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Estimating the temperature field of a building envelope could be a time-consuming task. The use of a reduced-order method is then proposed: the Proper Generalized Decomposition method. The solution of the transient heat equation is then re-written as a function of its parameters: the boundary conditions, the initial condition, etc. To avoid a tremendous number of parameters, the initial condition is parameterized. This is usually done by using the Proper Orthogonal Decomposition method to provide an optimal basis. Building this basis requires data and a learning strategy. As an alternative, the use of orthogonal polynomials (Chebyshev, Legendre) is here proposed.
Highlights
• Chebyshev and Legendre polynomials are used to approximate the initial condition
• Performance of Chebyshev and Legendre polynomials are compared to the POD basis
• Each basis combined with the PGD model is compared to laboratory measurements
• The influence of four different parameters on the accuracy of the basis is studied
• For each approximation basis, CPU calculation times are evaluated and compared

To read the full-text of this research,

you can request a copy directly from the authors.

... Image signals are subject to several processes such as transmission [1], enhancement [2], transformation [3], hiding [4], and compression [5,6]. Similarly to image processing, speech signals processing is also essential [7], and involves several stages such as transfer [8], acquisition [9], and coding [10]. Pattern recognition, which is considered an automated process, This is attributed to the fact that pitfalls may happen even when small errors in floating numbers occur. ...

... The computation of the initial values (K p 0 (0)) as a function of different polynomial sizes using Equation(8). ...

Krawtchouk polynomials (KPs) and their moments are promising techniques for applications of information theory, coding theory, and signal processing. This is due to the special capabilities of KPs in feature extraction and classification processes. The main challenge in existing KPs recurrence algorithms is that of numerical errors, which occur during the computation of the coefficients in large polynomial sizes, particularly when the KP parameter (p) values deviate away from 0.5 to 0 and 1. To this end, this paper proposes a new recurrence relation in order to compute the coefficients of KPs in high orders. In particular, this paper discusses the development of a new algorithm and
presents a new mathematical model for computing the initial value of the KP parameter. In addition, a new diagonal recurrence relation is introduced and used in the proposed algorithm. The diagonal recurrence algorithm was derived from the existing n direction and x-direction recurrence algorithms. The diagonal and existing recurrence algorithms were subsequently exploited to compute the KP coefficients. First, the KP coefficients were computed for one partition after dividing the KP plane into
four. To compute the KP coefficients in the other partitions, the symmetry relations were exploited. The performance evaluation of the proposed recurrence algorithm was determined through different comparisons which were carried out in state-of-the-art works in terms of reconstruction error, polynomial-size, and computation cost. The obtained results indicate that the proposed algorithm is reliable and computes lesser coefficients when compared to the existing algorithms across wide ranges of parameter values of p and polynomial sizes N. The results also show that the improvement ratio of the computed coefficients ranges from 18.64% to 81.55% in comparison to the existing algorithms. Besides this, the proposed algorithm can generate polynomials of an order ∼8.5 times larger than those generated using state-of-the-art algorithms.

In this paper, the spectral method is developed as a reduced-order model for the solution of parametric problems within the building refurbishment framework. We propose to use the spectral reduced-order method to solve parametric problems in an innovative way, integrating the unknown parameter as one of the coordinates of the decomposition. The residual is minimized combining the Tau–Galerkin method with the Collocation approach. The developed method is evaluated in terms of accuracy and reduction of the computational time in three different cases. The dynamic behaviour of unidimensional moisture diffusion is investigated. The cases focus on solving parametric problems in which the solution depends on space, time, diffusivity and material thickness. Results highlight that the parametric spectral reduced-order method provides accurate solutions and can reduce 10 times the degree of freedom of the solution. It allows efficient computation of the physical phenomena with a lower error when compared to traditional approaches.

Separated representations at the heart of Proper Generalized Decomposition are constructed incrementally by minimizing the problem residual. However, the modes involved in the resulting decomposition do not exhibit a clear multi-scale character. In order to recover a multi-scale description of the solution within a separated representation framework, we study the use of wavelets for approximating the functions involved in the separated representation of the solution. We will prove that such an approach allows separating the different scales as well as taking profit from its multi-resolution behavior for defining adaptive strategies.

