Orthomodular logic. A proposal of a logic for quantum physics

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2016, Director: Antoni Torrens TorrellClassical physics are widely known to be closely related to classical propositional calculus, whereas there does not exist a strongly settled analogue for quantum physics. We will focus on orthomodular logic and, in particular, we will study two different sentential logics that have been purposed with this aim over othomodular lattices. Thus, we will introduce their semantics from the foundations of quantum mechanics and presenting them by means of two different approaches whose equivalence will be shown. Additionally, we will give an adequate syntax for each proposal, the first one due to M. L. dalla Chiara and R. Giuntini and the last one to G. Kalmbach. Finally, completeness theorems will be discussed as well as the results that have been reached

From pure conduction to homogeneous isotropic turbulence

L’objectiu d’aquest treball és estudiar multiples casos relacionats amb la dinàmica de fluids i amb la transferència de calor. Més concretament, sis casos de complexitat creixent han estat abordats des de la perspectiva de la dinàmica de fluids computacional, resolent les equacions en derivades parcials associades a cada fenomen a partir del mètode de volums finits. Els casos que han estat simulats són: conducció pura, convecció-difusió, lid-driven cavity, differentially heated cavity, equació de Burgers en una i dues dimensions i turbulència isotròpica homogènia (implementant el model de turbulència de Smagorinsky).The aim of this work is to study multiple cases related to fluid dynamics and heat transfer. More concretely, six cases of increasing complexity will be approached from the perspective of computational fluid dynamics, solving the partial di↵erential equations related to each phenomenon by means of the finite volume method. The cases that have been simulated are: pure conduction, convection-di↵usion, lid-driven cavity, differentially heated cavity, one and two-dimensional Burgers equation and homogeneous isotropic turbulence (implementing the Smagorinsky turbulence model)

On Preconditioning Variable Poisson Equation with Extreme Contrasts in the Coefficients

It is well known that the solution by means of iterative methods of very ill-conditioned systems leads to very poor convergence rates. In this context, preconditioning becomes crucial in order to modify the spectrum of the system being solved and improve the performance of the solvers. A proper balance between the reduction in the number of iterations and the overhead of the construction and application of the preconditioner needs to be sought to actually decrease the total execution time of the solvers. This is particularly important when considering variable coefficients matrices as, in general, its preconditioners will also be variable and need to be updated regularly at an affordable cost. In this work we present a family of variable preconditioners designed for the effective solution of variable Poisson equation with extreme contrasts in the coefficients, which represents a particularly challenging case as it translates into a variable and extremely ill-conditioned system arising in many situations such as with multiphase flows presenting high density ratios or in the presence of highly-stretched adaptive mesh refinements. Finally, the results of the numerical experiments performed are presented and discussed, confirming our preconditioners as extremely affordable, highly-parallelizable and easy-to-implement alternatives to the more standard (and usually unfeasible) preconditioners, still showing great improvements in the rate of convergence of the solvers without requiring the variable coefficients matrix to be explicitly rebuilt at each iteration.Àdel Alsalti-Baldellou, F. Xavier Trias and Assensi Oliva have been financially supported by a competitive R+D project (ENE2017-88697-R) by the Spanish Research Agency. Àdel Alsalti-Baldellou is also supported by predoctoral grants DIN2018-010061 and 2019-DI-90, given by, respectively, the Spanish Ministry of Science, Innovation and Universities (MICINN) and the Catalan Agency for Management of University and Research Grants (AGAUR).Postprint (published version

Efficient strategies for solving the variable Poisson equation with large contrasts in the coefficients

Discrete versions of Poisson's equation with large contrasts in the coefficients result in very ill-conditioned systems. Thus, its iterative solution represents a major challenge, for instance, in porous media and multiphase flow simulations, where considerable permeability and density ratios are usually found. The existing strategies trying to remedy this are highly dependent on whether the coefficient matrix remains constant at each time iteration or not. In this regard, incompressible multiphase flows with high-density ratios are particularly demanding as their resulting Poisson equation varies along with the density field, making the reconstruction of complex preconditioners impractical. This work presents a strategy for solving such versions of the variable Poisson equation.Roughly, we first make it constant through an adequate approximation. Then, we block-diagonalise it through an inexpensive change of basis that takes advantage of mesh reflection symmetries, which are common in multiphase flows. Finally, we solve the resulting set of fully decoupled subsystems with virtually any solver. The numerical experiments conducted on a multiphase flow simulation prove the benefits of such an approach, resulting in up to 6.6x faster convergences

