Dear CSIAM-AM Reader:

On behalf of the editorial board and the China Society for Industrial and Applied Mathematics (CSIAM), it is our pleasure to welcome you to the first issue of the CSIAM Transactions on Applied Mathematics (CSIAM-AM). The goal of CSIAM-AM is to provide you, the valuable reader, with original research articles or reviews of high quality on a topic in applied mathematics that highlight the key role of mathematics in advancing modern science and technology.
The journal comes at the most exciting and critical time for applied mathematics as its relevance has never been so clear. The rapid developments of big data and artificially intelligence have greatly expanded the application scope of applied mathematics, and presented unprecedented opportunities and challenges to applied mathematics. CSIAM-AM grows out of the strong demand of a quality platform for communicating significant research findings in applied mathematics that address interesting and challenging scientific problems. It will also be the flagship journal of CSIAM, a fast-growing society with over 10,000 individual non-student members.
The objective of CSIAM-AM is to publish only quality manuscripts and to maintain the quality as we move on. As an important step to achieve the objective, we are fortunate to have a distinguished group of mathematicians as our founding members of the editorial board. Since the reception of the first submission in February 2019, the initial response of the journal has been overwhelming. We hope the journal will continue to reach to the right audience and expand our reader group.
Finally, we would like to take the opportunity to acknowledge the help of many individuals over the course of launching the journal. In particular, the inauguration issue would not be possible without the generosity of the board members and reviewers who handled the first wave of submissions in a timely fashion, the staffs of the Global Science Press and CSIAM for their professional assistance, and members of the CSIAM Committee on Publications for their efforts.
Please enjoy the issue and we look forward to your future submissions.

Sincerely,
Pingwen Zhang, Editor-in-Chief and President of CSIAM
Gang Bao, Managing Editor
CSIAM Transactions on Applied Mathematics

In this paper, we will build a roadmap for the growing literature of high order quadrature-based entropy stable discontinuous Galerkin (DG) methods, trying to elucidate the motivations and emphasize the contributions. Compared to the classic DG method which is only provably stable for the square entropy, these DG methods can be tailored to satisfy an arbitrary given entropy inequality, and do not require exact integration. The methodology is within the summation-by-parts (SBP) paradigm, such that the discrete operators collocated at the quadrature points should satisfy the SBP property. The construction is relatively easy for quadrature rules with collocated surface nodes. We use the flux differencing technique to ensure entropy balance within elements, and the simultaneous approximation terms (SATs) to produce entropy dissipation on element interfaces. The further extension to general quadrature rules is achieved through careful modifications of SATs.

The article is devoted to a review of the following new elements of the modern theory of solving inverse problems: (a) general theory of Tikhonov's regularization with practical examples is considered; (b) an overview of a-priori and a-posteriori error estimates for solutions of ill-posed problems is presented as well as a general scheme of a-posteriori error estimation; (c) a-posteriori error estimates for linear inverse problems and its finite-dimensional approximation are considered in detail together with practical a-posteriori error estimate algorithms; (d) optimality in order for the error estimator and extra-optimal regularizing algorithms are also discussed. In addition, the article contains applications of these theoretical results to solving two practical geophysical problems. First, for inverse problems of computer microtomography in microstructure analysis of shales, numerical experiments demonstrate that the use of functions with bounded $VH$-variation for a piecewise uniform regularization has a theoretical and practical advantage over methods using $BV$-variation. For these problems, a new algorithm of a-posteriori error estimation makes it possible to calculate the error of the solution in the form of a number. Second, in geophysical prospecting, Tikhonov's regularization is very effective in magnetic parameters inversion method with full tensor gradient data. In particular, the regularization algorithms allow to compare different models in this method and choose the best one, MGT-model.

We consider a two-species competition model in which both populations are identical except their movement strategies: One species moves upward along the fitness gradient, while the other does not diffuse. While both species can coexist in homogeneous environment, we show that the species with directed movement has some advantage over the non-diffusing species in certain measurement. In contrast, if one species moves by random dispersal while the other does not diffuse, then the non-diffusing population could have advantage. Understanding the full dynamics of these ODE-PDE hybrid systems poses challenging mathematical questions.

In this paper, we analyze an interior penalty discontinuous Galerkin method for solving the coupled Cahn–Hilliard and Navier–Stokes equations. We prove unconditional unique solvability of the discrete system, and we derive stability bounds without any restrictions on the chemical energy density function. The numerical solutions satisfy a discrete energy dissipation law and mass conservation laws. Convergence of the method is obtained by obtaining optimal a priori error estimates.

An implicit evaluation method of vector 2-norms is presented for function evaluations arising from sphere constrained quadratic optimizations. The efficiency of the method in terms of computational costs mainly comes from the well-known shifted conjugate gradient method, and the robustness of the method comes from the fact that it never suffers from cancellations when the coefficient matrix is symmetric positive definite. Numerical experiments indicates that the method is promising for reducing computational costs of Ye's hybrid method for solving sphere constrained quadratic optimizations.

We study coarse-grained models of some linear static lattice models with interactions up to second nearest neighbors. It will be demonstrated how nonlocal interactions, as described by a nonlocal kernel function, arise from a coarse-graining procedure. Some important properties of the nonlocal kernels will be established such as its decay rate and positivity. We also study the scaling behavior of the kernel functions as the level of coarse-graining changes. In addition, we suggest closure approximations of the nonlocal interactions that can be expressed in local PDE forms by introducing auxiliary variables.

Emerging infectious diseases are existential threats to human health and global stability. The recent outbreaks of the novel coronavirus COVID-19 have rapidly formed a global pandemic, causing hundreds of thousands of infections and huge economic loss. The WHO declares that more precise measures to track, detect and isolate infected people are among the most effective means to quickly contain the outbreak. Based on trajectory provided by the big data and the mean field theory, we establish an aggregated risk mean field that contains information of all risk-spreading particles by proposing a spatio-temporal model named HiRES risk map. It has dynamic fine spatial resolution and high computation efficiency enabling fast update. We then propose an objective individual epidemic risk scoring model named HiRES-p based on HiRES risk maps, and use it to develop statistical inference and machine learning methods for detecting suspected epidemic-infected individuals. We conduct numerical experiments by applying the proposed methods to study the early outbreak of COVID-19 in China. Results show that the HiRES risk map has strong ability in capturing global trend and local variability of the epidemic risk, thus can be applied to monitor epidemic risk at country, province, city and community levels, as well as at specific high-risk locations such as hospital and station. HiRES-p score seems to be an effective measurement of personal epidemic risk. The accuracy of both detecting methods are above 90% when the population infection rate is under 20%, which indicates great application potential in epidemic risk prevention and control practice.