Featured articles
  • A Unified Approach to Solving Some Inverse Problems for Evolution Equations by Using Observability Inequalities

    by Kaïs Ammari, Mourad Choulli & Faouzi Triki, CSIAM-AM 1 (2020), pp. 207-239.

    We survey some of our recent results on inverse problems for evolution equations. The goal is to provide a unified approach to solve various types of evolution equations. The inverse problems we consider consist in determining unknown coefficients from boundary measurements by varying initial conditions. Based on observability inequalities and a special choice of initial conditions, we provide uniqueness and stability estimates for the recovery of volume and boundary lower order coefficients in wave and heat equations. Some of the results presented here are slightly improved from their original versions.

  • Detecting Suspected Epidemic Cases Using Trajectory Big Data

    by Chuansai Zhou, Wen Yuan, Jun Wang, Haiyong Xu, Yong Jiang, Xinmin Wang, Qiuzi Han Wen & Pingwen Zhang, CSIAM-AM 1 (2020), pp. 186-206.

    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.

  • Review of Entropy Stable Discontinuous Galerkin Methods for Systems of Conservation Laws on Unstructured Simplex Meshes

    by Tianheng Chen & Chi-Wang Shu, CSIAM-AM 1 (2020), pp. 1-52.

    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.