We discuss the use of hierarchical collocation to approximate the numerical solution of parametric models. With respect to traditional projection-based reduced order modeling, the use of a collocation enables non-intrusive approach based on sparse adaptive sampling of the parametric space. This allows to recover the low-dimensional structure of the parametric solution subspace while also learning the functional dependency from the parameters in explicit form. A sparse low-rank approximate tensor representation of the parametric solution can be built through an incremental strategy that only needs to have access to the output of a deterministic solver. Non-intrusiveness makes this approach straightforwardly applicable to challenging problems characterized by nonlinearity or non affine weak forms. As we show in the various examples presented in the paper, the method can be interfaced with no particular effort to existing third party simulation software making the proposed approach particularly appealing and adapted to practical engineering problems of industrial interest.

It is of great concern to produce numerically efficient methods for moisture diffusion through porous media, capable of accurately calculate moisture distribution with a reduced computational effort. In this way, model reduction methods are promising approaches to bring a solution to this issue since they do not degrade the physical model and provide a significant reduction of computational cost. Therefore, this article explores in details the capabilities of two model-reduction techniques - the Spectral Reduced-Order Model (Spectral-ROM) and the Proper Generalised Decomposition (PGD) - to numerically solve moisture diffusive transfer through porous materials. Both approaches are applied to three different problems to provide clear examples of the construction and use of these reduced-order models. The methodology of both approaches is explained extensively so that the article can be used as a numerical benchmark by anyone interested in building a reduced-order model for diffusion problems in porous materials. Linear and non-linear unsteady behaviors of unidimensional moisture diffusion are investigated. The last case focuses on solving a parametric problem in which the solution depends on space, time and the diffusivity properties. Results have highlighted that both methods provide accurate solutions and enable to reduce significantly the order of the model around ten times lower than the large original model. It also allows an efficient computation of the physical phenomena with an error lower than 10^{-2} when compared to a reference solution.

The aims of this study are to highlight the microclimatic phenomena to which the energy consumption of a building is the most sensitive and to compare different approaches to coupling microclimate and building energy simulation models. We first present a study in which spatial variations in outdoor air temperature are not taken into account so as to compare the relative effect of convective and radiative heat flux on the outer surface energy
balance. Then, several coupling methods are implemented, for which energy consumption in winter and indoor temperature in summer are compared between insulated and non-insulated buildings. Results show that for the urban context, taking into account long-wave radiative heat fluxes is crucial. Moreover, representing local modifications in outdoor air temperature is necessary for non-insulated buildings in summer, but can be neglected in winter. In conclusion, an intermediate coupling method that can be used under certain assumptions is proposed.

Dynamic Data-Driven Application Systems—DDDAS—appear as a new paradigm in the field of applied sciences and engineering, and in particular in Simulation-based Engineering Sciences. By DDDAS we mean a set of techniques that allow the linkage of simulation tools with measurement devices for real-time control of systems and processes. One essential feature of DDDAS is the ability to dynamically incorporate additional data into an executing application, and in reverse, the ability of an application to dynamically control the measurement process. DDDAS need accurate and fast simulation tools using if possible off-line computations to limit as much as possible the on-line computations. With this aim, efficient solvers can be constructed by introducing all the sources of variability as extra-coordinates in order to solve the model off-line only once. This way, its most general solution is obtained and therefore it can be then considered in on-line purposes. So to speak, we introduce a physics-based meta-modeling technique without the need for prior computer experiments. However, such models, that must be solved off-line, are defined in highly multidimensional spaces suffering the so-called curse of dimensionality. We proposed recently a technique, the Proper Generalized Decomposition—PGD—able to circumvent the redoubtable curse of dimensionality. The marriage of DDDAS concepts and tools and PGD off-line computations could open unimaginable possibilities in the field of dynamic data-driven application systems. In this work we explore some possibilities in the context of on-line parameter estimation.

