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In this report, the recent and future contributions of experts and groups of Delft University of Technology (DUT) on the topic of smart grids (SG) are presented. This may provide guidelines on how DUT can contribute in current and future SG projects, such as the current TKI project on “Smart Grid Evolution (SGE): Developing a service platform for the Internet of Energy”. The SGE project is meant to prepare for the realization of a large-scale demonstration case in Netherlands, starting in June 2016. In this demonstration case, a smart grid with at least 100.000 connections will be constructed.1 The 3TU.Federation is involved to contribute to the development of a collaborative design and decision support environment based on state-of-the-art knowledge of smart grid concepts with appropriate tool support. The representatives of DUT are Jok Tang, Cees Witteveen and Kees Vuik. In the groups headed by Cees Witteveen and Kees Vuik, the topic of smart grids is under investigation. Some of the members of these groups are interviewed, and their input is the basis of this document. Also several experts from other groups and faculties of DUT are interviewed. The list of interviewed people is not exhaustive, but it can be interpreted as an overview of the total SG expertise that DUT possesses., Delft Institute of Applied Mathematics, Electrical Engineering, Mathematics and Computer Science

In this report, the recent and future contributions of experts and groups of Delft University of Technology (DUT) on the topic of smart grids (SG) are presented. This may provide guidelines on how DUT can contribute in current and future SG projects, such as the current TKI project on “Smart Grid Evolution (SGE): Developing a service platform for the Internet of Energy”. The SGE project is meant to prepare for the realization of a large-scale demonstration case in Netherlands, starting in June 2016. In this demonstration case, a smart grid with at least 100.000 connections will be constructed.1 The 3TU.Federation is involved to contribute to the development of a collaborative design and decision support environment based on state-of-the-art knowledge of smart grid concepts with appropriate tool support. The representatives of DUT are Jok Tang, Cees Witteveen and Kees Vuik. In the groups headed by Cees Witteveen and Kees Vuik, the topic of smart grids is under investigation. Some of the members of these groups are interviewed, and their input is the basis of this document. Also several experts from other groups and faculties of DUT are interviewed. The list of interviewed people is not exhaustive, but it can be interpreted as an overview of the total SG expertise that DUT possesses., Delft Institute of Applied Mathematics, Electrical Engineering, Mathematics and Computer Science

We consider the solution of large and sparse linear systems of equations by GMRES. Due to the appearance of unfavorable eigenvalues in the spectrum of the coefficient matrix, the convergence of GMRES may hamper. To overcome this, a deflated variant of GMRES can be used, which treats those unfavorable eigenvalues effectively. In the literature, several deflated GMRES variants are applied successfully to various problems, while a theoretical justification is often lacking. In contrast to deflated CG, the convergence of deflated GMRES seems to be harder to analyze and to understand. This paper presents some new theoretical insights into deflated GMRES based on A-invariant deflation subspaces. Fundamental results regarding the convergence of deflated GMRES are proved in order to show the effectiveness and robustness of this method. Numerical experiments are provided to illustrate the theoretical results and to show some further properties of deflated GMRES. Consequently, practical variants of deflated GMRES from the literature can be better understood based on the results presented in this paper., Electrical Engineering, Mathematics and Computer Science

It is well known that two-level and multilevel preconditioned conjugate gradient (PCG) methods provide efficient techniques for solving large and sparse linear systems whose coefficient matrices are symmetric and positive definite. A two-level PCG method combines a traditional (one-level) preconditioner, such as incomplete Cholesky, with a projection-type preconditioner to get rid of the effect of both small and large eigenvalues of the coefficient matrix; multilevel approaches arise by recursively applying the two-level technique within the projection step. In the literature, various such preconditioners are known, coming from the fields of deflation, domain decomposition, and multigrid (MG). At first glance, these methods seem to be quite distinct; however, from an abstract point of view, they are closely related. The aim of this paper is to relate two-level PCG methods with symmetric two-grid (V(1,1)-cycle) preconditioners (derived from MG approaches), in their abstract form, to deflation methods and a two-level domain-decomposition approach inspired by the balancing Neumann–Neumann method. The MG-based preconditioner is often expected to be more effective than these other two-level preconditioners, but this is shown to be not always true. For common choices of the parameters, MG leads to larger error reductions in each iteration, but the work per iteration is more expensive, which makes this comparison unfair. We show that, for special choices of the underlying one-level preconditioners in the deflation or domain-decomposition methods, the work per iteration of these preconditioners is approximately the same as that for the MG preconditioner, and the convergence properties of the resulting two-level PCG methods will also be (approximately) the same. This means that, in this respect, the particular choice of the two-level preconditioner is less important than the choice of the parameters. Numerical experiments are presented to emphasize the theoretical results., Delft Institute of Applied Mathematics, Electrical Engineering, Mathematics and Computer Science

