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Shaoshuai Mou - One of the best experts on this subject based on the ideXlab platform.

  • a distributed algorithm for solving a Linear Algebraic Equation
    2015
    Co-Authors: Shaoshuai Mou, Ji Liu, Stephen A Morse
    Abstract:

    A distributed algorithm is described for solving a Linear Algebraic Equation of the form $Ax = b$ assuming the Equation has at least one solution. The Equation is simultaneously solved by $m$ agents assuming each agent knows only a subset of the rows of the partitioned matrix $[\matrix{A & b}]$ , the current estimates of the Equation's solution generated by its neighbors, and nothing more. Each agent recursively updates its estimate by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a time-dependent directed graph $\BBN(t)$ whose vertices correspond to agents and whose arcs depict neighbor relations. It is shown that for any matrix $A$ for which the Equation has a solution and any sequence of “repeatedly jointly strongly connected graphs” $\BBN(t)$ , $t = 1, 2, \ldots$ , the algorithm causes all agents' estimates to converge exponentially fast to the same solution to $Ax = b$ . It is also shown that, under mild assumptions, the neighbor graph sequence must actually be repeatedly jointly strongly connected if exponential convergence is to be assured. A worst case convergence rate bound is derived for the case when $Ax = b$ has a unique solution. It is demonstrated that with minor modification, the algorithm can track the solution to $Ax = b$ , even if $A$ and $b$ are changing with time, provided the rates of change of $A$ and $b$ are sufficiently small. It is also shown that in the absence of communication delays, exponential convergence to a solution occurs even if the times at which each agent updates its estimates are not synchronized with the update times of its neighbors. A modification of the algorithm is outlined which enables it to obtain a least squares solution to $Ax = b$ in a distributed manner, even if $Ax = b$ does not have a solution.

  • a distributed algorithm for solving a Linear Algebraic Equation
    2015
    Co-Authors: Shaoshuai Mou, Ji Liu, Stephen A Morse
    Abstract:

    A distributed algorithm is described for solving a Linear Algebraic Equation of the form $Ax=b$ assuming the Equation has at least one solution. The Equation is simultaneously solved by $m$ agents assuming each agent knows only a subset of the rows of the partitioned matrix $(A,b)$, the current estimates of the Equation's solution generated by its neighbors, and nothing more. Each agent recursively updates its estimate by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a time-dependent directed graph $\mathbb{N}(t)$ whose vertices correspond to agents and whose arcs depict neighbor relations. It is shown that for any matrix $A$ for which the Equation has a solution and any sequence of "repeatedly jointly strongly connected graphs" $\mathbb{N}(t)$, $t=1,2,\ldots$, the algorithm causes all agents' estimates to converge exponentially fast to the same solution to $Ax=b$. It is also shown that the neighbor graph sequence must actually be repeatedly jointly strongly connected if exponential convergence is to be assured. A worst case convergence rate bound is derived for the case when $Ax=b$ has a unique solution. It is demonstrated that with minor modification, the algorithm can track the solution to $Ax = b$, even if $A$ and $b$ are changing with time, provided the rates of change of $A$ and $b$ are sufficiently small. It is also shown that in the absence of communication delays, exponential convergence to a solution occurs even if the times at which each agent updates its estimates are not synchronized with the update times of its neighbors. A modification of the algorithm is outlined which enables it to obtain a least squares solution to $Ax=b$ in a distributed manner, even if $Ax=b$ does not have a solution.