Many of the popular building energy simulation programs around the world are reaching maturity — some use simulation methods (and even code) that originated in the 1960s. For more than two decades, the US government supported development of two hourly building energy simulation programs, BLAST and DOE-2. Designed in the days of mainframe computers, expanding their capabilities further has become difficult, time-consuming, and expensive. At the same time, the 30 years have seen significant advances in analysis and computational methods and power — providing an opportunity for significant improvement in these tools.In 1996, a US federal agency began developing a new building energy simulation tool, EnergyPlus, building on development experience with two existing programs: DOE-2 and BLAST. EnergyPlus includes a number of innovative simulation features — such as variable time steps, user-configurable modular systems that are integrated with a heat and mass balance-based zone simulation — and input and output data structures tailored to facilitate third party module and interface development. Other planned simulation capabilities include multizone airflow, and electric power and solar thermal and photovoltaic simulation. Beta testing of EnergyPlus began in late 1999 and the first release is scheduled for early 2001.

Contents PREFACE x Acknowledgments xiv Errata and Extended-Bibliography xvi 1 Introduction 1 1.1 Series expansions .................................. 1 1.2 First Example .................................... 2 1.3 Comparison with finite element methods .................... 4 1.4 Comparisons with Finite Differences ....................... 6 1.5 Parallel Computers ................................. 9 1.6 Choice of basis functions .............................. 9 1.7 Boundary conditions ................................ 10 1.8 Non-Interpolating and Pseudospectral ...................... 12 1.9 Nonlinearity ..................................... 13 1.10 Time-dependent problems ............................. 15 1.11 FAQ: Frequently Asked Questions ........................ 16 1.12 The Chrysalis .................................... 17 2 Chebyshev & Fourier Series 19 2.1 Introduction ..................................... 19 2.2 Fourier series ...........

An assessment of physics-based and data-driven reduced-order models (ROMs) is presented for the study of convective heat transfer in a rectangular cavity. Despite the simple geometrical configuration, the current setup offers increasingly rich dynamics as the thermal forcing is increased, thus making it a suitable candidate to evaluate the performance of ROMs. First, flow simulations are performed using a high-order spectral element method that will feed the ROMs with well-resolved temporal and spatial information. Proper orthogonal decomposition (POD) is applied to reduce the problem dimensionality for all models. The class of tested physics-based models include the Galerkin and least-squares Petrov-Galerkin (LSPG) methods that rely on projection of the Navier-Stokes and energy equations being solved. On the other hand, the data-driven methods applied in this work rely on regression of the governing equations, which are treated as a nonlinear dynamical system. The data-driven methods tested here include the sparse identification of nonlinear dynamics (SINDy) approach and a method recently proposed in literature based on deep neural networks (DNNs). All ROMs are able to represent the periodical temporal dynamics of a low Rayleigh number flow. However, the physics-based approaches demonstrate a better performance for a moderate Rayleigh number case with more complex flow dynamics, when several frequencies are excited in a non-periodical fashion. ARTICLE HISTORY

The urban context is often simplified or neglected in Building Energy Models (BEMs) due to the difficulties of taking accurately into account all the heat fluxes emanating from the environment. Oversimplifying the urban context can impact the accuracy of the BEM predictions. Nevertheless, several approaches can be used to allow for the impact of the urban environment on the dynamic behavior of a building, its heating and cooling demands, and thermal comfort. This state of the art review provides a critical overview of the different methods currently used to take into account the urban microclimate in building design simulations. First, both the microclimate and building models are presented, focusing on their assumptions and capabilities. Second, a few examples of coupling, performed between both modeling scales are analyzed. Last, the discussion highlights the differences obtained between simulations that take the urban context into consideration and those that simplify or neglect urban heat fluxes. The remaining scientific obstacles to a more effective consideration of the urban context impacting the BEMs are indicated.

A parametric soil model has been developed to improve the computational time of microclimate simulation tools. It combines the use of two methods: the Proper Generalized Decomposition and the Proper Orthogonal Decomposition. Offline, a learning process is required to build the model, before its use on-line. A methodology to select a short and representative learning process needs to be developed. The k-means clustering method is used to build a training climate made of 24 days representative of a full climate. The offline computation costs are reduced by 94.4% for an error of 0.8%.