We consider the solution of large and sparse linear systems of equations by GMRES. Due to the appearance of unfavorable eigenvalues in the spectrum of the coefficient matrix, the convergence of GMRES may hamper. To overcome this, a deflated variant of GMRES can be used, which treats those unfavorable eigenvalues effectively. In the literature, several deflated GMRES variants are applied successfully to various problems, while a theoretical justification is often lacking. In contrast to deflated CG, the convergence of deflated GMRES seems to be harder to analyze and to understand. This paper presents some new theoretical insights into deflated GMRES based on A-invariant deflation subspaces. Fundamental results regarding the convergence of deflated GMRES are proved in order to show the effectiveness and robustness of this method. Numerical experiments are provided to illustrate the theoretical results and to show some further properties of deflated GMRES. Consequently, practical variants of deflated GMRES from the literature can be better understood based on the results presented in this paper., Electrical Engineering, Mathematics and Computer Science

It is well known that two-level and multilevel preconditioned conjugate gradient (PCG) methods provide efficient techniques for solving large and sparse linear systems whose coefficient matrices are symmetric and positive definite. A two-level PCG method combines a traditional (one-level) preconditioner, such as incomplete Cholesky, with a projection-type preconditioner to get rid of the effect of both small and large eigenvalues of the coefficient matrix; multilevel approaches arise by recursively applying the two-level technique within the projection step. In the literature, various such preconditioners are known, coming from the fields of deflation, domain decomposition, and multigrid (MG). At first glance, these methods seem to be quite distinct; however, from an abstract point of view, they are closely related. The aim of this paper is to relate two-level PCG methods with symmetric two-grid (V(1,1)-cycle) preconditioners (derived from MG approaches), in their abstract form, to deflation methods and a two-level domain-decomposition approach inspired by the balancing Neumann–Neumann method. The MG-based preconditioner is often expected to be more effective than these other two-level preconditioners, but this is shown to be not always true. For common choices of the parameters, MG leads to larger error reductions in each iteration, but the work per iteration is more expensive, which makes this comparison unfair. We show that, for special choices of the underlying one-level preconditioners in the deflation or domain-decomposition methods, the work per iteration of these preconditioners is approximately the same as that for the MG preconditioner, and the convergence properties of the resulting two-level PCG methods will also be (approximately) the same. This means that, in this respect, the particular choice of the two-level preconditioner is less important than the choice of the parameters. Numerical experiments are presented to emphasize the theoretical results., Delft Institute of Applied Mathematics, Electrical Engineering, Mathematics and Computer Science

For various applications, it is well-known that a multi-level, in particular two-level, preconditioned CG (PCG) method is an efficient method for solving large and sparse linear systems with a coefficient matrix that is symmetric positive definite. The corresponding two-level preconditioner combines traditional and projection-type preconditioners to get rid of the effect of both small and large eigenvalues of the coefficient matrix. In the literature, various two-level PCG methods are known, coming from the fields of deflation, domain decomposition and multigrid. Even though these two-level methods differ a lot in their specific components, it can be shown that from an abstract point of view they are closely related to each other. We investigate their equivalences, robustness, spectral and convergence properties, by accounting for their implementation, the effect of roundoff errors and their sensitivity to inexact coarse solves, severe termination criteria and perturbed starting vectors, Microelectronics & Computer Engineering, Electrical Engineering, Mathematics and Computer Science

