A relatively new preconditioning technique called support graph preconditioning has many merits over the traditional incomplete factorization based methods. A major limitation of this technique is that it is applicable to symmetric diagonally dominant matrices only. This work presents a technique that can be used to transform the symmetric positive definite matrices arising from elliptic finite element problems into symmetric diagonally dominant M-matrices. The basic idea is to approximate the element gradient matrix by taking the gradients along chosen edges, whose unit vectors form a new coordinate system. For Lagrangian elements, the rows of the element gradient matrix in this new coordinate system are scaled edge vectors, thus a diagonally dominant symmetric semidefinite M-matrix can be generated to approximate the element stiffness matrix. Depending on the element type, one or more such coordinate systems are required to obtain a global nonsingular M-matrix. Since such approximation takes place at the element level, the degradation in the quality of the preconditioner is only a small constant factor independent of the size of the problem. This technique of element coordinate transformations applies to a variety of first order Lagrangian elements. Combination of this technique and other techniques enables us to construct an M-matrix preconditioner for a wide range of second order elliptic problems even with higher order elements. Another contribution of this work is the proposal of a new variant of Vaidya’s support graph preconditioning technique called modified domain partitioned support graph preconditioners. Numerical experiments are conducted for various second order elliptic finite element problems, along with performance comparison to the incomplete factorization based preconditioners. Results show that these support graph preconditioners are superior when solving ill-conditioned problems. In addition, the domain partition feature provides inherent parallelism, and initial experiments show a good potential of parallelization and scalability of these preconditioners.

This thesis studies the Reach Control Problem (RCP) for affine systems defined on simplices. The thesis focuses on cases when it is known that the problem is not solvable by continuous state feedback. Previous work has proposed (discontinuous) piecewise affine feedback to resolve the gap between solvability by open-loop controls and solvability by feedbacks. The first results on solvability by time-varying feedback are presented. Time-varying feedback has the advantage to be more robust to measurement errors circumventing problems of discontinuous controllers. The results are theoretically appealing in light of the strong analogies with the theory of stabilization for linear control systems. The method is shown to solve RCP for all cases in the literature where continuous state feedback fails, provided it is solvable by open loop control. Textbook examples are provided. The motivation for studying RCP and its relevance to complex control specifications is illustrated using a material transfer system.

This thesis studies the Reach Control Problem (RCP) for affine systems defined on simplices. The thesis focuses on cases when it is known that the problem is not solvable by continuous state feedback. Previous work has proposed (discontinuous) piecewise affine feedback to resolve the gap between solvability by open-loop controls and solvability by feedbacks. The first results on solvability by time-varying feedback are presented. Time-varying feedback has the advantage to be more robust to measurement errors circumventing problems of discontinuous controllers. The results are theoretically appealing in light of the strong analogies with the theory of stabilization for linear control systems. The method is shown to solve RCP for all cases in the literature where continuous state feedback fails, provided it is solvable by open loop control. Textbook examples are provided. The motivation for studying RCP and its relevance to complex control specifications is illustrated using a material transfer system.

Universiẗat, Diss., 2001--Osnabrück.

Universiẗat, Diss., 2004--Osnabrück.

Seja Uln sistema econômico, que envolve n indústrias interdependentes tais que cada indústria produz um único tipo de artigo. Denotemos com t ij a quantidade da entrada (insumo) da iêsima mercadoria que a economia necessita para produzir uma unidade da mercadoria} de saída (produto). A matriz T := [ tlj ] de insumo-produto de Leontief é uma matriz não-negativa. Descreveremos as propriedades das matrizes não-negativas, necessárias para uma análise matemática do modelo de Leontief. Se esse modelo descreve uma economia viável, a soma dos elementos em cada coluna de T será menor ou igual a l. Suponhamos mais que o sistema econômico modelado contenha um setor aberto, onde trabalho, lucro, etc. entram como segue. Seja x, o produto total que a indústria i requer para atender à demanda do setor aberto e das n indústrias. Então x = Tx + d, onde d := [ d,] é o vetor das demandas, isto é, d; é a demanda do s~tor aberto sobre a indústria iésúna. Aqui l;JXj representa o insumo que a j ésima indústria necessita da i•s•ma indústria. Os níveis de produção requeridos pela totalidade das n indústrias, a fim de poder atender a essas demandas, constituem o vetor-solução do sistema linear Ax = d, com A := I- T. Como a soma dos elementos de cada coluna de T é menor ou igual a I; o raio espectral de T também é menor ou igual a 1. Quando o raio espectral é menor que 1, T é convergente e A tem um inversa com todos os elementos não-negativos (matriz não-negativa). Discutiremos as matrizes não-negativas. Além disso, os elementos não-diagonais de A := I - T são todos negativos ou nulos. Matrizes com esse quadro de sinais, cujas inversas são não-negativas, são ditas matrizes-M não-singulares. Discutjremos também as matrizes-M não-singulares e singulares. O objetivo principal deste trabalho é a apl icação interessante da teoria das matrizes nãonegativas e matrizes-M, na análise do modelo de Leontief descrito muito brevemente acima, resultando um método elegante de análise de insumo-produto. / Let us consid~r an economic system, that involves n interdependent industries, assuming that each industry produces only one type of commodities. Let tij denote the amount of input ofthe ith commodity needed by the economy to produce a unit output o f commodity j. The Leontief input-output matrix T := [ tij] is a nonnegative matrix. We will describe the properties of nonnegative matrices, necessary for a mathematical analysis ofthe Leontiefs model. Ifthat model describes an economically feasible situation, the sum of the elements in each column of T does not exceed I. Let us further suppose that the modeled economic system contains an open sector, where labor, profit, etc. enter in the following way. Let x, be the total output o f the industry i required to meet the demand o f the open sector and ali n industries. Then x = Tx + d, where d := [ d; ], is the vector ofthe demands, that is, d; is the demand of the open sector from the ith industry. Here li]Xj represents the input requirement of the jth industry from the ith. The output leveis required o f the totality o f the n industries, in order to meet these demands, are the solution vector x ofthe linear system Ax = d, with A :=I- T. As the sum ofthe elements of each column ofT is at most I, it follows that the spectral radius ofT is also at most I. When the spectral radius is less than 1, T is convergent and A is inverse-positive, that is, A'1 is a nonnegative matrix. We will discuss the nonnegative matrices. Besides, A:= I - T has ali its off-diagonal entries nonpositive. Jnverse-positive matrices with this sign pattem are called nonsingular M-matrices. We will also discuss nonsingular and singular M-matrices. The main goal of this work is the interesting appl ication of the nonnegative matrices and M-matrices theory to the analysis ofthe Leontiefs model, described very shortly above, resulting in an elegant method o f input-output analysis.

