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At least 199 records · Page 11

Statistical Analysis of Large Simulated Yield Datasets for Studying Climate Effects

Many studies have been carried out during the last decade to study the effect of climate change on crop yields and other key crop characteristics. In these studies, one or several crop models were used to simulate crop growth and development for different climate scenarios that correspond to different projections of atmospheric CO2 concentration, temperature, and rainfall changes (Semenov et al., 1996; Tubiello and Ewert, 2002; White et al., 2011). The Agricultural Model Intercomparison and Improvement Project (AgMIP; Rosenzweig et al., 2013) builds on these studies with the goal of using an ensemble of multiple crop models in order to assess effects of climate change scenarios for several crops in contrasting environments. These studies generate large datasets, including thousands of simulated crop yield data. They include series of yield values obtained by combining several crop models with different climate scenarios that are defined by several climatic variables (temperature, CO2, rainfall, etc.). Such datasets potentially provide useful information on the possible effects of different climate change scenarios on crop yields. However, it is sometimes difficult to analyze these datasets and to summarize them in a useful way due to their structural complexity; simulated yield data can differ among contrasting climate scenarios, sites, and crop models. Another issue is that it is not straightforward to extrapolate the results obtained for the scenarios to alternative climate change scenarios not initially included in the simulation protocols. Additional dynamic crop model simulations for new climate change scenarios are an option but this approach is costly, especially when a large number of crop models are used to generate the simulated data, as in AgMIP. Statistical models have been used to analyze responses of measured yield data to climate variables in past studies (Lobell et al., 2011), but the use of a statistical model to analyze yields simulated by complex process-based crop models is a rather new idea. We demonstrate herewith that statistical methods can play an important role in analyzing simulated yield data sets obtained from the ensembles of process-based crop models. Formal statistical analysis is helpful to estimate the effects of different climatic variables on yield, and to describe the between-model variability of these effects.

climate↗

A Clustering-based biased Monte Carlo Approach to Protein Titration Curve Prediction

We develop and implement a novel approach to computing the ensemble averages in systems characterized by pair-wise interactions between the entities. Methods involving full enumeration of the configuration space result in exponential complexity. Sampling methods such as Markov Chain Monte Carlo (MCMC) algorithms have been proposed to tackle the exponential complexity of these problems. In certain scenarios where significant energetic coupling exists between the entities, the accuracy of the such algorithms can be diminished. We propose a strategy to improve the accuracy of the MCMC runs by taking advantage of the cluster structure in the interaction energy matrix. We propose two different schemes for performing the biased MCMC runs on the partitioned systems and show that they are valid MCMC schemes. We then apply these algorithms to the problem of computing the protonation fractions and hence the titration curves of titratable protein residues that constitute a given protein. We leverage both synthesized and real-world systems and show the improved performance of our biased MCMC methods when compared to the regular MCMC method.

Visweswara Sathanur, Arun↗

SEC ‐ SAXS / MC Ensemble Structural Studies of the Microtubule Binding Protein Cdt1 Show Monomeric, Folded‐Over Conformations

ABSTRACT Cdt1 is a mixed folded protein critical for DNA replication licensing and it also has a “moonlighting” role at the kinetochore via direct binding to microtubules and the Ndc80 complex. However, it is unknown how the structure and conformations of Cdt1 could allow it to participate in these multiple, unique sets of protein complexes. While robust methods exist to study entirely folded or unfolded proteins, structure–function studies of combined, mixed folded/disordered proteins remain challenging. In this work, we employ orthogonal biophysical and computational techniques to provide structural characterization of mitosis‐competent human Cdt1. Thermal stability analyses shows that both folded winged helix domains1 are unstable. CD and NMR show that the N‐terminal and linker regions are intrinsically disordered. DLS shows that Cdt1 is monomeric and polydisperse, while SEC‐MALS confirms that it is monomeric at high concentrations, but without any apparent inter‐molecular self‐association. SEC‐SAXS enabled computational modeling of the protein structures. Using the program SASSIE, we performed rigid body Monte Carlo simulations to generate a conformational ensemble of structures. We observe that neither fully extended nor extremely compact Cdt1 conformations are consistent with SAXS. The best‐fit models have the N‐terminal and linker disordered regions extended into the solution and the two folded domains close to each other in apparent “folded over” conformations. We hypothesize the best‐fit Cdt1 conformations could be consistent with a function as a scaffold protein that may be sterically blocked without binding partners. Our study also provides a template for combining experimental and computational techniques to study mixed‐folded proteins.

