Mathematics

In the current study a meshfree Lagrangian particle method for the Landau-Lifshitz Navier-Stokes (LLNS) equations is developed. The LLNS equations incorporate thermal fluctuation into macroscopic hydrodynamics by the addition of white noise fluxes whose magnitudes are set by a fluctuationdissipation theorem. The study focuses on capturing the correct variance and correlations computed at equilibrium flows, which are compared with available theoretical values. Moreover, a numerical test for the random walk of standing shock wave has been considered for capturing the shock location.

A meshfree Lagrangian method for the fluctuating hydrodynamic equations (FHEs) with fluid-structure interactions is presented. Brownian motion of the particle is investigated by direct numerical simulation of the fluctuating hydrodynamic equations. In this framework a bidirectional coupling has been introduced between the fluctuating fluid and the solid object. The force governing the motion of the solid object is solely due to the surrounding fluid particles. Since a meshfree formulation is used, the method can be extended to many real applications involving complex fluid flows. A threedimensional implementation is presented. In particular, we observe the short and long-time behaviour of the velocity autocorrelation function (VACF) of Brownian particles and compare it with the analytical expression. Moreover, the Stokes-Einstein relation is reproduced to ensure the correct long-time behaviour of Brownian dynamics.

The current study aimed to develop a meshfree Lagrangian particle method for the Landau-Lifshitz Navier-Stokes (LLNS) equations. The LLNS equations incorporate thermal fluctuation into macroscopic hydrodynamics by addition of white noise fluxes whose magnitudes are set by a fluctuation-dissipation theorem. Moreover, the study focuses on capturing correct variance and correlation computed at equilibrium flows, which are compared with available theoretical values and found very good agreement.

We interpret uncertainty in a model for seismic wave propagation by treating the model parameters as random variables, and apply the Multilevel Monte Carlo (MLMC) method to reduce the cost of approximating expected values of selected, physically relevant, quantities of interest (QoI) with respect to the random variables. Targeting source inversion problems, where the source of an earthquake is inferred from ground motion recordings on the Earth's surface, we consider two QoIs that measure the discrepancies between computed seismic signals and given reference signals: one QoI, Q E , is defined in terms of the L 2-misfit, which is directly related to maximum likelihood estimates of the source parameters; the other, Q W , is based on the quadratic Wasserstein distance between probability distributions, and represents one possible choice in a class of such misfit functions that have become increasingly popular to solve seismic inversion in recent years.

Generalized maximum dynamic flow, earliest arrival flow and contraflow problems have been extensively discussed as core problems in the research of evacuation planning. In this work, we formulate and justify the solution procedure of two problems. First, we present the generalized maximum dynamic contraflow problem where a maximum amount of evacuees can be sent to a safe destination from the dangerous place within a given time horizon with minimum loss by allowing reversing the direction of the roads. Second, the generalized earliest arrival contraflow is formulated where as many evacuees as possible should be sent in every time period to a safe destination from dangerous place through highest gain path allowing reversing the direction of the roads. A pseudopolynomial time algorithm for both problems will be presented on a single source and a single sink lossy network having capacities, transit times and gain factors on the arcs.

 Both the earliest arrival flow and the contraflow problems have been highly focused in the research of evacuation planning. We formulated the earliest arrival contraflow problem, where as many evacuees as possible should be sent from dangerous places to safe destinations in every time period by reversing the direction of roads at time zero. A strongly polynomial time algorithm for the earliest arrival contraflow problem on a two-terminal series-parallel graph model having capacities and transit times on the arcs has been presented.

Contraflow technique has gained a considerable focus in evacuation planning research over the past several years. In this work, we design efficient algorithms to solve the maximum, lex-maximum, earliest arrival, and quickest dynamic flow problems having constant attributes and their generalizations with partial contraflow reconfiguration in the context of evacuation planning. The partial static contraflow problems, that are foundations to the dynamic flows, are also studied. Moreover, the contraflow model with inflow-dependent transit time on arcs is introduced. A strongly polynomial time algorithm to compute approximate solution of the quickest partial contraflow problem on two terminal networks is presented, which is substantiated by numerical computations considering Kathmandu road network as an evacuation network. Our results show that the quickest time to evacuate a flow of value 100,000 units is reduced by more than 42% using the partial contraflow technique, and the differenc...