  • an asynchronous distributed algorithm for solving a Linear Algebraic Equation
    2013
    Co-Authors: Ji Liu, Shaoshuai Mou, Stephen A Morse
    Abstract:

    A distributed algorithm is described for solving a Linear Algebraic Equation of the form Ax = b where A is a matrix for which the Equation has at least one solution. The Equation is simultaneously and asynchronously solved by m agents assuming each agent knows only a subset of the rows of the partitioned matrix [A b], the estimates of the Equation's solution generated by its neighbors, and nothing more. Each agent recursively updates its estimate of a solution at its own event times by utilizing estimates generated by each of its neighbors which are transmitted with delays. Each agent has its own event time sequence and the event time sequences of different agents are not assumed to be synchronized. Neighbor relations are characterized by a time-dependent directed graph whose vertices correspond to agents and whose arcs depict neighbor relations. It is shown that for any matrix A for which the Equation has a solution and any repeatedly jointly strongly connected sequence of neighbor graphs defined on the merged sequence of all agents' event times, the algorithm causes all agents' estimates to converge exponentially fast to the same solution to Ax = b.

  • a distributed algorithm for solving a Linear Algebraic Equation
    2013
    Co-Authors: Shaoshuai Mou, Ji Liu, A S Morse
    Abstract:

    A distributed algorithm is described for solving a Linear Algebraic Equation of the form Ax = b where A is a matrix for which the Equation has at least one solution. The Equation is simultaneously solved by m agents assuming each agent knows only a subset of the rows of the partitioned matrix [A b], the current estimates of the Equation's solution generated by its neighbors, and nothing more. Each agent recursively updates its estimate of a solution by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a time-dependent directed graph N(t) whose vertices correspond to agents and whose arcs depict neighbor relations. It is shown that for any matrix A for which the Equation has a solution and any sequence of “repeatedly jointly strongly connected graphs” N(t), t = 1, 2, ..., the algorithm causes all agents' estimates to converge exponentially fast to the same solution to Ax = b. It is also shown that in the absence of transmission delays, convergence to a solution occurs even if the times at which each agent updates its estimates are not synchronized with the update times of its neighbors.

  • a fixed neighbor distributed algorithm for solving a Linear Algebraic Equation
    2013
    Co-Authors: Shaoshuai Mou, A S Morse
    Abstract:

    This paper presents a distributed algorithm for solving a Linear Algebraic Equation of the form Ax = b where A is an n × n nonsingular matrix and b is an n-vector. The Equation is solved by a network of n agents assuming that each agent knows exactly one distinct row of the partitioned matrix [A b], the current estimates of the Equation's solution generated by its neighbors, and nothing more. Each agent recursively updates its estimate of A-1b by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a simple, undirected graph G whose vertices correspond to agents and whose edges depict neighbor relations. It is shown that for any nonsingular matrix A and any connected graph G, the proposed algorithm causes all agents' estimates to converge exponentially fast to the desired solution A-1b.

A S Morse - One of the best experts on this subject based on the ideXlab platform.

  • a distributed algorithm for solving a Linear Algebraic Equation
    2013
    Co-Authors: Shaoshuai Mou, Ji Liu, A S Morse
    Abstract:

    A distributed algorithm is described for solving a Linear Algebraic Equation of the form Ax = b where A is a matrix for which the Equation has at least one solution. The Equation is simultaneously solved by m agents assuming each agent knows only a subset of the rows of the partitioned matrix [A b], the current estimates of the Equation's solution generated by its neighbors, and nothing more. Each agent recursively updates its estimate of a solution by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a time-dependent directed graph N(t) whose vertices correspond to agents and whose arcs depict neighbor relations. It is shown that for any matrix A for which the Equation has a solution and any sequence of “repeatedly jointly strongly connected graphs” N(t), t = 1, 2, ..., the algorithm causes all agents' estimates to converge exponentially fast to the same solution to Ax = b. It is also shown that in the absence of transmission delays, convergence to a solution occurs even if the times at which each agent updates its estimates are not synchronized with the update times of its neighbors.

  • a fixed neighbor distributed algorithm for solving a Linear Algebraic Equation
    2013
    Co-Authors: Shaoshuai Mou, A S Morse
    Abstract:

    This paper presents a distributed algorithm for solving a Linear Algebraic Equation of the form Ax = b where A is an n × n nonsingular matrix and b is an n-vector. The Equation is solved by a network of n agents assuming that each agent knows exactly one distinct row of the partitioned matrix [A b], the current estimates of the Equation's solution generated by its neighbors, and nothing more. Each agent recursively updates its estimate of A-1b by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a simple, undirected graph G whose vertices correspond to agents and whose edges depict neighbor relations. It is shown that for any nonsingular matrix A and any connected graph G, the proposed algorithm causes all agents' estimates to converge exponentially fast to the desired solution A-1b.