Cell Biology↗

Urban Flood Modeling: Uncertainty Quantification and Physics‐Informed Gaussian Processes Regression Forecasting

Abstract Estimating uncertainty in flood model predictions is important for many applications, including risk assessment and flood forecasting. We focus on uncertainty in physics‐based urban flooding models. We consider the effects of the model's complexity and uncertainty in key input parameters. The effect of rainfall intensity on the uncertainty in water depth predictions is also studied. As a test study, we choose the Interconnected Channel and Pond Routing (ICPR) model of a part of the city of Minneapolis. The uncertainty in the ICPR model's predictions of the floodwater depth is quantified in terms of the ensemble variance using the multilevel Monte Carlo (MC) simulation method. Our results show that uncertainties in the studied domain are highly localized. Model simplifications, such as disregarding the groundwater flow, lead to overly confident predictions, that is, predictions that are both less accurate and uncertain than those of the more complex model. We find that for the same number of uncertain parameters, increasing the model resolution reduces uncertainty in the model predictions (and increases the MC method's computational cost). We employ the multilevel MC method to reduce the cost of estimating uncertainty in a high‐resolution ICPR model. Finally, we use the ensemble estimates of the mean and covariance of the flood depth for real‐time flood depth forecasting using the physics‐informed Gaussian process regression method. We show that even with few measurements, the proposed framework results in a more accurate forecast than that provided by the mean prediction of the ICPR model.

Kohanpur, Amir H.↗

Biochemical and structural insights into SARS-CoV-2 polyprotein processing by Mpro

SARS-CoV-2, a human coronavirus, is the causative agent of the COVID-19 pandemic. Its genome is translated into two large polyproteins subsequently cleaved by viral papain-like protease and main protease (Mpro). Polyprotein processing is essential yet incompletely understood. We studied Mpro-mediated processing of the nsp7-11 polyprotein, whose mature products include cofactors of the viral replicase, and identified the order of cleavages. Integrative modeling based on mass spectrometry (including hydrogen-deuterium exchange and cross-linking) and x-ray scattering yielded a nsp7-11 structural ensemble, demonstrating shared secondary structural elements with individual nsps. The pattern of cross-links and HDX footprint of the C145A Mpro and nsp7-11 complex demonstrate preferential binding of the enzyme active site to the polyprotein junction sites and additional transient contacts to help orient the enzyme on its substrate for cleavage. Last, proteolysis assays were used to characterize the effect of inhibitors/binders on Mpro processing/inhibition using the nsp7-11 polyprotein as substrate.

