions. The latter are problems emerging in robust optimization and mathematical economics, which can not be cast in traditional generalized equations. The analysis here reported focuses on several quantitative forms of semicontinuity for set-valued mappings, widely investigated in variational analysis, which include, among others, calmness. Sufficient conditions for the occurrence of these properties in the case of the solution mapping to a parameterized set-valued inclusion are established. Consequences on the calmness of the optimal value function, in the context of parametric optimization, are explored. Some specific tools for the analysis of the sufficient conditions, in the case of set-valued inclusion with concave multifunction term, are provided in a Banach space setting.

Decomposition Algorithms for Some Deterministic and Two-Stage Stochastic Single-Leader Multi-Follower Games

We consider a certain class of hierarchical decision problems that can be viewed as single-leader multi-follower games, and be represented by a virtual market coordinator trying to set a price system for traded goods, according to some criterion that balances supply and demand. The objective function of the market coordinator involves the decisions of many agents, which are taken independently by solving convex optimization problems that depend on the price configuration and on realizations of future states of the economy. One traditional way of solving this problem is via a mixed complementarity formulation. However, this approach can become impractical when the numbers of agents and/or scenarios become large. This work concerns agent-wise and scenario-wise decomposition algorithms to solve the equilibrium problems in question, assuming that the solutions of the agents’ problems are unique, which is natural in many applications (when solutions are not unique, the approximating problems are still well-defined, but the convergence properties of the algorithm are not established). The algorithm is based on a previous work of the authors, where a suitable regularization of solution mappings of fully parameterized convex problems is developed. Here, we show one specific strategy to manage the regularization parameter, extend some theoretical results to the current setting, and prove that the smooth approximations of the market coordinator’s problem converge epigraphically to the original problem. Numerical experiments and some comparisons with the complementarity solver PATH are shown for the two-stage stochastic Walrasian equilibrium problem.

Particle finite element method implementation for large deformation analysis using Abaqus

In this study, a simple PFEM approach for analyzing large deformation problems in geotechnical practice is implemented in the commercial FEM package Abaqus. The main feature of the proposed Abaqus-PFEM approach lies in its capability to absorb the advantages of the functionality available in Abaqus and integrate them into PFEM with a single Python script, which leads to a considerable reduction in coding work. By utilizing the built-in functions in Abaqus to fulfil the standard incremental FEM analysis, as well as the powerful mesh-to-mesh solution mapping technique, the proposed Abaqus-PFEM approach allows for the large deformation analysis automatically running with a single Python script and requires no intervention from the user. The accuracy of the proposed Abaqus-PFEM approach is firstly validated through a simple elastic cantilever beam bending problem. Then, the performance and robustness of the proposed Abaqus-PFEM approach are further examined by three illustrative numerical examples: penetration of rigid footing, penetration of T-bar and pipeline–soil interaction problem. The numerical results demonstrate that the proposed Abaqus-PFEM approach as a powerful and easily extensible numerical tool is capable of handling large deformation and soil–structure interaction problems in geotechnical engineering, and consequently, it offers an alternative way to tackle such problems.

A note on the Cauchy problem for the two-component Novikov system

Considered herein is the initial value problem for the two-component Novikov system with peakons. Based on the local well-posedness results for this problem, it is shown that the solution map $$z_{0}\mapsto z(t)$$ of this problem in the periodic case is not uniformly continuous in Besov spaces $$B^{s}_{p,r}({\mathbb {T}})\times B^{s}_{p,r}({\mathbb {T}}) $$ with $$s>\max \{5/2,2+1/p\}, 1\le p,r\le \infty $$ and $$B^{5/2}_{2,1}({\mathbb {T}})\times B^{5/2}_{2,1}({\mathbb {T}})$$ through the method of approximate solutions. Furthermore, it is in the non-periodic case that the non-uniform continuity of this solution map in Besov spaces $$B^{s}_{p,r}({\mathbb {R}})\times B^{s}_{p,r}({\mathbb {R}})$$ with $$s>\max \{5/2,2+1/p\}, 1\le p,r\le \infty $$ is discussed by constructing new subtle initial data. Finally, the Holder continuity of the solution map in Besov spaces is proved.

Multilevel Fine-Tuning: Closing Generalization Gaps in Approximation of Solution Maps under a Limited Budget for Training Data

In scientific machine learning, regression networks have been recently applied to approximate solution maps (e.g., the potential-ground state map of the Schrodinger equation). In this paper, we aim...

Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis

This paper concerns a distributed optimal control problem for a tumor growth model of Cahn–Hilliard type including chemotaxis with possibly singular potentials, where the control and state variables are nonlinearly coupled. First, we discuss the weak well-posedness of the system under very general assumptions for the potentials, which may be singular and nonsmooth. Then, we establish the strong well-posedness of the system in a reduced setting, which however admits the logarithmic potential: this analysis will lay the foundation for the study of the corresponding optimal control problem. Concerning the optimization problem, we address the existence of minimizers and establish both first-order necessary and second-order sufficient conditions for optimality. The mathematically challenging second-order analysis is completely performed here, after showing that the solution mapping is twice continuously differentiable between suitable Banach spaces via the implicit function theorem. Then, we completely identify the second-order Fréchet derivative of the control-to-state operator and carry out a thorough and detailed investigation about the related properties.

Damped Klein-Gordon equation with variable diffusion coefficient

<p style='text-indent:20px;'>We consider a damped Klein-Gordon equation with a variable diffusion coefficient. This problem is challenging because of the equation's unbounded nonlinearity. First, we study the nonlinearity's continuity properties. Then the existence and the uniqueness of the solutions is established. The main result is the continuity of the solution map on the set of admissible parameters. Its application to the parameter identification problem is considered.</p>

Calmness of the solution mapping of parametric variational relation problems

We consider a class of variational relation problems, depending on two parameters from metric spaces. The issue under investigation is the behaviour of the solution in a neighborhood of a fixed pair of parameters, more precisely, the H?lder calmness of the solution mapping. After establishing some sufficient condition for calmness in a general framework, we particularize the result for a variational inclusion and for an equilibrium problem.