Stephen A Morse - One of the best experts on this subject based on the ideXlab platform.

  • a distributed algorithm with an arbitrary initialization for solving a Linear Algebraic Equation
    2016
    Co-Authors: Lili Wang, Daniel Fullmer, Stephen A Morse
    Abstract:

    A consensus-based distributed algorithm is described for solving a Linear Algebraic Equation. Unlike a previously proposed algorithm for solving the same problem, the one proposed in this paper does so with any arbitrary initialization. In addition, the algorithm contains design parameters which may be chosen to optimize convergence rate.

  • a distributed algorithm for solving a Linear Algebraic Equation
    2015
    Co-Authors: Shaoshuai Mou, Ji Liu, Stephen A Morse
    Abstract:

    A distributed algorithm is described for solving a Linear Algebraic Equation of the form $Ax = b$ assuming the Equation has at least one solution. The Equation is simultaneously solved by $m$ agents assuming each agent knows only a subset of the rows of the partitioned matrix $[\matrix{A & b}]$ , the current estimates of the Equation's solution generated by its neighbors, and nothing more. Each agent recursively updates its estimate by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a time-dependent directed graph $\BBN(t)$ whose vertices correspond to agents and whose arcs depict neighbor relations. It is shown that for any matrix $A$ for which the Equation has a solution and any sequence of “repeatedly jointly strongly connected graphs” $\BBN(t)$ , $t = 1, 2, \ldots$ , the algorithm causes all agents' estimates to converge exponentially fast to the same solution to $Ax = b$ . It is also shown that, under mild assumptions, the neighbor graph sequence must actually be repeatedly jointly strongly connected if exponential convergence is to be assured. A worst case convergence rate bound is derived for the case when $Ax = b$ has a unique solution. It is demonstrated that with minor modification, the algorithm can track the solution to $Ax = b$ , even if $A$ and $b$ are changing with time, provided the rates of change of $A$ and $b$ are sufficiently small. It is also shown that in the absence of communication delays, exponential convergence to a solution occurs even if the times at which each agent updates its estimates are not synchronized with the update times of its neighbors. A modification of the algorithm is outlined which enables it to obtain a least squares solution to $Ax = b$ in a distributed manner, even if $Ax = b$ does not have a solution.

  • a distributed algorithm for solving a Linear Algebraic Equation
    2015
    Co-Authors: Shaoshuai Mou, Ji Liu, Stephen A Morse
    Abstract:

    A distributed algorithm is described for solving a Linear Algebraic Equation of the form $Ax=b$ assuming the Equation has at least one solution. The Equation is simultaneously solved by $m$ agents assuming each agent knows only a subset of the rows of the partitioned matrix $(A,b)$, the current estimates of the Equation's solution generated by its neighbors, and nothing more. Each agent recursively updates its estimate by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a time-dependent directed graph $\mathbb{N}(t)$ whose vertices correspond to agents and whose arcs depict neighbor relations. It is shown that for any matrix $A$ for which the Equation has a solution and any sequence of "repeatedly jointly strongly connected graphs" $\mathbb{N}(t)$, $t=1,2,\ldots$, the algorithm causes all agents' estimates to converge exponentially fast to the same solution to $Ax=b$. It is also shown that the neighbor graph sequence must actually be repeatedly jointly strongly connected if exponential convergence is to be assured. A worst case convergence rate bound is derived for the case when $Ax=b$ has a unique solution. It is demonstrated that with minor modification, the algorithm can track the solution to $Ax = b$, even if $A$ and $b$ are changing with time, provided the rates of change of $A$ and $b$ are sufficiently small. It is also shown that in the absence of communication delays, exponential convergence to a solution occurs even if the times at which each agent updates its estimates are not synchronized with the update times of its neighbors. A modification of the algorithm is outlined which enables it to obtain a least squares solution to $Ax=b$ in a distributed manner, even if $Ax=b$ does not have a solution.