60 APPLIED LIFE SCIENCES↗

Mechanism and dynamics of fatty acid photodecarboxylase

Photoenzymes are rare biocatalysts driven by absorption of a photon at each catalytic cycle; they inspire development of artificial photoenzymes with valuable activities. Fatty acid photodecarboxylase (FAP) is a natural photoenzyme that has potential applications in the bio-based production of hydrocarbons, yet its mechanism is far from fully understood. RATIONALE To elucidate the mechanism of FAP, we studied the wild-type (WT) enzyme from Chlorella variabilis (CvFAP) and variants with altered active-site residues using a wealth of techniques, including static and time-resolved crystallography and spectroscopy, as well as biochemical and computational approaches. RESULTS A 1.8-Å-resolution CvFAP x-ray crystal structure revealed a dense hydrogen-bonding network positioning the fatty acid carboxyl group in the vicinity of the flavin adenine dinucleotide (FAD) cofactor. Structures solved from free electron laser and low-dose synchrotron x-ray crystal data further highlighted an unusual bent shape of the oxidized flavin chromophore, and showed that the bending angle (14°) did not change upon photon absorption (step 1) or throughout the photocycle. Calculations showed that bending substantially affected the energy levels of the flavin. Structural and spectroscopic analysis of WT and mutant proteins targeting two conserved active-site residues, R451 and C432, demonstrated that both residues were crucial for proper positioning of the substrate and water molecules and for oxidation of the fatty acid carboxylate by 1 FAD* (~300 ps in WT FAP) to form FAD ∙– (step 2). Time-resolved infrared spectroscopy demonstrated that decarboxylation occured quasi-instantaneously upon this forward electron transfer, consistent with barrierless bond cleavage predicted by quantum chemistry calculations and with snapshots obtained by time-resolved crystallography. Transient absorption spectroscopy in H 2 O and D 2 O buffers indicated that back electron transfer from FAD ∙– was coupled to and limited by transfer of an exchangeable proton or hydrogen atom (step 3). Unexpectedly, concomitant with FAD ∙– reoxidation (to a red-shifted form FAD RS ) in 100 ns, most of the CO 2 product was converted, most likely into bicarbonate (as inferred from FTIR spectra of the cryotrapped FAD RS intermediate). Calculations indicated that this catalytic transformation involved an active-site water molecule. Cryo-Fourier transform infrared spectroscopy studies suggested that bicarbonate formation (step 4) was preceded by deprotonation of an arginine residue (step 3). At room temperature, the remaining CO 2 left the protein in 1.5 μs (step 4'). The observation of residual electron density close to C432 in electron density maps derived from time-resolved and cryocrystallography data suggests that this residue may play a role in stabilizing CO 2 and/or bicarbonate. Three routes for alkane formation were identified by quantum chemistry calculations; the one shown in the figure is favored by the ensemble of experimental data. CONCLUSION Finally, we provide a detailed and comprehensive characterization of light-driven hydrocarbon formation by FAP, which uses a remarkably complex mechanism including unique catalytic steps. We anticipate that our results will help to expand the green chemistry toolkit.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

MADA: Multi-Agent Design Assistant

MADA (Multi-Agent Design Assistant) is a Large Language Model (LLM) powered multi-agent framework that coordinates specialized agents for complex design workflows. The system was designed for HPC workflows with the following agents in mind: 1) A Job Management Agent (JMA) launches and manages ensemble simulations on HPC systems, 2) a Geometry Agent (GA) generates meshes, and 3) an Inverse Design Agent (IDA) proposes new designs informed by simulation outcomes. Our framework reduces cumbersome manual workflow setup, and enables automated design exploration at scale. However, the software also enables users to rapidly create new multi-agent systems. Simply define new agents in a configuration file, giving each their own set of tools (via MCP), and then chat and prompt your new multi-agent system. Is

Gunnarson, BrianS [Lawrence Livermore National Lab↗

Automated calculation and convergence of defect transport tensors

Defect diffusion is a key process in materials science and catalysis, but as migration mechanisms are often too complex to enumerate a priori, calculation of transport tensors typically have no measure of convergence and require significant end-user intervention. These two bottlenecks prevent high-throughput implementations essential to propagate model-form uncertainty from interatomic interactions to predictive simulations. In order to address these issues, we extend a massively parallel accelerated sampling scheme, autonomously controlled by Bayesian estimators of statewide sampling completeness, to build atomistic kinetic Monte Carlo models on a state-space irreducible under exchange and space group symmetries. Focusing on isolated defects, we derive analytic expressions for drift and diffusion coefficients, providing a convergence metric by calculating the Kullback–Leibler divergence across the ensemble of diffusion processes consistent with the sampling uncertainty. The autonomy and efficacy of the method is demonstrated on surface trimers in tungsten and Hexa-interstitials in magnesium oxide, both of which exhibit complex, correlated migration mechanisms.

36 MATERIALS SCIENCE↗

Evaluating the Importance of Conformers for Understanding the Vacuum-Ultraviolet Spectra of Oxiranes: Experiment and Theory

Vacuum-ultraviolet (VUV) absorption spectroscopy enables electronic transitions that offer the unambiguous identification of molecules. As target molecules become more complex, multifunctional species present a great challenge to both experimental and computational spectroscopy. This research reports both experimental and theoretical studies of oxiranes. Computationally, the nuclear ensemble approach has been used to accurately predict experimental spectra for a variety of molecules. However, this approach incurs great computational cost, as ensembles generally consist of thousands of geometries. The present study aims to drastically reduce the ensemble by evaluating the significance of the conformers to the predicted spectra. This approach was applied to 11 substituted oxiranes using the Conformer Rotamer Ensemble Sampling Tool (CREST) of Grimme to generate an ensemble of unique conformers determined by their Boltzmann populations. Five TD-DFT functionals (BMK, CAM-B3LYP, M06-2X, MN15, ωB97X-D) and EOM-CCSD were used to simulate the spectrum of each substituted oxirane ensemble. Computed spectra were then compared to the experiment using both qualitative and quantitative metrics. Based on these metrics, it was observed that certain conformers may not be necessary to characterize this set of oxiranes despite the temperature (323 K) of the experiment. A single conformer can then be used with TD-DFT and EOM-CCSD to replicate the experimental spectra of these medium-sized combustion species.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Cold Season Performance of the NU-WRF Regional Climate Model in the Great Lakes Region