Numerical simulation is a powerful tool for assessing the causes of an Urban Heat Island (UHI) effect or quantifying the impact of mitigation solutions on local climatic conditions. However, the numerical cost associated with such a tool is quite significant at the scale of an entire district. Today, the main challenge consists of achieving both a proper representation of the physical phenomena and a critical reduction in the numerical costs of running simulations. This paper presents a combined parametric urban soil model that accurately reproduces thermal heat flux exchanges between the soil and the urban environment with a reduced computational time. For this purpose, the use of a combination of two reduced-order methods is proposed herein: the Proper Orthogonal Decomposition method, and the Proper Generalized Decomposition method. The developed model is applied to two case studies in order to establish a practical evaluation: an open area independent of the influences of the surrounding surface, and a theoretical urban scene with two canyon streets. The error due to the model reduction remains below 0.2 °C on the mean surface temperature for a reduced computational cost of 80%. Compared to in situ measurements the error remains bellow 1.24 °C at the surface.

This work presents an efficient numerical method based on spectral expansions for simulation of heat and moisture diffusive transfers through multilayered porous materials.
Traditionally, by using the finite-difference approach, the problem is discretized in time and
space domains (Method of lines) to obtain a large system of coupled Ordinary Differential
Equations (ODEs), which is computationally expensive. To avoid such a cost, this paper
proposes a reduced-order method that is faster and accurate, using a much smaller system of ODEs. To demonstrate the benefits of this approach, tree case studies are presented. The first one considers nonlinear heat and moisture transfer through one material layer. The second case - highly nonlinear - imposes a high moisture content gradient - simulating a rain like condition - over a two-layered domain, while the last one compares the numerical prediction against experimental data for validation purposes. Results show how the nonlinearities and the interface between materials are easily and naturally treated with the spectral reduced-order method. Concerning the reliability part, predictions show a good agreement with experimental results, which confirm robustness, calculation efficiency and high accuracy of the proposed approach for predicting the coupled heat and moisture transfer through porous materials.

One possibility to improve the accuracy of building performance simulation (BPS) tools is via co-simulation techniques, where more accurate mathematical models representing particular and complex physical phenomena are employed through data exchanging between the BPS and a specialized software where those models are available. This article performs a deeper investigation of a recently proposed co-simulation technique that presents as novelty the employment of artificial intelligence as a strategy to reduce the computational burden generally required by co-simulations. Basically, the strategy, known as intelligentco-simulation, constructs new mathematical models through a learning procedure (training period) that is performed using the input–output data generated by a standard co-simulation, where the models of specialized software are employed. Once the learning phase is complete, the specialized software is disconnected from the BPS and the simulation goes on by using the synthesized models, requiring a much lower computational cost and with a low impact on the accuracy of the results. The synthesis of accurate-and-fast models is performed through machine learning techniques and the purpose of this paper is precisely a deep investigation of two techniques – recurrent neural networks and proper orthogonal decomposition reduction method, whose main goal is to reduce the training time period and to improve the accuracy. The case study focuses on a co-simulation between Domus and CFX programs, performing a two-dimensional diffusive heat transfer problem through a building envelope. The results show that for a standard co-simulation of 14 h, the intelligent co-simulation provided a reduction of 90% in the computer run time with accuracy error at the order of .

Local model order reduction methods provide better results than global ones to problems with intricate manifold solution structure. A posteriori methods (e.g. Proper Orthogonal Decomposition) have been many times applied locally, but a priori methods (e.g. Proper Generalized Decomposition) have the difficulty of determining the manifold structure of the solution in a previous way. We propose three strategies for estimating the appropriate size of the local sub-domains where afterwards local PGD (ℓ-PGD) is applied. It can be seen as a sort of a priori manifold learning or non-linear dimensionality reduction technique. Finally, three examples support the work.

This work presents an alternative view on the numerical simulation of diffusion processes applied to the heat and moisture transfer through multilayered porous building materials. Traditionally, by using the finite-difference approach, the discretization follows the Method Of Lines (MOL), when the problem is first discretized in space to obtain a large system of coupled Ordinary Differential Equations (ODEs). This paper proposes to change this
viewpoint. First, we discretize in time to obtain a small system of coupled ODEs, which means instead of having a Cauchy (Initial Value) Problem (IVP), we have a Boundary Value Problem (BVP). Fortunately, BVPs can be solved efficiently today using adaptive collocation finite-difference methods of high order. To demonstrate the benefits of this new approach, three case studies are presented. The first one considers nonlinear heat and moisture transfer through one material layer. The second case includes the rain effect, while the last one considers two material layers. Results show how the nonlinearities and the interface between materials are easily treated, by reasonably using a fourth-order adaptative method. In our numerical simulations, we use adaptive methods of the fourth order which in most practical situations is more than enough.