  • an asynchronous distributed algorithm for solving a Linear Algebraic Equation
    2013
    Co-Authors: Ji Liu, Shaoshuai Mou, Stephen A Morse
    Abstract:

    A distributed algorithm is described for solving a Linear Algebraic Equation of the form Ax = b where A is a matrix for which the Equation has at least one solution. The Equation is simultaneously and asynchronously solved by m agents assuming each agent knows only a subset of the rows of the partitioned matrix [A b], the estimates of the Equation's solution generated by its neighbors, and nothing more. Each agent recursively updates its estimate of a solution at its own event times by utilizing estimates generated by each of its neighbors which are transmitted with delays. Each agent has its own event time sequence and the event time sequences of different agents are not assumed to be synchronized. Neighbor relations are characterized by a time-dependent directed graph whose vertices correspond to agents and whose arcs depict neighbor relations. It is shown that for any matrix A for which the Equation has a solution and any repeatedly jointly strongly connected sequence of neighbor graphs defined on the merged sequence of all agents' event times, the algorithm causes all agents' estimates to converge exponentially fast to the same solution to Ax = b.

Uwe Helmke - One of the best experts on this subject based on the ideXlab platform.

  • network flows that solve Linear Equations
    2017
    Co-Authors: Brian D O Anderson, Uwe Helmke
    Abstract:

    We study distributed network flows as solvers in continuous time for the Linear Algebraic Equation $\mathbf{z}=\mathbf{H}\mathbf{y}$ . Each node $i$ has access to a row ${\mathbf{h}}_{i}^{\mathrm{T}}$ of the matrix $\mathbf{H}$ and the corresponding entry $z_i$ in the vector $\mathbf{z}$ . The first “consensus + projection” flow under investigation consists of two terms, one from standard consensus dynamics and the other contributing to projection onto each affine subspace specified by the $\mathbf{h}_{i}$ and $z_i$ . The second “projection consensus” flow on the other hand simply replaces the relative state feedback in consensus dynamics with projected relative state feedback. Without dwell-time assumption on switching graphs, we prove that all node states converge to a common solution of the Linear Algebraic Equation, if there is any. The convergence is global for the “consensus + projection” flow while local for the “projection consensus” flow in the sense that the initial values must lie on the affine subspaces. If the Linear Equation has no exact solutions, we show that the node states can converge to a ball around the least-squares solution whose radius can be made arbitrarily small through selecting a sufficiently large gain for the “consensus + projection” flow for a fixed bidirectional graph. Semi-global convergence to approximate least-squares solutions is also demonstrated for switching balanced directed graphs under suitable conditions. It is also shown that the “projection consensus” flow drives the average of the node states to the least-squares solution with a complete graph. Numerical examples are provided as illustrations of the established results.

  • network flows that solve Linear Equations
    2015
    Co-Authors: Brian D O Anderson, Uwe Helmke
    Abstract:

    We study distributed network flows as solvers in continuous time for the Linear Algebraic Equation $\mathbf{z}=\mathbf{H}\mathbf{y}$. Each node $i$ has access to a row $\mathbf{h}_i^{\rm T}$ of the matrix $\mathbf{H}$ and the corresponding entry $z_i$ in the vector $\mathbf{z}$. The first "consensus + projection" flow under investigation consists of two terms, one from standard consensus dynamics and the other contributing to projection onto each affine subspace specified by the $\mathbf{h}_i$ and $z_i$. The second "projection consensus" flow on the other hand simply replaces the relative state feedback in consensus dynamics with projected relative state feedback. Without dwell-time assumption on switching graphs as well as without positively lower bounded assumption on arc weights, we prove that all node states converge to a common solution of the Linear Algebraic Equation, if there is any. The convergence is global for the "consensus + projection" flow while local for the "projection consensus" flow in the sense that the initial values must lie on the affine subspaces. If the Linear Equation has no exact solutions, we show that the node states can converge to a ball around the least squares solution whose radius can be made arbitrarily small through selecting a sufficiently large gain for the "consensus + projection" flow under fixed bidirectional graphs. Semi-global convergence to approximate least squares solutions is demonstrated for general switching directed graphs under suitable conditions. It is also shown that the "projection consensus" flow drives the average of the node states to the least squares solution with complete graph. Numerical examples are provided as illustrations of the established results.