As Earth’s largest collection of freshwater, the Laurentian Great Lakes have enormous ecological and socio-economic value. Their basin has become a regional hotspot of climatic and limnological change, potentially threatening its vital natural resources. Consequentially, there is a need to assess the current state of climate models regarding their performance across the Great Lakes region and develop the next generation of high-resolution regional climate models to address complex limnological processes and lake-atmosphere interactions. In response to this need, the current paper focuses on the generation and analysis of a 20-member ensemble of 3-km National Aeronautics and Space Administration (NASA)-Unified Weather Research and Forecasting (NU-WRF) simulations for the 2014-2015 cold season. The study aims to identify the model’s strengths and weaknesses; optimal configuration for the region; and the impacts of different physics parameterizations, coupling to a 1D lake model, time-variant lake-surface temperatures, and spectral nudging. Several key biases are identified in the cold-season simulations for the Great Lakes region, including an atmospheric cold bias that is amplified by coupling to a 1D lake model but diminished by applying the Community Atmosphere Model radiation scheme and Morrison microphysics scheme; an excess precipitation bias; anomalously early initiation of fall lake turnover and subsequent cold lake bias; excessive and overly persistent lake ice cover; and insufficient evaporation over Lakes Superior and Huron. The research team is currently addressing these key limitations by coupling NU-WRF to a 3D lake model in support of the next generation of regional climate models for the critical Great Lakes Basin.

Michael Notaro↗

Structural basis of differential gene expression at eQTLs loci from high-resolution ensemble models of 3D single-cell chromatin conformations

Abstract Motivation Techniques such as high-throughput chromosome conformation capture (Hi-C) have provided a wealth of information on nucleus organization and genome important for understanding gene expression regulation. Genome-Wide Association Studies have identified numerous loci associated with complex traits. Expression quantitative trait loci (eQTL) studies have further linked the genetic variants to alteration in expression levels of associated target genes across individuals. However, the functional roles of many eQTLs in noncoding regions remain unclear. Current joint analyses of Hi-C and eQTLs data lack advanced computational tools, limiting what can be learned from these data. Results We developed a computational method for simultaneous analysis of Hi-C and eQTL data, capable of identifying a small set of nonrandom interactions from all Hi-C interactions. Using these nonrandom interactions, we reconstructed large ensembles (×105) of high-resolution single-cell 3D chromatin conformations with thorough sampling, accurately replicating Hi-C measurements. Our results revealed many-body interactions in chromatin conformation at the single-cell level within eQTL loci, providing a detailed view of how 3D chromatin structures form the physical foundation for gene regulation, including how genetic variants of eQTLs affect the expression of associated eGenes. Furthermore, our method can deconvolve chromatin heterogeneity and investigate the spatial associations of eQTLs and eGenes at subpopulation level, revealing their regulatory impacts on gene expression. Together, ensemble modeling of thoroughly sampled single-cell chromatin conformations combined with eQTL data, helps decipher how 3D chromatin structures provide the physical basis for gene regulation, expression control, and aid in understanding the overall structure-function relationships of genome organization. Availability and implementation It is available at https://github.com/uic-liang-lab/3DChromFolding-eQTL-Loci.