Seja Uln sistema econômico, que envolve n indústrias interdependentes tais que cada indústria produz um único tipo de artigo. Denotemos com t ij a quantidade da entrada (insumo) da iêsima mercadoria que a economia necessita para produzir uma unidade da mercadoria} de saída (produto). A matriz T := [ tlj ] de insumo-produto de Leontief é uma matriz não-negativa. Descreveremos as propriedades das matrizes não-negativas, necessárias para uma análise matemática do modelo de Leontief. Se esse modelo descreve uma economia viável, a soma dos elementos em cada coluna de T será menor ou igual a l. Suponhamos mais que o sistema econômico modelado contenha um setor aberto, onde trabalho, lucro, etc. entram como segue. Seja x, o produto total que a indústria i requer para atender à demanda do setor aberto e das n indústrias. Então x = Tx + d, onde d := [ d,] é o vetor das demandas, isto é, d; é a demanda do s~tor aberto sobre a indústria iésúna. Aqui l;JXj representa o insumo que a j ésima indústria necessita da i•s•ma indústria. Os níveis de produção requeridos pela totalidade das n indústrias, a fim de poder atender a essas demandas, constituem o vetor-solução do sistema linear Ax = d, com A := I- T. Como a soma dos elementos de cada coluna de T é menor ou igual a I; o raio espectral de T também é menor ou igual a 1. Quando o raio espectral é menor que 1, T é convergente e A tem um inversa com todos os elementos não-negativos (matriz não-negativa). Discutiremos as matrizes não-negativas. Além disso, os elementos não-diagonais de A := I - T são todos negativos ou nulos. Matrizes com esse quadro de sinais, cujas inversas são não-negativas, são ditas matrizes-M não-singulares. Discutjremos também as matrizes-M não-singulares e singulares. O objetivo principal deste trabalho é a apl icação interessante da teoria das matrizes nãonegativas e matrizes-M, na análise do modelo de Leontief descrito muito brevemente acima, resultando um método elegante de análise de insumo-produto. / Let us consid~r an economic system, that involves n interdependent industries, assuming that each industry produces only one type of commodities. Let tij denote the amount of input ofthe ith commodity needed by the economy to produce a unit output o f commodity j. The Leontief input-output matrix T := [ tij] is a nonnegative matrix. We will describe the properties of nonnegative matrices, necessary for a mathematical analysis ofthe Leontiefs model. Ifthat model describes an economically feasible situation, the sum of the elements in each column of T does not exceed I. Let us further suppose that the modeled economic system contains an open sector, where labor, profit, etc. enter in the following way. Let x, be the total output o f the industry i required to meet the demand o f the open sector and ali n industries. Then x = Tx + d, where d := [ d; ], is the vector ofthe demands, that is, d; is the demand of the open sector from the ith industry. Here li]Xj represents the input requirement of the jth industry from the ith. The output leveis required o f the totality o f the n industries, in order to meet these demands, are the solution vector x ofthe linear system Ax = d, with A :=I- T. As the sum ofthe elements of each column ofT is at most I, it follows that the spectral radius ofT is also at most I. When the spectral radius is less than 1, T is convergent and A is inverse-positive, that is, A'1 is a nonnegative matrix. We will discuss the nonnegative matrices. Besides, A:= I - T has ali its off-diagonal entries nonpositive. Jnverse-positive matrices with this sign pattem are called nonsingular M-matrices. We will also discuss nonsingular and singular M-matrices. The main goal of this work is the interesting appl ication of the nonnegative matrices and M-matrices theory to the analysis ofthe Leontiefs model, described very shortly above, resulting in an elegant method o f input-output analysis.