Exploiting spatial symmetries for solving Poisson's equation

This paper presents a strategy to accelerate virtually any Poisson solver by taking advantage of s spatial reflection symmetries. More precisely, we have proved the existence of an inexpensive block diagonalisation that transforms the original Poisson equation into a set of 2s fully decoupled subsystems then solved concurrently. This block diagonalisation is identical regardless of the mesh connectivity (structured or unstructured) and the geometric complexity of the problem, therefore applying to a wide range of academic and industrial configurations. In fact, it simplifies the task of discretising complex geometries since it only requires meshing a portion of the domain that is then mirrored implicitly by the symmetries’ hyperplanes. Thus, the resulting meshes naturally inherit the exploited symmetries, and their memory footprint becomes 2s times smaller. Thanks to the subsystems’ better spectral properties, iterative solvers converge significantly faster. Additionally, imposing an adequate grid points’ ordering allows reducing the operators’ footprint and replacing the standard sparse matrix-vector products with the sparse matrixmatrix product, a higher arithmetic intensity kernel. As a result, matrix multiplications are accelerated, and massive simulations become more affordable. Finally, we include numerical experiments based on a turbulent flow simulation and making state-of-theart solvers exploit a varying number of symmetries. On the one hand, algebraic multigrid and preconditioned Krylov subspace methods require up to 23% and 72% fewer iterations, resulting in up to 1.7x and 5.6x overall speedups, respectively. On the other, sparse direct solvers’ memory footprint, setup and solution costs are reduced by up to 48%, 58% and 46%, respectively.This work has been financially supported by two competitive R+D projects: RETOtwin (PDC2021-120970-I00), given by MCIN/AEI/10.13039/501100011033 and European Union Next GenerationEU/PRTR, and FusionCAT (001-P-001722), given by Generalitat de Catalunya RIS3CAT-FEDER. Àdel Alsalti-Baldellou has also been supported by the predoctoral grants DIN2018-010061 and 2019-DI-90, given by MCIN/AEI/10.13039/501100011033 and the Catalan Agency for Management of University and Research Grants (AGAUR), respectively.Peer ReviewedPostprint (published version

DNS and LES on unstructured grids: playing with matrices to preserve symmetries using a minimal set of algebraic kernels

The essence of turbulence are the smallest scales of motion. They result from a subtle balance between two differential operators differing in symmetry: the convective operator is skew-symmetric, whereas the diffusive is symmetric and negative-definite. On the other hand, accuracy and stability need to be reconciled for numerical simulations of turbulent flows in complex configurations. With this in mind, a fully-conservative discretization method for collocated unstructured grids was proposed [Trias et al., J.Comp.Phys. 258, 246-267, 2014]: it preserves the symmetries of the differential operators and it has shown to be a very suitable approach for DNS and LES. On the other hand, an efficient cross-platform portability is nowadays one of the greatest challenges for CFD codes. In this regard, our leitmotiv reads: relying on a minimal set of (algebraic) kernels is crucial for code portability and maintenance! In this context, this work focuses on the computation of eigenbounds for the above-mentioned convection and diffusion matrices which are needed to determine the time-step `a la CFL. A new inexpensive method that allows this, without explicitly constructing these time-dependent matrices is proposed and tested. It only requires a sparse-matrix vector product where only the vector changes on time. Hence, apart from being significantly more efficient than the standard CFL condition, cross-platform portability is straightforward

DNS and LES on unstructured grids: playing with matrices to preserve symmetries using a minimal set of algebraic kernels

The essence of turbulence are the smallest scales of motion. They result from asubtle balance between two differential operators differing in symmetry: The convective operator is skew-symmetric, whereas the diffusive is symmetric and negative-definite. On the other hand, accuracy and stability need to be reconciled for numerical simulations of turbulent flows in complex configurations. With this in mind, a fully-conservative discretization method for collocated unstructured grids was proposed [Trias et al., J.Comp.Phys. 258, 246-267, 2014]: It preserves the symmetries of the differential operators and it has shown to be a very suitable approach for DNS and LES. On the other hand, an efficient cross-platform portability is nowadays one of the greatest challenges for CFD codes. In this regard, our leitmotiv reads: Relying on a minimal set of (algebraic) kernels is crucial for code portability and maintenance! In this context, this work focuses on the computation of eigenbounds for the above-mentioned convection and diffusion matrices which are needed to determine the time-stepa la CFL. A new inexpensive method that allows this, without explicitly constructing these time-dependent matrices is proposed and tested. It only requires a sparse-matrix vector product where only the vector changes on time. Hence, apart from being significantly more efficient than the standard CFL condition, cross-platform portability is straightforward.F.X.T., X.A-F., A.A-B. and A.O. are supported by the Ministerio de Economía y Competitividad, Spain, RETOtwin project (PDC2021-120970-I00). F.X.T. and A.O. are supported by the Generalitat de Catalunya RIS3CAT-FEDER, FusionCAT project (001-P-001722). A.A-B. is supported by the predoctoral grants DIN2018-010061 and 2019-DI-90, by MCIN/AEI/10.13039/501100011033 and the AGAUR. A.G. is supported by the RSF project 19-11-00299. Calculations were carried out on MareNostrum 4 supercomputer at BSC. The authors thankfully acknowledge these institutions.Peer ReviewedPostprint (published version

Efficient strategies for solving the variable Poisson equation with large contrasts in the coefficients

