A new phase space method for recovering index of refraction from travel times

We develop a new phase space method for reconstructing the index of refraction of a medium from travel time measurements. The method is based on the so-called Stefanov–Uhlmann identity which links two Riemannian metrics with their travel time information. We design a numerical algorithm to solve the resulting inverse problem. The new algorithm is a hybrid approach that combines both Lagrangian and Eulerian formulations. In particular the Lagrangian formulation in phase space can take into account multiple arrival times naturally, while the Eulerian formulation for the index of refraction allows us to compute the solution in physical space. Numerical examples including isotropic metrics and the Marmousi synthetic model are shown to validate the new method

An Efficient Operator-Splitting Method for the Eigenvalue Problem of the Monge-Amp\`{e}re Equation

We develop an efficient operator-splitting method for the eigenvalue problem of the Monge-Amp\`{e}re operator in the Aleksandrov sense. The backbone of our method relies on a convergent Rayleigh inverse iterative formulation proposed by Abedin and Kitagawa (Inverse iteration for the {M}onge-{A}mp{\`e}re eigenvalue problem, {\it Proceedings of the American Mathematical Society}, 148 (2020), no. 11, 4975-4886). Modifying the theoretical formulation, we develop an efficient algorithm for computing the eigenvalue and eigenfunction of the Monge-Amp\`{e}re operator by solving a constrained Monge-Amp\`{e}re equation during each iteration. Our method consists of four essential steps: (i) Formulate the Monge-Amp\`{e}re eigenvalue problem as an optimization problem with a constraint; (ii) Adopt an indicator function to treat the constraint; (iii) Introduce an auxiliary variable to decouple the original constrained optimization problem into simpler optimization subproblems and associate the resulting new optimization problem with an initial value problem; and (iv) Discretize the resulting initial-value problem by an operator-splitting method in time and a mixed finite element method in space. The performance of our method is demonstrated by several experiments. Compared to existing methods, the new method is more efficient in terms of computational cost and has a comparable rate of convergence in terms of accuracy

On the Numerical Solution of Nonlinear Eigenvalue Problems for the Monge-Amp\`{e}re Operator

In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Amp\`{e}re operator $v\rightarrow \det \mathbf{D}^2 v$. The methodology we employ relies on the following ingredients: (i) A divergence formulation of the eigenvalue problems under consideration. (ii) The time discretization by operator-splitting of an initial value problem (a kind of gradient flow) associated with each eigenvalue problem. (iii) A finite element approximation relying on spaces of continuous piecewise affine functions. To validate the above methodology, we applied it to the solution of problems with known exact solutions: The results we obtained suggest convergence to the exact solution when the space discretization step $h\rightarrow 0$. We considered also test problems with no known exact solutions

Learning Rays via Deep Neural Network in a Ray-based IPDG Method for High-Frequency Helmholtz Equations in Inhomogeneous Media

We develop a deep learning approach to extract ray directions at discrete locations by analyzing highly oscillatory wave fields. A deep neural network is trained on a set of local plane-wave fields to predict ray directions at discrete locations. The resulting deep neural network is then applied to a reduced-frequency Helmholtz solution to extract the directions, which are further incorporated into a ray-based interior-penalty discontinuous Galerkin (IPDG) method to solve the Helmholtz equations at higher frequencies. In this way, we observe no apparent pollution effects in the resulting Helmholtz solutions in inhomogeneous media. Our 2D and 3D numerical results show that the proposed scheme is very efficient and yields highly accurate solutions.Comment: 30 page

A phase-space formulation for elastic-wave traveltime tomography

We adopt a recent work in Chung, Qian, Uhlmann and Zhao (Inverse Problems, 23(2007) 309-329) to develop a phase space method for reconstructing pressure wave speed and shear wave speed of an elastic medium from travel time measurements. The method is based on the so-called Stefanov-Uhlmann identity which links two Riemannian metrics with their travel time information. We design a numerical algorithm to solve the resulting inverse problem. The algorithm is a hybrid approach that combines both Lagrangian and Eulerian formulations. In particular the Lagrangian formulation in phase space can take into account multiple arrival times naturally, while the Eulerian formulation for wave speeds allows us to compute the solution in physical space. Numerical examples are shown to validate the method

Phospholemman: a novel cardiac stress protein.

Phospholemman (PLM), a member of the FXYD family of regulators of ion transport, is a major sarcolemmal substrate for protein kinases A and C in cardiac and skeletal muscle. In the heart, PLM co-localizes and co-immunoprecipitates with Na(+)-K(+)-ATPase, Na(+)/Ca(2+) exchanger, and L-type Ca(2+) channel. Functionally, when phosphorylated at serine(68), PLM stimulates Na(+)-K(+)-ATPase but inhibits Na(+)/Ca(2+) exchanger in cardiac myocytes. In heterologous expression systems, PLM modulates the gating of cardiac L-type Ca(2+) channel. Therefore, PLM occupies a key modulatory role in intracellular Na(+) and Ca(2+) homeostasis and is intimately involved in regulation of excitation-contraction (EC) coupling. Genetic ablation of PLM results in a slight increase in baseline cardiac contractility and prolongation of action potential duration. When hearts are subjected to catecholamine stress, PLM minimizes the risks of arrhythmogenesis by reducing Na(+) overload and simultaneously preserves inotropy by inhibiting Na(+)/Ca(2+) exchanger. In heart failure, both expression and phosphorylation state of PLM are altered and may partly account for abnormalities in EC coupling. The unique role of PLM in regulation of Na(+)-K(+)-ATPase, Na(+)/Ca(2+) exchanger, and potentially L-type Ca(2+) channel in the heart, together with the changes in its expression and phosphorylation in heart failure, make PLM a rational and novel target for development of drugs in our armamentarium against heart failure. Clin Trans Sci 2010; Volume 3: 189-196

Epidemiology and clinical course of COVID-19 in Shanghai, China.

Background: Novel coronavirus pneumonia (COVID-19) is prevalent around the world. We aimed to describe epidemiological features and clinical course in Shanghai. Methods: We retrospectively analysed 325 cases admitted at Shanghai Public Health Clinical Center, between January 20 and February 29, 2020. Results: 47.4% (154/325) had visited Wuhan within 2 weeks of illness onset. 57.2% occurred in 67 clusters; 40% were situated within 53 family clusters. 83.7% developed fever during the disease course. Median times from onset to first medical care, hospitalization and negative detection of nucleic acid by nasopharyngeal swab were 1, 4 and 8 days. Patients with mild disease using glucocorticoid tended to have longer viral shedding in blood and feces. At admission, 69.8% presented with lymphopenia and 38.8% had elevated D-dimers. Pneumonia was identified in 97.5% (314/322) of cases by chest CT scan. Severe-critical patients were 8% with a median time from onset to critical disease of 10.5 days. Half required oxygen therapy and 7.1% high-flow nasal oxygen. The case fatality rate was 0.92% with median time from onset to death of 16 days. Conclusion: COVID-19 cases in Shanghai were imported. Rapid identification, and effective control measures helped to contain the outbreak and prevent community transmission