This paper proposes the use of a Spectral method to simulate diffusive moisture transfer through porous materials as a Reduced-Order Model (ROM). The Spectral approach is an a priori method assuming a separated representation of the solution. The method is compared with both classical Euler implicit and Crank-Nicolson schemes, considered as large original models. Their performance - in terms of accuracy, complexity reduction and CPU time reduction - are discussed for linear and nonlinear cases of moisture diffusive transfer through single and multi-layered one-dimensional domains, considering highly moisture-dependent properties. Results show that the Spectral reduced-order model approach enables to simulate accurately the field of interest. Furthermore, numerical gains become particularly interesting for nonlinear cases since the proposed method can drastically reduce the computer run time, by a factor of 100, when compared to the traditional Crank-Nicolson scheme for one-dimensional applications.

In this paper, an innovative method to minimise energy losses through building envelopes is presented, using the Proper Generalised Decomposition (PGD), written in terms of space x, time t, thermal diffusivity and envelope thickness L. The physical phenomenon is solved at once, contrarily to classical numerical methods that cannot create a parameter dependent model. First, the PGD solution is validated with an analytical solution to prove its accuracy. Then a complex case study of a multi-layer wall submitted to transient boundary conditions is investigated. The parametric solution is computed as a function of the space and time coordinates, as well as the thermal insulation thickness and the load material thermal diffusivity. Physical behaviour and conduction loads are analysed for 76 values of thermal insulation thickness and 100 types of load material properties. Furthermore, the reduced computational cost of the PGD is highlighted. The method computes the solution 100 times faster than standard numerical approaches. In addition, the PGD solution has a low storage cost, providing interesting development of parametric solutions for real-time applications of energy management in buildings.

This book is intended to help researchers overcome the entrance barrier to Proper Generalized Decomposition (PGD), by providing a valuable tool to begin the programming task. Detailed Matlab Codes are included for every chapter in the book, in which the theory previously described is translated into practice. Examples include parametric problems, non-linear model order reduction and real-time simulation, among others.
Proper Generalized Decomposition (PGD) is a method for numerical simulation in many fields of applied science and engineering. As a generalization of Proper Orthogonal Decomposition or Principal Component Analysis to an arbitrary number of dimensions, PGD is able to provide the analyst with very accurate solutions for problems defined in high dimensional spaces, parametric problems and even real-time simulation.

In this chapter we review the basics of classical model order reduction techniques based on Proper Orthogonal Decomposition (POD), Principal Component Analysis (PCA) or Karhunen-Loève decompositions.

SUMMARYA new method is developed here for the real-time integration of the equations of solid dynamics based on the use of proper orthogonal decomposition (POD)–proper generalized decomposition (PGD) approaches and direct time integration. The method is based upon the formulation of solid dynamics equations as a parametric problem, depending on their initial conditions. A sort of black-box integrator that takes the resulting displacement field of the current time step as input and (via POD) provides the result for the subsequent time step at feedback rates on the order of 1 kHz is obtained. To avoid the so-called curse of dimensionality produced by the large amount of parameters in the formulation (one per degree of freedom of the full model), a combined POD–PGD strategy is implemented. Examples that show the promising results of this technique are included. Copyright © 2014 John Wiley & Sons, Ltd.

Fine modeling of the structure and mechanics of materials from the nanometric to the micrometric scales uses descriptions ranging from quantum to statistical mechanics. Most of these models consist of a partial differential equation defined in a highly multidimensional domain (e.g. Schrodinger equation, Fokker-Planck equations among many others). The main challenge related to these models is their associated curse of dimensionality. We proposed in some of our former works a new strategy able to circumvent the curse of dimensionality based on the use of separated representations (also known as finite sums decomposition). This technique proceeds by computing at each iteration a new sum that consists of a product of functions each one defined in one of the model coordinates. The issue related to error estimation has never been addressed. This paper presents a first attempt on the accuracy evaluation of such a kind of discretization techniques.