For various applications, it is well-known that a multi-level, in particular two-level, preconditioned CG (PCG) method is an efficient method for solving large and sparse linear systems with a coefficient matrix that is symmetric positive definite. The corresponding two-level preconditioner combines traditional and projection-type preconditioners to get rid of the effect of both small and large eigenvalues of the coefficient matrix. In the literature, various two-level PCG methods are known, coming from the fields of deflation, domain decomposition and multigrid. Even though these two-level methods differ a lot in their specific components, it can be shown that from an abstract point of view they are closely related to each other. We investigate their equivalences, robustness, spectral and convergence properties, by accounting for their implementation, the effect of roundoff errors and their sensitivity to inexact coarse solves, severe termination criteria and perturbed starting vectors, Microelectronics & Computer Engineering, Electrical Engineering, Mathematics and Computer Science

The Preconditioned Conjugate Gradient (PCG) method is one of the most popular iterative methods for solving large linear systems with a symmetric and positive semi-definite coefficient matrix. However, if the preconditioned coefficient matrix is ill-conditioned, the convergence of the PCG method typically deteriorates. Instead, a two-level PCG method can be used. The corresponding two-level preconditioner usually treats unfavorable eigenvalues of the coefficient matrix effectively, so that the two-level PCG method is expected to converge faster than the original PCG method. Many two-level preconditioners are known in the fields of deflation, multigrid and domain decomposition methods. Several of them are discussed in this thesis, where the main focus is on the deflation method. We show some theoretical properties of the deflation method, which give insights into the effectiveness of this method. A crucial component of the deflation preconditioner is the choice of projection vectors. Several choices are discussed and examined. We advocate that subdomain projection vectors, which are based on disjoint and piecewise-constant vectors, are among the best choices for a class of problems. Subsequently, we examine the application of the deflation method to linear systems with singular coefficient matrices. Several mathematically equivalent variants of the original deflation method are proposed to deal with the possible singularity of this coefficient matrix. In addition, two approaches are discussed in order to handle coarse linear systems with a Galerkin matrix, which are involved in each iteration of the deflation method. After the discussion of the implementation and efficiency issues of the deflation method, it is demonstrated that this method is usually faster than the original PCG method. Moreover, we present a comparison between the deflation method and other well-known two-level PCG methods, among them the balancing-Neumann-Neumann, additive coarse-grid correction, Electrical Engineering, Mathematics and Computer Science

Electrical Engineering, Mathematics and Computer Science

Electrical Engineering, Mathematics and Computer Science

The Preconditioned Conjugate Gradient (PCG) method is one of the most popular iterative methods for solving large linear systems with a symmetric and positive semi-definite coefficient matrix. However, if the preconditioned coefficient matrix is ill-conditioned, the convergence of the PCG method typically deteriorates. Instead, a two-level PCG method can be used. The corresponding two-level preconditioner usually treats unfavorable eigenvalues of the coefficient matrix effectively, so that the two-level PCG method is expected to converge faster than the original PCG method. Many two-level preconditioners are known in the fields of deflation, multigrid and domain decomposition methods. Several of them are discussed in this thesis, where the main focus is on the deflation method. We show some theoretical properties of the deflation method, which give insights into the effectiveness of this method. A crucial component of the deflation preconditioner is the choice of projection vectors. Several choices are discussed and examined. We advocate that subdomain projection vectors, which are based on disjoint and piecewise-constant vectors, are among the best choices for a class of problems. Subsequently, we examine the application of the deflation method to linear systems with singular coefficient matrices. Several mathematically equivalent variants of the original deflation method are proposed to deal with the possible singularity of this coefficient matrix. In addition, two approaches are discussed in order to handle coarse linear systems with a Galerkin matrix, which are involved in each iteration of the deflation method. After the discussion of the implementation and efficiency issues of the deflation method, it is demonstrated that this method is usually faster than the original PCG method. Moreover, we present a comparison between the deflation method and other well-known two-level PCG methods, among them the balancing-Neumann-Neumann, additive coarse-grid correction, Electrical Engineering, Mathematics and Computer Science

