The convergence of Markov chain--based Monte Carlo linear solvers using the Ulam--von Neumann algorithm for a linear system of the form $x = Hx + b$ is investigated in this paper. We analyze the convergence of the Monte Carlo solver based on the original Ulam--von Neumann algorithm under the conditions that $\|H\|<1$ as well as $\rho(H)<1$, where $\rho(H)$ is the spectral radius of $H$. We find that although the Monte Carlo solver is based on sampling the Neumann series, the convergence of Neumann series is not a sufficient condition for the convergence of the Monte Carlo solver. Actually, properties of $H$ are not the only factors determining the convergence of the Monte Carlo solver; the underlying transition probability matrix plays an important role. An improper selection of the transition matrix may result in divergence even though the condition $\|H\|<1$ holds. However, if the condition $\|H\|<1$ is satisfied, we show that there always exist certain transition matrices that guarantee convergence of ...

Shallow lakes, one of the most widespread water bodies in the world, are easily shifted to a new trophic state due to external interferences. Shifting hydrologic conditions and climate change can cause cyanobacterial harmful algal blooms (CyanoHABs) in shallow lakes, which pose serious threats to ecological integrity and human health. This study analyzed the effects of hydrologic and meteorological variables on cyanobacterial blooms in Yangtze-connected lakes (Lake Dongting and Poyang) and isolated lakes (Lake Chao and Tai). The results show that (i) chlorophyll-a (Chl-a) concentration tends to decrease exponentially with increasing relative lake level fluctuations (RLLF) and precipitation, but to increase linearly with increasing wind speed and air temperature; (ii) Chl-a concentrations in lakes were significantly higher when RLLF < 100, precipitation < 2.6 mm, wind speed > 2.6 m s⁻¹, or air temperature > 17.8 °C; (iii) the Chl-a concentration of Yangtze-isolated lakes was more significantly affected by water level amplitude, precipitation, wind speed and air temperature than the Yangtze-connected lakes; (iv) the RLLF and the ratio of wind speed to mean water depth could be innovative coupling factors to examine variation characteristics of Chl-a in shallow lakes with greater correlation than single factors.

Stochasticity and small system size effects in complex biochemical reaction networks can greatly alter transient and steady-state system properties. A common approach to modeling reaction networks, which accounts for system size, is the chemical master equation that governs the dynamics of the joint probability distribution for molecular copy number. However, calculation of the stationary distribution is often prohibitive, due to the large state-space associated with most biochemical reaction networks. Here, we analyze a network representing a luminal calcium release site model and investigate to what extent small system size effects and calcium fluctuations, driven by ion channel gating, influx and diffusion, alter steady-state ion channel properties including open probability. For a physiological ion channel gating model and number of channels, the state-space may be between approximately $10^6-10^8$ elements, and a novel modified block power method is used to solve the associated dominant eigenvector problem required to calculate the stationary distribution. We demonstrate that both small local cytosolic domain volume and a small number of ion channels drive calcium fluctuations that result in deviation from the corresponding model that neglects small system size effects.