The research on evacuation planning problem is promoted by the very challenging emergency issues due to large scale natural or man-created disasters. It is the process of shifting the maximum number of evacuees from the disastrous areas to the safe destinations as quickly and efficiently as possible. Contraflow is a widely accepted model for good solution of evacuation planning problem. It increases the outbound road capacity by reversing the direction of roads towards the safe destination. The continuous dynamic contraflow problem sends the maximum number of flow as a flow rate from the source to the sink in every moment of time unit. We propose the mathematical model for the continuous dynamic contraflow problem. We present efficient algorithms to solve the maximum continuous dynamic contraflow and quickest continuous contraflow problems on single source single sink arbitrary networks and continuous earliest arrival contraflow problem on single source single sink series-parallel n...

Generalized maximum dynamic flow, earliest arrival flow and contraflow problems have been extensively discussed as core problems in the research of evacuation planning. In this work, we formulate and justify the solution procedure of two problems. First, we present the generalized maximum dynamic contraflow problem where a maximum amount of evacuees can be sent to a safe destination from the dangerous place within a given time horizon with minimum loss by allowing reversing the direction of the roads. Second, the generalized earliest arrival contraflow is formulated where as many evacuees as possible should be sent in every time period to a safe destination from dangerous place through highest gain path allowing reversing the direction of the roads. A pseudopolynomial time algorithm for both problems will be presented on a single source and a single sink lossy network having capacities, transit times and gain factors on the arcs.

 Network flow over time is an important area for the researcher relating to the traffic assignment problem. Transmission times of the vehicles directly depend on the number of vehicles entering the road. Flow over time with fixed transit times can be solved by using classical (static) flow algorithms in a corresponding time expanded network which is not exactly applicable for flow over time with inflow dependent transit times. In this paper we discuss the time expanded graph for inflow-dependent transit times and non-existence of earliest arrival flow on it. Flow over time with inflow-dependent transit times are turned to inflow-preserving flow by pushing the flow from slower arc to the fast flow carrying arc. We gave an example to show that time horizon of quickest flow in bow graph GB was strictly smaller than time horizon of any inflow-preserving flow over time in GB satisfying the same demand. The relaxation in the modified bow graph turns the problem into the linear programming...

Contraflow technique has gained a considerable focus in evacuation planning research over the past several years. In this work, we design efficient algorithms to solve the maximum, lex-maximum, earliest arrival, and quickest dynamic flow problems having constant attributes and their generalizations with partial contraflow reconfiguration in the context of evacuation planning. The partial static contraflow problems, that are foundations to the dynamic flows, are also studied. Moreover, the contraflow model with inflow-dependent transit time on arcs is introduced. A strongly polynomial time algorithm to compute approximate solution of the quickest partial contraflow problem on two terminal networks is presented, which is substantiated by numerical computations considering Kathmandu road network as an evacuation network. Our results show that the quickest time to evacuate a flow of value 100,000 units is reduced by more than 42% using the partial contraflow technique, and the differenc...

The abstract flow model deals with the flow  paths (routes) that satisfy the switching property. Contraflow is a widely accepted solution approach that increases the flow and decreases the evacuation time making the traffic smooth during evacuation by reversing the required road directions from the risk areas to the safe places. In this paper, we integrate the concepts of abstract flow and contraflow, give mathematical formulations of these models and present efficient algorithms for solving the abstract contraflow problems. The efficient solution procedures are presented for  maximum dynamic, lexicographically maximum and earliest arrival abstract contraflow  problems. This approach maximizes the flow value in given time and seeks to eliminate the crossing conflicts.

Despite of implicit flow conservation on every arc of traditional network flow model, the generalized network flow model assumes the proportional and symmetric loss factor on each arc. This paper considers two terminal lossy networks specifying the portion of flow entering an arc at its tail node that reaches to its head node. We studied the generalized maximum continuous dynamic contraflow (GMCDCF) and generalized continuous earliest arrival contraflow (GCEACF) problems. The problems are efficiently solved with pseudo-polynomial time algorithms. Moreover, a fully polynomial time approximation scheme (FPTAS) is proposed in polynomial time.