Electrical Engineering, Mathematics and Computer Science

Electrical Engineering, Mathematics and Computer Science

Electrical Engineering, Mathematics and Computer Science

Electrical Engineering, Mathematics and Computer Science

Electrical Engineering, Mathematics and Computer Science

Electrical Engineering, Mathematics and Computer Science

Simulating bubbly flows is a very popular topic in CFD. These bubbly flows are governed by the Navier-Stokes equations. In many popular operator splitting formulations for these equations, solving the linear system coming from the discontinuous diffusion equation takes the most computational time, despite of its elliptic origins. Sometimes these singular linear systems are forced to be invertible leading to a worse (effective) condition number. If ICCG is used to solve this problem, the convergence is significantly slower than for the case of the original singular problem. In this talk, we show that applying the deflation technique, which leads to the DICCG method, remedies the worse condition number and the worse convergence of ICCG. Moreover, some useful equalities are derived from the deflated variants of the singular and invertible matrices, which are also generalized to preconditioned methods. It appears that solving the invertible and singular linear systems with DICCG leads to exactly the same convergence results. Numerical experiments considering air-bubbles in water emphasize these theoretical results. This means that the deflation method is well-applicable for singular linear systems. In addition, from the numerical experiments it appears that DICCG is insensitive for the geometry of the density field, which is an important advantage of the deflation method.

For various applications, it is well-known that deflated ICCG is an efficient method for solving linear systems with invertible and singular co-efficient matrix. This deflated ICCG with subdomain deflation vectors is used by us to solve linear systems with singular coefficient matrix, arising from a discretization of the Poisson equation with Neumann boundary conditions and discontinuous coefficients. However, we have not explained the fact that the use of subdomain deflation vectors is effective. In this paper, we investigate this by doing some perturbed eigenvector analysis and we make it plausible that the use of subdomain deflation vectors is beneficial for the number of iterations and the amount of computational time. Moreover, we introduce new variants of the deflation technique which can deal with bubbly flow problems and with two-phase flow problems in general. The most simple variant is the so-called levelset deflation technique which chooses deflation vectors corresponding to the bubbles in the domain. Another variant is to combine the subdomain and levelset deflation technique which makes profit of the advantages of the original subdomain deflation and the levelset deflation. Numerical experiments show the good performance of these new deflation variants. Finally, the undeflated and deflated MICCG and RICCG methods are also investigated, which apply the modified and relaxed IC preconditioners instead of the original IC preconditioner, respectively. We show that the extension of the deflation technique to these methods is only beneficial for test problems with smooth coefficients. In our bubbly flow problems, the original ICCG and DICCG appear to be the most efficient methods., Electrical Engineering, Mathematics and Computer Science

Simulating bubbly flows is a very popular topic in CFD. These bubbly flows are governed by the Navier-Stokes equations. In many popular operator splitting formulations for these equations, solving the linear system coming from the discontinuous Poisson equation takes the most computational time, despite of its elliptic nature. ICCG is widely used for this purpose, but for complex bubbly flows this method shows slow convergence. As alternative for ICCG, we apply a deflated variant of ICCG which is called DICCG. This new method incorporates the eigenmodes corresponding to the components which caused the slow convergence of ICCG. Some implementation issues of DICCG are discussed in this paper and some considerations about the singularity are made. Moreover, coarse linear systems have to be solved within DICCG. We discuss methods to do this efficiently which results in two approaches DICCG1 and DICCG2. In exact arithmetics, we prove that these variants lead to the same convergence results. Thereafter we show with numerical experiments that both DICCG approaches are very efficient. Compared to ICCG, DICCG decreases significantly the number of iterations and the computational time as well, which are required for solving Poisson equation in applications of 2-D and 3-D bubbly flows., Electrical Engineering, Mathematics and Computer Science