Seja Uln sistema econômico, que envolve n indústrias interdependentes tais que cada indústria produz um único tipo de artigo. Denotemos com t ij a quantidade da entrada (insumo) da iêsima mercadoria que a economia necessita para produzir uma unidade da mercadoria} de saída (produto). A matriz T := [ tlj ] de insumo-produto de Leontief é uma matriz não-negativa. Descreveremos as propriedades das matrizes não-negativas, necessárias para uma análise matemática do modelo de Leontief. Se esse modelo descreve uma economia viável, a soma dos elementos em cada coluna de T será menor ou igual a l. Suponhamos mais que o sistema econômico modelado contenha um setor aberto, onde trabalho, lucro, etc. entram como segue. Seja x, o produto total que a indústria i requer para atender à demanda do setor aberto e das n indústrias. Então x = Tx + d, onde d := [ d,] é o vetor das demandas, isto é, d; é a demanda do s~tor aberto sobre a indústria iésúna. Aqui l;JXj representa o insumo que a j ésima indústria necessita da i•s•ma indústria. Os níveis de produção requeridos pela totalidade das n indústrias, a fim de poder atender a essas demandas, constituem o vetor-solução do sistema linear Ax = d, com A := I- T. Como a soma dos elementos de cada coluna de T é menor ou igual a I; o raio espectral de T também é menor ou igual a 1. Quando o raio espectral é menor que 1, T é convergente e A tem um inversa com todos os elementos não-negativos (matriz não-negativa). Discutiremos as matrizes não-negativas. Além disso, os elementos não-diagonais de A := I - T são todos negativos ou nulos. Matrizes com esse quadro de sinais, cujas inversas são não-negativas, são ditas matrizes-M não-singulares. Discutjremos também as matrizes-M não-singulares e singulares. O objetivo principal deste trabalho é a apl icação interessante da teoria das matrizes nãonegativas e matrizes-M, na análise do modelo de Leontief descrito muito brevemente acima, resultando um método elegante de análise de insumo-produto. / Let us consid~r an economic system, that involves n interdependent industries, assuming that each industry produces only one type of commodities. Let tij denote the amount of input ofthe ith commodity needed by the economy to produce a unit output o f commodity j. The Leontief input-output matrix T := [ tij] is a nonnegative matrix. We will describe the properties of nonnegative matrices, necessary for a mathematical analysis ofthe Leontiefs model. Ifthat model describes an economically feasible situation, the sum of the elements in each column of T does not exceed I. Let us further suppose that the modeled economic system contains an open sector, where labor, profit, etc. enter in the following way. Let x, be the total output o f the industry i required to meet the demand o f the open sector and ali n industries. Then x = Tx + d, where d := [ d; ], is the vector ofthe demands, that is, d; is the demand of the open sector from the ith industry. Here li]Xj represents the input requirement of the jth industry from the ith. The output leveis required o f the totality o f the n industries, in order to meet these demands, are the solution vector x ofthe linear system Ax = d, with A :=I- T. As the sum ofthe elements of each column ofT is at most I, it follows that the spectral radius ofT is also at most I. When the spectral radius is less than 1, T is convergent and A is inverse-positive, that is, A'1 is a nonnegative matrix. We will discuss the nonnegative matrices. Besides, A:= I - T has ali its off-diagonal entries nonpositive. Jnverse-positive matrices with this sign pattem are called nonsingular M-matrices. We will also discuss nonsingular and singular M-matrices. The main goal of this work is the interesting appl ication of the nonnegative matrices and M-matrices theory to the analysis ofthe Leontiefs model, described very shortly above, resulting in an elegant method o f input-output analysis.

In the first part of this thesis the spectrum of a matrix operator is determined. For this the coefficients of the matrix operator are assumed to satisfy rather general properties which combine smoothness and decay. With this the asymptotics of the eigenfunctions can be determined. This in turn leads to properties of the spectra with the aid of the M-matrix. In the second part it will be shown that if a discrete canonical system has absolutely continuous spectrum of a certain multiplicity, then there is a corresponding number of linearly independent solutions y which are bounded in a weak sense.

In this thesis the spectral properties of differential operators generated by the formally self-adjoint differential expression Τy = w⁻₁[(ry″)″ - (py′)′ + qy] are investigated. The main tools to be used are the theory of asymptotic integration and the Titchmarsh--Weyl M-matrix. Subject to certain regularity conditions on the coefficients asymptotic integration leads to estimates for the eigenfunctions of the corresponding differential equation Τy = zy. According to the theory of asymptotic integration the regularity conditions combine smoothness with decay, i.e. admissible coefficients are (in an appropriate sense) either short range or slowly varying. Knowledge of the asymptotics (x → ∞) of the solutions will then be used to determine the deficiency index and to derive properties of the M-matrix which is closely related to the spectral measure of an associated self-adjoint realization Τ. Consequently we can compute the multiplicity of the spectrum, locate the absolutely continuous spectrum and give conditions for the singular continuous spectrum to be empty. This generalizes classical results on second order operators.