Du, Lin (ORCID:0009000289869812)↗

A new spatially distributed added value index for regional climate models: the EURO-CORDEX and the CORDEX-CORE highest resolution ensembles

The added value of using regional climate models (RCMs) to downscale data from general circulation models (GCMs) has often been questioned and researched. Although several studies have used different methods to identify (and in some cases quantify) the added value, there is still a need to find a general metric that quantifies the added value of any variable. This paper builds on past studies to propose a new metric of added value in the simulation of present-day climate which measures the difference in the probability density functions (PDFs) at each grid-cell between a model and an observation source, and then compares the results of the RCM and GCM in order to spatially compute the added value index. The same method is also adapted to quantify the climate change downscaling signal in a way that is consistent with the present-day metric. These new metrics are tested on the daily precipitation output from the EURO-CORDEX and CORDEX-CORE projection ensembles and reveal an overall positive added value of RCMs, especially at the tail-end of the distribution. Higher added value is obtained in areas of complex topography and coast-lines, as well as in tropical regions. Areas with large added value in present-day climate are consistent with areas of significant climate change downscaling signal in the RCP 8.5 far future simulations, and when the analysis is repeated at a low-resolution. The use of different resolution observations shows that the added value tends to decrease when models are compared to low-resolution observation datasets.

54 ENVIRONMENTAL SCIENCES↗

Inverse Thermodynamics: Designing Interactions for Targeted Phase Behavior

The traditional goal of inverse self-assembly is to design interactions that drive particles toward a desired target structure. However, achieving successful self-assembly also requires tuning the thermodynamic conditions under which the structure is stable. In this work, we extend the inverse design paradigm to explicitly address this challenge by developing a framework for inverse thermodynamics, i.e., the design of interaction potentials that realize specific thermodynamic behavior. As a step in this direction, using patchy particle mixtures as a model system, we demonstrate how precise control over both bonding topology and bond energetics enables the programming of targeted phase behavior. In particular, we establish design principles for azeotropic demixing and show how to create mixtures that exhibit azeotropy at any prescribed composition. Our predictions are validated through Gibbs-ensemble simulations [Panagiotopoulos, Mol. Phys. 1987, 61, 813−826]. These results highlight the necessity of coupling structural design with thermodynamic engineering, and provide a blueprint for controlling complex phase behavior in multicomponent systems.

Azeotropes↗

Characterizing and quantifying uncertainty in projections of climate change impacts on air quality

Climate change can aggravate air pollution, with important public health and environmental consequences. While major sources of uncertainty in climate change projections - greenhouse gas (GHG) emissions scenario, model response, and internal variability - have been investigated extensively, their propagation to estimates of air quality impacts has not been systematically assessed. Here, we compare these uncertainties using a coupled modeling framework that includes a human activity model, an Earth system model of intermediate complexity, and a global atmospheric chemistry model. Uncertainties in projections of U.S. air quality under 21st century climate change are quantified based on a climate-chemistry ensemble that includes multiple initializations, representations of climate sensitivity, and climate policy scenarios, under constant air pollution emissions. We find that climate-related uncertainties are comparable at mid-century, making it difficult to distinguish the impact of variations in GHG emissions on ozone and particulate matter pollution. While GHG emissions scenario eventually becomes the dominant uncertainty based on the scenarios considered, all sources of uncertainty are significant through the end of the century. The results provide insights into intrinsically different uncertainties in projections of air pollution impacts and the potential for large ensembles to better capture them.

54 ENVIRONMENTAL SCIENCES↗

A Practical Philosophy of Complex Climate Modelling

We give an overview of the practice of developing and using complex climate models, as seen from experiences in a major climate modelling center and through participation in the Coupled Model Intercomparison Project (CMIP).We discuss the construction and calibration of models; their evaluation, especially through use of out-of-sample tests; and their exploitation in multi-model ensembles to identify biases and make predictions. We stress that adequacy or utility of climate models is best assessed via their skill against more naive predictions. The framework we use for making inferences about reality using simulations is naturally Bayesian (in an informal sense), and has many points of contact with more familiar examples of scientific epistemology. While the use of complex simulations in science is a development that changes much in how science is done in practice, we argue that the concepts being applied fit very much into traditional practices of the scientific method, albeit those more often associated with laboratory work.

complex simulation↗

Optimal Power Management of Battery Energy Storage Systems via Ensemble Kalman Inversion