Simulating bubbly flows is a very popular topic in CFD. These bubbly flows are governed by the Navier-Stokes equations. In many popular operator splitting formulations for these equations, solving the linear system coming from the discontinuous diffusion equation takes the most computational time, despite of its elliptic origins. Sometimes these singular linear systems are forced to be invertible leading to a worse (effective) condition number. If ICCG is used to solve this problem, the convergence is significantly slower than for the case of the original singular problem. In this talk, we show that applying the deflation technique, which leads to the DICCG method, remedies the worse condition number and the worse convergence of ICCG. Moreover, some useful equalities are derived from the deflated variants of the singular and invertible matrices, which are also generalized to preconditioned methods. It appears that solving the invertible and singular linear systems with DICCG leads to exactly the same convergence results. Numerical experiments considering air-bubbles in water emphasize these theoretical results. This means that the deflation method is well-applicable for singular linear systems. In addition, from the numerical experiments it appears that DICCG is insensitive for the geometry of the density field, which is an important advantage of the deflation method., Electrical Engineering, Mathematics and Computer Science

Simulating bubbly flows is a very popular topic in CFD. These bubbly flows are governed by the Navier-Stokes equations. In many popular operator splitting formulations for these equations, solving the linear system coming from the discontinuous diffusion equation takes the most computational time, despite of its elliptic origins. Sometimes these singular linear systems are forced to be invertible leading to a worse (effective) condition number. If ICCG is used to solve this problem, the convergence is significantly slower than for the case of the original singular problem. In this talk, we show that applying the deflation technique, which leads to the DICCG method, remedies the worse condition number and the worse convergence of ICCG. Moreover, some useful equalities are derived from the deflated variants of the singular and invertible matrices, which are also generalized to preconditioned methods. It appears that solving the invertible and singular linear systems with DICCG leads to exactly the same convergence results. Numerical experiments considering air-bubbles in water emphasize these theoretical results. This means that the deflation method is well-applicable for singular linear systems. In addition, from the numerical experiments it appears that DICCG is insensitive for the geometry of the density field, which is an important advantage of the deflation method.

For various applications, it is well-known that deflated ICCG is an efficient method for solving linear systems with invertible and singular co-efficient matrix. This deflated ICCG with subdomain deflation vectors is used by us to solve linear systems with singular coefficient matrix, arising from a discretization of the Poisson equation with Neumann boundary conditions and discontinuous coefficients. However, we have not explained the fact that the use of subdomain deflation vectors is effective. In this paper, we investigate this by doing some perturbed eigenvector analysis and we make it plausible that the use of subdomain deflation vectors is beneficial for the number of iterations and the amount of computational time. Moreover, we introduce new variants of the deflation technique which can deal with bubbly flow problems and with two-phase flow problems in general. The most simple variant is the so-called levelset deflation technique which chooses deflation vectors corresponding to the bubbles in the domain. Another variant is to combine the subdomain and levelset deflation technique which makes profit of the advantages of the original subdomain deflation and the levelset deflation. Numerical experiments show the good performance of these new deflation variants. Finally, the undeflated and deflated MICCG and RICCG methods are also investigated, which apply the modified and relaxed IC preconditioners instead of the original IC preconditioner, respectively. We show that the extension of the deflation technique to these methods is only beneficial for test problems with smooth coefficients. In our bubbly flow problems, the original ICCG and DICCG appear to be the most efficient methods., Electrical Engineering, Mathematics and Computer Science

Simulating bubbly flows is a very popular topic in CFD. These bubbly flows are governed by the Navier-Stokes equations. In many popular operator splitting formulations for these equations, solving the linear system coming from the discontinuous Poisson equation takes the most computational time, despite of its elliptic nature. ICCG is widely used for this purpose, but for complex bubbly flows this method shows slow convergence. As alternative for ICCG, we apply a deflated variant of ICCG which is called DICCG. This new method incorporates the eigenmodes corresponding to the components which caused the slow convergence of ICCG. Some implementation issues of DICCG are discussed in this paper and some considerations about the singularity are made. Moreover, coarse linear systems have to be solved within DICCG. We discuss methods to do this efficiently which results in two approaches DICCG1 and DICCG2. In exact arithmetics, we prove that these variants lead to the same convergence results. Thereafter we show with numerical experiments that both DICCG approaches are very efficient. Compared to ICCG, DICCG decreases significantly the number of iterations and the computational time as well, which are required for solving Poisson equation in applications of 2-D and 3-D bubbly flows., Electrical Engineering, Mathematics and Computer Science