This paper proposes a new method for solving the heat transfer equation based on a parallelisation in time of the computation. A parametric multidimensional model is solved within the context of the Proper Generalised Decomposition (PGD). The initial field of temperature and the boundary conditions of the problem are treated as extra-coordinates, similar to time and space. Two main approaches are exposed: a "full" parallelisation based on an off-line parallel computation and a "partial" parallelisation based on a decomposition of the original problem. Thanks to an optimised overlapping strategy, the reattachment of the local solutions at the interfaces of the time subdomains can be improved. For large problems, the parallel execution of the algorithm provides an interesting speedup and opens new perspectives regarding real-time simulation.

This paper revisits a new model reduction methodology based on the use of separated representations, the so called Proper Generalized Decomposition-PGD. Space and time separated representations generalize Proper Orthogonal Decompositions-POD-avoiding any a priori knowledge on the solution in contrast to the vast majority of POD based model reduction technologies as well as reduced bases approaches. Moreover, PGD allows to treat efficiently models defined in degenerated domains as well as the multidimensional models arising from multidimensional physics (quantum chemistry, kinetic theory descriptions, ... ) or from the standard ones when some sources of variability are introduced in the model as extra-coordinates.

In view of the increasing popularity of the application of proper orthogonal decomposition (POD) methods in engineering "elds and the loose description of connections among the POD methods, the purpose of this paper is to give a summary of the POD methods and to show the connections among these methods. Firstly, the derivation and the performance of the three POD methods:Karhunen}Loè ve decomposition (KLD), principal component analysis (PCA), and singular value decomposition (SVD) are summarized, then the equivalence problem is discussed via a theoretical comparison among the three methods. The equivalence of the matrices for processing, the objective functions, the optimal basis vectors, the mean-square errors, and the asymptotic connections of the three methods are demonstrated and proved when the methods are used to handle the POD of discrete random vectors.

This paper focuses on the efficient solution of models defined in high dimensional spaces. Those models involve numerous numerical challenges because of their associated curse of dimensionality. It is well known that in mesh-based discrete models the complexity (degrees of freedom) scales exponentially with the dimension of the space. Many models encountered in computational science and engineering involve numerous dimensions called configurational coordinates. Some examples are the models encountered in biology making use of the chemical master equation, quantum chemistry involving the solution of the Schrödinger or Dirac equations, kinetic theory descriptions of complex systems based on the solution of the so-called Fokker–Planck equation, stochastic models in which the random variables are included as new coordinates, financial mathematics, etc. This paper revisits the curse of dimensionality and proposes an efficient strategy for circumventing such challenging issue. This strategy, based on the use of a Proper Generalized Decomposition, is specially well suited to treat the multidimensional parametric equations.

Proper 727 generalized decomposition based dynamic data driven inverse identification. Mathematics and Computers 728 in Simulation

- David González
- F Masson
- Adrien Poulhaon
- Elías Leygue
- Francisco Cueto
- Chinesta

David González, F Masson, F Poulhaon, Adrien Leygue, Elías Cueto, and Francisco Chinesta. Proper
727
generalized decomposition based dynamic data driven inverse identification. Mathematics and Computers
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in Simulation, 82(9):1677-1695, 2012.

Assessment of reduced-order modeling

- Victor Zucatti
- F S Hugo
- Lui
- B Diogo
- William R Pitz
- Wolf

Victor Zucatti, Hugo FS Lui, Diogo B Pitz, and William R Wolf. Assessment of reduced-order modeling

How to Perform an Efficient Learning Process for a Combined Pod and pgd Soil Urban Thermal Model? Application to Canyon Streets

- Marie-Hélène Azam
- Marjoriemusy Sihemguernouti

Proper Generalized Decompositions

- Elías Cueto

Solving Nonlinear Diffusive Problems in Buildings by Means of a Spectral Reduced-order Model

- Gasparin
- Suelen
- Denysdutykh Julienberger

Assessment of Reduced-order Modeling Strategies for Convective Heat Transfer

- Zucatti
- Victor
- F S Hugo
- Diogo Lui
- William Bpitz
- Rwolf