Optimal power management of battery energy storage systems (BESS) is crucial for their safe and efficient operation. Numerical optimization techniques are frequently utilized to solve the optimal power management problems. However, these techniques often fall short of delivering real-time solutions for large-scale BESS due to their computational complexity. To address this issue, this paper proposes a computationally efficient approach. We introduce a new set of decision variables called power-sharing ratios corresponding to each cell, indicating their allocated power share from the output power demand. We then formulate an optimal power management problem to minimize the system-wide power losses while ensuring compliance with safety, balancing, and power supply-demand match constraints. To efficiently solve this problem, a parametrized control policy is designed and leveraged to transform the optimal power management problem into a parameter estimation problem. We then implement the ensemble Kalman inversion to estimate the optimal parameter set. The proposed approach significantly reduces computational requirements due to 1) the much lower dimensionality of the decision parameters and 2) the estimation treatment of the optimal power management problem. Finally, we conduct extensive simulations to validate the effectiveness of the proposed approach. The results show promise in accuracy and computation time compared with explored numerical optimization techniques.

Farakhor, Amir↗

Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemble

In this work we explore the capability of physics-informed neural networks (PINNs) to discover multiple solutions. Many real-world phenomena governed by nonlinear differential equations (DEs), such as fluid flow, exhibit multiple solutions under the same conditions, yet capturing this solution multiplicity remains a significant challenge. A key difficulty lies in providing appropriate initial conditions or guesses, as widely used time-marching schemes and Newton’s method are highly sensitive to these choices when solving complex computational problems. While machine learning models, particularly PINNs, have shown promise in solving DEs, their ability to capture multiple solutions remains underexplored. In this work, we propose a simple and practical approach using PINNs to learn and discover multiple solutions. We first demonstrate that PINNs, when combined with random initialization and deep ensemble method—originally developed for uncertainty quantification—can effectively uncover multiple solutions to nonlinear ordinary and partial DEs. Although training large ensembles of PINNs may appear computationally demanding, this can be done efficiently using vectorization techniques supported by modern deep learning frameworks, allowing many networks to be trained simultaneously. Our approach highlights the critical role of initialization in shaping solution diversity, addressing an often-overlooked aspect of machine learning for scientific computing. Furthermore, we propose utilizing PINN-generated solutions as initial conditions or initial guesses for conventional numerical solvers to enhance accuracy and efficiency in capturing multiple solutions. Extensive numerical experiments, including the Allen–Cahn equation and cavity flow, where our approach successfully identifies both stable and unstable solutions, validate the effectiveness of our method. These findings establish a general and efficient framework for addressing solution multiplicity in nonlinear DEs.

97 MATHEMATICS AND COMPUTING↗

A computational pipeline to generate a synthetic dataset of metal ion sorption to oxides for AI/ML exploration

The charged mineral/electrolyte interfaces are ubiquitous in the surface and subsurface–including the surroundings of the geological disposal sites for radioactive waste. Therefore, understanding how ions interact with charged surfaces is critically important for predicting radionuclide mobility in the case of waste leakage. At present, the Surface Complexation Models (SCMs) are the most successful thermodynamic frameworks to describe ion retention by mineral surfaces. SCMs are interfacial speciation models that account for the effect of the electric field generated by charged surfaces on sorption equilibria. These models have been successfully used to analyze and interpret a broad range of experimental observations including potentiometric and electrokinetic titrations or spectroscopy. Unfortunately, many of the current procedures to solve and fit SCM to experimental data are not optimal, which leads to a non-transferable or non-unique description of interfacial electrostatics and consequently of the strength and extent of ion retention by mineral surfaces. Recent developments in Artificial Intelligence (AI) offer a new avenue to replace SCM solvers and fitting algorithms with trained AI surrogates. Unfortunately, there is a lack of a standardized dataset covering a wide range of SCM parameter values available for AI exploration and training–a gap filled by this study. Here, we described the computational pipeline to generate synthetic SCM data and discussed approaches to transform this dataset into AI-learnable input. First, we used this pipeline to generate a synthetic dataset of electrostatic properties for a broad range of the prototypical oxide/electrolyte interfaces. The next step is to extend this dataset to include complex radionuclide sorption and complexation, and finally, to provide trained AI architectures able to infer SCMs parameter values rapidly from experimental data. Here, we illustrated the AI-surrogate development using the ensemble learning algorithms, such as Random Forest and Gradient Boosting. These surrogate models allow a rapid prediction of the SCM model parameters, do not rely on an initial guess, and guarantee convergence in all cases.

Li, Chunhui↗