Ramesh B Kudenatti - One of the best experts on this subject based on the ideXlab platform.

  • impact of temperature dependant viscosity and thermal conductivity on mhd boundary layer flow of two phase dusty fluid through permeable medium
    2019
    Co-Authors: G Kalpana, K R Madhura, Ramesh B Kudenatti
    Abstract:

    Abstract The distribution of solid particles in a fluid leading to two-phase nature as in micro-propulsion, aerosol filtration, conveying of powdered materials, petroleum industry and lunar ash flow can be described as an open question. In this study, boundary layer flow of an electrically conducting magnetohydrodynamic dusty fluid in a porous medium is presented. Fluid viscosity and thermal conductivity are assumed to be an inverse Linear function of temperature. The governing nonLinear partial differential Equations and their physically realistic boundary conditions are reduced into dimensionless form by using appropriate transformations. The resultant coupled system of Equations is discretized using finite difference scheme and Thomas algorithm is implemented to solve the Linear Algebraic Equation. A representative set of numerical results are plotted to visualize the impact of existing fluid interaction parameters. Skin-friction, heat and mass transfer coefficients are tabulated and results obtained are in good agreement with the literature. From this investigation, velocity, temperature and concentration fields are observed to be an increasing function in accordance with the raise in both viscosity variation parameter ( θ r ) and thermal conductivity variation parameter ( θ c ). This computation illustrates that the friction factor coefficient is found to be less for low thermal expansion coefficient irrespective of all emerging parameters of the flow. Furthermore, enhancement in θ r reduces the heat transfer rate and strengthens the mass transfer rate whereas the reverse trend is observed for θ c .

  • Impact of temperature-dependant viscosity and thermal conductivity on MHD boundary layer flow of two-phase dusty fluid through permeable medium
    2019
    Co-Authors: G Kalpana, K R Madhura, Ramesh B Kudenatti
    Abstract:

    The distribution of solid particles in a fluid leading to two-phase nature as in micro-propulsion, aerosol filtration, conveying of powdered materials, petroleum industry and lunar ash flow can be described as an open question. In this study, boundary layer flow of an electrically conducting magnetohydrodynamic dusty fluid in a porous medium is presented. Fluid viscosity and thermal conductivity are assumed to be an inverse Linear function of temperature. The governing nonLinear partial differential Equations and their physically realistic boundary conditions are reduced into dimensionless form by using appropriate transformations. The resultant coupled system of Equations is discretized using finite difference scheme and Thomas algorithm is implemented to solve the Linear Algebraic Equation. A representative set of numerical results are plotted to visualize the impact of existing fluid interaction parameters. Skin-friction, heat and mass transfer coefficients are tabulated and results obtained are in good agreement with the literature. From this investigation, velocity, temperature and concentration fields are observed to be an increasing function in accordance with the raise in both viscosity variation parameter (θr) and thermal conductivity variation parameter (θc). This computation illustrates that the friction factor coefficient is found to be less for low thermal expansion coefficient irrespective of all emerging parameters of the flow. Furthermore, enhancement in θr reduces the heat transfer rate and strengthens the mass transfer rate whereas the reverse trend is observed for θc. Keywords: Boundary layer, Dusty fluid, Variable viscosity, Variable thermal conductivity, Heat and mass transfe