Behavior of different numerical schemes for population genetic drift problems

In this paper, we focus on numerical methods for the genetic drift problems, which is governed by a degenerated convection-dominated parabolic equation. Due to the degeneration and convection, Dirac singularities will always be developed at boundary points as time evolves. In order to find a \emph{complete solution} which should keep the conservation of total probability and expectation, three different schemes based on finite volume methods are used to solve the equation numerically: one is a upwind scheme, the other two are different central schemes. We observed that all the methods are stable and can keep the total probability, but have totally different long-time behaviors concerning with the conservation of expectation. We prove that any extra infinitesimal diffusion leads to a same artificial steady state. So upwind scheme does not work due to its intrinsic numerical viscosity. We find one of the central schemes introduces a numerical viscosity term too, which is beyond the common understanding in the convection-diffusion community. Careful analysis is presented to prove that the other central scheme does work. Our study shows that the numerical methods should be carefully chosen and any method with intrinsic numerical viscosity must be avoided.Comment: 17 pages, 8 figure

Solving multiscale elliptic problems by sparse radial basis function neural networks

Machine learning has been successfully applied to various fields of scientific computing in recent years. In this work, we propose a sparse radial basis function neural network method to solve elliptic partial differential equations (PDEs) with multiscale coefficients. Inspired by the deep mixed residual method, we rewrite the second-order problem into a first-order system and employ multiple radial basis function neural networks (RBFNNs) to approximate unknown functions in the system. To aviod the overfitting due to the simplicity of RBFNN, an additional regularization is introduced in the loss function. Thus the loss function contains two parts: the $L_2$ loss for the residual of the first-order system and boundary conditions, and the $\ell_1$ regularization term for the weights of radial basis functions (RBFs). An algorithm for optimizing the specific loss function is introduced to accelerate the training process. The accuracy and effectiveness of the proposed method are demonstrated through a collection of multiscale problems with scale separation, discontinuity and multiple scales from one to three dimensions. Notably, the $\ell_1$ regularization can achieve the goal of representing the solution by fewer RBFs. As a consequence, the total number of RBFs scales like $\mathcal{O}(\varepsilon^{-n\tau})$, where $\varepsilon$ is the smallest scale, $n$ is the dimensionality, and $\tau$ is typically smaller than $1$. It is worth mentioning that the proposed method not only has the numerical convergence and thus provides a reliable numerical solution in three dimensions when a classical method is typically not affordable, but also outperforms most other available machine learning methods in terms of accuracy and robustness.Comment: 23 pages, 12 figure

The prognostic value of postoperative radiotherapy in right tumor for lung related death: based on SEER database and real-world data

BackgroundPostoperative radiotherapy (PORT) is a therapeutic strategy for patients with non-small cell lung cancer (NSCLC). Nevertheless, some studies suggesting PORT does not improve overall survival (OS) including Lung ART phase III trial. The role of PORT and high-risk groups need to be confirmed.MethodsPatients from the Surveillance, Epidemiology, and End Results program (SEER) from 2004 to 2015 were eligible. Aged ≥18 years with stage IIIA-N2 NSCLC, accepted PORT or not were considered for the study. Cox regression analyses and multivariate competing risk model were performed. Propensity score matching (PSM) was conducted. Data from a single-center study in China were used for validation.ResultsIn all patients with IIIA-N2 NSCLC, death from respiratory illness increased year by year, with right lung-related deaths accounting for the main proportion. In SEER database, PORT was detrimental for OS after PSM (hazard ratio [HR], 1.088; 95% CI, 1.088–1.174; P = 0.031), with a same trend for death from the lungs (HR, 1.13; 95% CI, 1.04–1.22; P = 0.005). Right tumor receiving PORT were prone to death from lung disease(HR, 1.14; 95% CI, 1.02–1.27; P = 0.018). In China single-center cohort, PORT was significantly correlated with deteriorated OS (HR 1.356; 95% CI 1.127–1.632; P <0.01), especially in the right laterality (HR 1.365; 95% CI 1.062–1.755; P = 0.015).ConclusionsPORT was a risk factor for stage IIIA-N2 NSCLC patients, particularly with characters of right laterality, male sex, age ≥65 years, and advanced tumor stage. These patients are more likely to death from lung disease after PORT

A six-generation Chinese family in haplogroup B4C1C exhibits high penetrance of 1555A > G-induced hearing Loss

<p>Abstract</p> <p>Background</p> <p>The 1555A > G mutation is the most common cause of aminoglycoside-induced and non-syndromic deafness. However, the variable clinical phenotype and incomplete penetrance of A1555G-induced hearing loss complicate our understanding of this mutation. Environmental factors, nuclear genes, mitochondrial haplotypes/variants and a possible threshold effect have been reported to may be involved in its manifestation.</p> <p>Methods</p> <p>Here, we performed a clinical, molecular, genetic and phylogenic analysis in a six-generation Chinese family.</p> <p>Results</p> <p>A clinical evaluation revealed that affected individuals without aminoglycoside exposure developed hearing loss extending gradually from 12000 Hz to 8000 Hz and then to 4000 Hz. Using pyrosequencing, we detected an identical homoplasmic 1555A > G mutation in all individuals except one. We did not find any correlation between the mutation load and the severity of hearing loss. T123N coexisted with the 1555A > G mutation in six affected subjects in our pedigree. Analysis of the complete mtDNA genome of this family revealed that this family belonged to haplotype B4C1C and exhibited high penetrance. Upon the inclusion of subjects that had been exposed to aminoglycosides, the penetrance of the hearing loss was 63.6%.; without exposure to aminoglycosides, it was 51.5%. This pedigree and another reported Chinese pedigree share the same haplotype (B4C1C) and lack functionally significant mitochondrial tRNA variants, but nevertheless they exhibit a different penetrance of hearing loss.</p> <p>Conclusions</p> <p>Our results imply that the factors responsible for the higher penetrance and variable expression of the deafness associated with the 1555A > G mutation in this pedigree may not be mtDNA haplotype/variants, but rather nuclear genes and/or aminoglycosides.</p