Simulating bubbly flows is a very popular topic in CFD. These bubbly flows are governed by the Navier-Stokes equations. In many popular operator splitting formulations for these equations, solving the linear system coming from the discontinuous diffusion equation takes the most computational time, despite of its elliptic origins. Sometimes these singular linear systems are forced to be invertible leading to a worse (effective) condition number. If ICCG is used to solve this problem, the convergence is significantly slower than for the case of the original singular problem. In this talk, we show that applying the deflation technique, which leads to the DICCG method, remedies the worse condition number and the worse convergence of ICCG. Moreover, some useful equalities are derived from the deflated variants of the singular and invertible matrices, which are also generalized to preconditioned methods. It appears that solving the invertible and singular linear systems with DICCG leads to exactly the same convergence results. Numerical experiments considering air-bubbles in water emphasize these theoretical results. This means that the deflation method is well-applicable for singular linear systems. In addition, from the numerical experiments it appears that DICCG is insensitive for the geometry of the density field, which is an important advantage of the deflation method., Electrical Engineering, Mathematics and Computer Science

Electrical Engineering, Mathematics and Computer Science

In this report we give new insights into the properties of invertible and singular deflated and preconditioned linear systems where the coefficient matrices are also symmetric and positive (semi-) definite. First we prove that the invertible de ated matrix has always a more favorable effeective condition number compared to the original matrix. So, in theory, the solution of the deflated linear system converges faster in iterative methods than the original one. Thereafter, some results are presented considering the singular systems originally from the Poisson equation with Neumann boundary conditions. In practice these linear systems are forced to be invertible leading to a worse (eective) condition number. We show that applying the deflation technique remedies this problem of a worse condition number. Moreover, we derive some useful equalities between the de ated variants of the singular and invertible matrices. Then we prove that the de ated singular matrix has always a more favorable effective condition number compared by the original matrix., Electrical Engineering, Mathematics and Computer Science

In this report we give a short overview of aspects on parallel deflated conjugate gradient method which is applied on large, sparse, symmetric and semi-positive definite linear systems obtained from moving boundary problems. Moreover, we present some results of small numerical experiments. After introducing the Navier-Stokes equations for multiphase flows, the pressure correction method is described shortly. In this method the Poisson equation dominates and leads to an ill-conditioned and huge linear sytem. This system is solved with the conjugate gradient method preconditioned with incomplete Cholesky or block-Jacobi. To accelerate the convergence of the iterative method a deflation technique is incorporated. We end with some aspects on implementing the whole technique in a parallel environment., Electrical Engineering, Mathematics and Computer Science

Electrical Engineering, Mathematics and Computer Science

In this report we give new insights into the properties of invertible and singular deflated and preconditioned linear systems where the coefficient matrices are also symmetric and positive (semi-) definite. First we prove that the invertible de ated matrix has always a more favorable effeective condition number compared to the original matrix. So, in theory, the solution of the deflated linear system converges faster in iterative methods than the original one. Thereafter, some results are presented considering the singular systems originally from the Poisson equation with Neumann boundary conditions. In practice these linear systems are forced to be invertible leading to a worse (eective) condition number. We show that applying the deflation technique remedies this problem of a worse condition number. Moreover, we derive some useful equalities between the de ated variants of the singular and invertible matrices. Then we prove that the de ated singular matrix has always a more favorable effective condition number compared by the original matrix., Electrical Engineering, Mathematics and Computer Science

In this report we give a short overview of aspects on parallel deflated conjugate gradient method which is applied on large, sparse, symmetric and semi-positive definite linear systems obtained from moving boundary problems. Moreover, we present some results of small numerical experiments. After introducing the Navier-Stokes equations for multiphase flows, the pressure correction method is described shortly. In this method the Poisson equation dominates and leads to an ill-conditioned and huge linear sytem. This system is solved with the conjugate gradient method preconditioned with incomplete Cholesky or block-Jacobi. To accelerate the convergence of the iterative method a deflation technique is incorporated. We end with some aspects on implementing the whole technique in a parallel environment., Electrical Engineering, Mathematics and Computer Science