On the existence and stability of solutions to stochastic equilibrium problems

In this paper we consider stochastic equilibrium problems involving parameter of probability measures. Employing the fixed point theorem of Knaster, Kuratowski, Mazurkiewicz, and Fan (KKM-Fan), conditions for the existence of solutions to such problems are established. We then propose new metric concepts on the underlying stochastic spaces and study some properties corresponding to these metrics. Afterwards, we study sufficient conditions for the solution mappings of such problems, that are closed, upper (lower) semicontinuous and continuous with respect to the mentioned metrics. Finally, the special cases of stochastic optimization problems are taken into account as the applications.

Stability of Vector Optimization Problems Under Approximate Equilibrium Constraints via Free-Disposal Sets

The stability of vector optimization problems under approximate equilibrium constraints (AOPVF) via free-disposal sets was discussed. Firstly, the Berge-semicontinuity of the constraint set mapping and the closedness, the convexity and the compactness of the constraint set were obtained with the weaker convexity assumption. Moreover, under the assumption of Gamma-convergence for the objective functional sequences, the lower Painlevé-Kuratowski convergence of the weak efficient solution set and the Berge-semicontinuity of weak efficient solution mappings for AOPVF were obtained respectively. Some examples illustrate that the results are new and meaningful.

The Random Feature Model for Input-Output Maps between Banach Spaces

Well known to the machine learning community, the random feature model is a parametric approximation to kernel interpolation or regression methods. It is typically used to approximate functions mapping a finite-dimensional input space to the real line. In this paper, we instead propose a methodology for use of the random feature model as a data-driven surrogate for operators that map an input Banach space to an output Banach space. Although the methodology is quite general, we consider operators defined by partial differential equations (PDEs); here, the inputs and outputs are themselves functions, with the input parameters being functions required to specify the problem, such as initial data or coefficients, and the outputs being solutions of the problem. Upon discretization, the model inherits several desirable attributes from this infinite-dimensional viewpoint, including mesh-invariant approximation error with respect to the true PDE solution map and the capability to be trained at one mesh resolution and then deployed at different mesh resolutions. We view the random feature model as a non-intrusive data-driven emulator, provide a mathematical framework for its interpretation, and demonstrate its ability to efficiently and accurately approximate the nonlinear parameter-to-solution maps of two prototypical PDEs arising in physical science and engineering applications: viscous Burgers' equation and a variable coefficient elliptic equation.

Multiobjective optimal control of a non-smooth semilinear elliptic partial differential equation

This paper is concerned with the derivation and analysis of first-order necessary optimality conditions for a class of multiobjective optimal control problems governed by an elliptic non-smooth semilinear partial differential equation. Using an adjoint calculus for the inverse of the non-linear and non-differentiable directional derivative of the solution map of the considered PDE, we extend the concept of strong stationarity to the multiobjective setting and demonstrate that the properties of weak and proper Pareto stationarity can also be characterized by suitable multiplier systems that involve both primal and dual quantities. The established optimality conditions imply in particular that Pareto stationary points possess additional regularity properties and that mollification approaches are – in a certain sense – exact for the studied problem class. We further show that the obtained results are closely related to rather peculiar hidden regularization effects that only reveal themselves when the control is eliminated and the problem is reduced to the state. This observation is also new for the case of a single objective function. The paper concludes with numerical experiments that illustrate that the derived optimality systems are amenable to numerical solution procedures.

On Sobolev rough paths

We introduce the space of rough paths with Sobolev regularity and the corresponding concept of controlled Sobolev paths. Based on these notions, we study rough path integration and rough differential equations. As main result, we prove that the solution map associated to differential equations driven by rough paths is a locally Lipschitz continuous map on the Sobolev rough path space for any arbitrary low regularity α and integrability p provided α>1/p.

Stability of the set of solutions for generalized vector equilibrium problems with cone constraints

In this paper, the Hausdorff quasi-upper and Hausdorff quasi-lower semicontinuity for set-valued mappings are introduced to derive the upper semicontinuity and lower semicontinuity of cone constraint mapping in normed vector spaces. We discuss existence of solutions for generalized vector equilibrium problems with cone constraints. Moreover, we establish the lower semicontinuity of (weakly) efficient approximate solution mappings, upper semicontinuity of weakly efficient approximate solution mappings, and Hausdorff upper semicontinuity of efficient approximate solution mappings to parametric generalized vector equilibrium problems with cone constraints under some suitable conditions. Some examples are also given to illustrate our main results.

Confidence regions of stochastic variational inequalities: error bound approach

ABSTRACT In this paper, we aim to build confidence regions of the true solution to the stochastic variational inequalities problem (SVIP) when the sample average approximation (SAA) scheme is implemented. A new approach based on error bound conditions admitted by the SVIP is proposed. This so-called error bound approach provides an upper bound of the distance between SAA solutions and the true solution set through the distance between the SAA function and the true counterpart at the SAA solutions. Certain statistical tools such as central limit theorem and Owen's empirical likelihood theorem are then employed to construct the asymptotic confidence regions of the solutions to SVIP. In particular, if the SVIP admits a global error bound condition, the non-asymptotic (uniform) confidence regions of the solutions are also approachable. Different from the conventional normal map approach, our error bound approach does not require any information regarding the derivative of the solution mapping with respect to perturbations of involved functions in SVIP. For constructing component-wise confidence regions, the validity of the error bound approach is guaranteed for those cases where the functions own separable structures.