Contraflow technique is the widely accepted model on network optimization. It allows arc reversal that increases the arc capacities. The earliest arrival transshipment contraflow is an important model that transship the given flow value by sending the maximum amount at each time point from the beginning within given time period by reversing the direction arcs from the sources to the sinks at time zero. This problem has not been solved polynomially on complex networks, i.e., multi-terminal networks yet. However, its 2-value-approximation solution has been found by Pyakurel and Dhamala [13] in pseudopolynomial time complexity. Moreover, they have claimed that for the special case of zero transit time on each arc, the 2-value-approximation solution can be computed in polynomial time complexity. In this paper, we solve their claim presenting an efficient algorithm.

Multi-commodity flow problem appears when several distinct commodities are shipped from supply nodes to the demand nodes through a network without violating the capacity constraints. The quickest multi-commodity flow problem deals with the minimization of time satisfying given demand. Ingeneral, the quickest multi-commodity flow problems are computationally hard. The outbound lane capacities can be increased through reverting the orientation of lanes towards the demand nodes. We present two approximation algorithms by introducing partial contraow technique in the continuous-time quick estmulti-commodity ow problem: one polynomial-time with the help of length-bounded flow and another FPTAS by using _-condensed time-expanded graph. Both algorithms reverse only necessary arc capacities to get the optimal solutions and save unused arc capacities which may be used for other purposes.   

Due to different disasters, natural or men made, world is facing the problem of massive damage of life and property. To save the life of maximum number of evacuees, an efficient evacuation planning is essential. Prioritization is the process of deciding the relative importance or urgency of things or objects. It helps to focus on the objectives and goals in an efficient way. Prioritized flow problems are applicable for large scale disaster management problems. Prioritization of evacuees on the basis of case sensitive or risk of life is one of the best and reliable way of evacuation planning. In this paper, we introduce the maximum evacuation planning problem with commodity (evacuee) priority ordering and develop multi-commodity flow model for priority based evacuation planning problem. We present polynomial time algorithms to solve the problem in static as well as dynamic flow models.

Every models in the network flow theory aim to increase flow value from the sources to the sinks and reduce time or cost satisfying the capacity and flow conservation constraints. Recently, the network flow model without flow conservation constraints at the intermediate nodes has been investigated by Pyakurel and Dempe [13]. In this model, if the incoming flow to a intermediate node is greater than the outgoing flow, then the excess flow can be stored at the node respecting its capacity. Moreover, the model works if sum of the outgoing arc capacities from the source of a network is greater than the minimum cut capacity and intermediate node has storage capacity. The excess flow has been sent as far as possible from the source. In two terminal network, the maximum static flow and maximum dynamic flow problems with intermediate storage have been solved in polynomial time complexities. Motivated by this work, we study the lexicographic maximum flow problem with intermediate storage in ...

In this paper, we investigate the minimum cost flow problem in two terminal series parallel network. We present modified minimum cost flow algorithm that computes the maximum dynamic and the earliest arrival flows in strongly polynomial time and also preserves all unused arc capacities. We also present strongly polynomial time minimum cost partial contraflow algorithm that solves both problems with partial reversals of arc capacities on two terminal series parallel networks.

Motivated by applications in evacuation planning, we consider a problem of optimizing flow with arc reversals in which the transit time depends on the orientation of the arc. In the considered problems, the transit time on an arc may change when it is reversed, contrary to the problems considered in the existing literature. Extending the existing idea of auxiliary network construction to allow asymmetric transit time on arcs, we present strongly polynomial time algorithms for solving single-source-single-sink maximum dynamic contraflow problem and quickest contraflow problem. +e results are substantiated by a computational experiment in a Kathmandu road network. An algorithm to solve the corresponding earliest arrival contraflow problem with a pseudo-polynomial-time complexity is also presented. +e partial contraflow approach for the corresponding problems has also been discussed.