Discrete versions of Poisson’s equation with large contrasts in the coefficients result in very ill-conditioned systems. Thus, its iterative solution represents a major challenge, for instance, in porous media and multiphase flow simulations, where considerable permeability and density ratios are usually found. The existing strategies trying to remedy this are highly dependent on whether the coefficient matrix remains constant at each time iteration or not. In this regard, incompressible multiphase flows with high-density ratios are particularly demanding as their resulting Poisson equation varies along with the density field, making the reconstruction of complex preconditioners impractical. This work presents a strategy for solving such versions of the variable Poisson equation. Roughly, we first make it constant through an adequate approximation. Then, we block-diagonalise it through an inexpensive change of basis that takes advantage of mesh reflection symmetries, which are common in multiphase flows. Finally, we solve the resulting set of fully decoupled subsystems with virtually any solver. The numerical experiments conducted on a multiphase flow simulation prove the benefits of such an approach, resulting in up to 6.6x faster convergences.Adel Alsalti-Baldellou, Xavier Àlvarez-Farré, F. Xavier Trias and Assensi Oliva have been ´ financially supported by two competitive R+D projects: RETOtwin (PDC2021-120970-I00), given by MCIN/AEI/10.13039/501100011033 and European Union Next GenerationEU/PRTR, and FusionCAT (001-P-001722), given by Generalitat de Catalunya RIS3CAT-FEDER. Adel ` Alsalti-Baldellou has also been supported by the predoctoral grants DIN2018-010061 and 2019- DI-90, given by MCIN/AEI/10.13039/501100011033 and the Catalan Agency for Management of University and Research Grants (AGAUR), respectively. Andrey Gorobets has been supported by the RSF project 19-11-00299.Peer ReviewedPostprint (published version

Development of a low-level, algebra-based library to provide platform portability on hybrid supercomputers

Continuous enhancement in hardware technologies enables scientific computing to advance incessantly and reach further aims. Since the start of the global race for exascale high-performance computing, massively-parallel devices of various architectures have been incorporated into the newest supercomputers, leading to an increasing hybridization of compute nodes. In this context of accelerated innovation, software portability and efficiency become crucial. Traditionally, scientific computing software development using mesh methods is based on calculations in iterative stencil loops over a discretized geometry—the mesh. Despite being intuitive and versatile, the interdependency between algorithms and their computational implementations in stencil applications usually results in a large number of subroutines and introduces an inevitable complexity when it comes to portability and sustainability. An alternative is to break the interdependency between the algorithm and its implementation, and then to cast the calculations into a minimalist set of kernels. Algebra-based implementations rely on a reduced set of basic linear algebra subroutines, which simplifies the deployment of software in hybrid computing systems. In this work, we tackle the development of a fully-portable, algebraic library that can be coupled beneath other high-level, algebra-oriented framework. Namely, this library provides platform portability in the simplest possible manner (i.e., the user develops applications in a purely sequential style). Internally, algebraic objects are distributed among computing devices using a multilevel decomposition approach. Data exchanges between computing units or between nodes are hidden by a multithreaded overlapping scheme.The work of X.A.F, A.A.B, A.O., and F.X.T. has been financially supported by the following R+D projects: RETOtwin (PDC2021-120970-I00), given by MCIN/AEI/10.13039/501100011033 and European Union Next Generation EU/PRTR, FusionCAT (001-P-001722), given by Generalitat de Catalunya RIS3CAT-FEDER. X. A. F. has also been supported by a predoctoral contract (2019FI B2-00076) by the Government of Catalonia. A.A.B has also been supported by the predoctoral grants DIN2018-010061 and 2019-DI-90, given by MCIN/AEI/10.13039/501100011033 and the Catalan Agency for Management of University and Research Grants (AGAUR), respectively. The work of A. G. has been funded by the Russian Science Foundation, project 19-11-00299. The studies of this work have been carried out using computational resources of the Barcelona Supercomputing Center (IM-2020-3-0030 and IM-2022-1-0015). The authors thankfully acknowledge these institutions.Peer ReviewedPostprint (published version

Development of a low-level, algebra-based library to provide platform portability on hybrid supercomputers

Continuous enhancement in hardware technologies enables scientific computing to advance incessantly and reach further aims. Since the start of the global race for exascale high-performance computing, massively-parallel devices of various architectures have been incorporated into the newest supercomputers, leading to an increasing hybridization of compute nodes. In this context of accelerated innovation, software portability and efficiency become crucial. Traditionally, scientific computing software development using mesh methods is based on calculations in iterative stencil loops over a discretized geometry--the mesh. Despite being intuitive and versatile, the interdependency between algorithms and their computational implementations in stencil applications usually results in a large number of subroutines and introduces an inevitable complexity when it comes to portability and sustainability. An alternative is to break the interdependency between the algorithm and its implementation, and then to cast the calculations into a minimalist set of kernels. Algebra-based implementations rely on a reduced set of basic linear algebra subroutines, which simplifies the deployment of software in hybrid computing systems. In this work, we tackle the development of a fully-portable, algebraic library that can be coupled beneath other high-level, algebra-oriented framework. Namely, this library provides platform portability in the simplest possible manner (i.e., the user develops applications in a purely sequential style). Internally, algebraic objects are distributed among computing devices using a multilevel decomposition approach. Data exchanges between computing units or between nodes are hidden by a multithreaded overlapping scheme