Adaptive greedy algorithms based on parameter-domain decomposition and reconstruction for the reduced basis method

The reduced basis method (RBM) empowers repeated and rapid evaluation of parametrized partial differential equations through an offline-online decomposition, a.k.a. a learning-execution process. A key feature of the method is a greedy algorithm repeatedly scanning the training set, a fine discretization of the parameter domain, to identify the next dimension of the parameter-induced solution manifold along which we expand the surrogate solution space. Although successfully applied to problems with fairly high parametric dimensions, the challenge is that this scanning cost dominates the offline cost due to it being proportional to the cardinality of the training set which is exponential with respect to the parameter dimension. In this work, we review three recent attempts in effectively delaying this curse of dimensionality, and propose two new hybrid strategies through successive refinement and multilevel maximization of the error estimate over the training set. All five offline-enhanced methods and the original greedy algorithm are tested and compared on {two types of problems: the thermal block problem and the geometrically parameterized Helmholtz problem

Proper generalized decomposition solutions within a domain decomposition strategy

Domain decomposition strategies and proper generalized decomposition are efficiently combined to obtain a fast evaluation of the solution approximation in parameterized elliptic problems with complex geometries. The classical difficulties associated to the combination of layered domains with arbitrarily oriented midsurfaces, which may require in‐plane–out‐of‐plane techniques, are now dismissed. More generally, solutions on large domains can now be confronted within a domain decomposition approach. This is done with a reduced cost in the offline phase because the proper generalized decomposition gives an explicit description of the solution in each subdomain in terms of the solution at the interface. Thus, the evaluation of the approximation in each subdomain is a simple function evaluation given the interface values (and the other problem parameters). The interface solution can be characterized by any a priori user‐defined approximation. Here, for illustration purposes, hierarchical polynomials are used. The repetitiveness of the subdomains is exploited to reduce drastically the offline computational effort. The online phase requires solving a nonlinear problem to determine all the interface solutions. However, this problem only has degrees of freedom on the interfaces and the Jacobian matrix is explicitly determined. Obviously, other parameters characterizing the solution (material constants, external loads, and geometry) can also be incorporated in the explicit description of the solution

Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries

The aim of this work is to solve parametrized partial differential equations in computational domains represented by networks of repetitive geometries by combining reduced basis and domain decomposition techniques. The main idea behind this approach is to compute once, locally and for few reference shapes, some representative finite element solutions for different values of the parameters and with a set of different suitable boundary conditions on the boundaries: these functions will represent the basis of a reduced space where the global solution is sought for. The continuity of the latter is assured by a classical domain decomposition approach. Test results on Poisson problem show the flexibility of the proposed method in which accuracy and computational time may be tuned by varying the number of reduced basis functions employed, or the set of boundary conditions used for defining locally the basis functions. The proposed approach simplifies the pre-computation of the reduced basis space by splitting the global problem into smaller local subproblems. Thanks to this feature, it allows dealing with arbitrarily complex network and features more flexibility than a classical global reduced basis approximation where the topology of the geometry is fixed

Localized model reduction for parameterized problems

In this contribution we present a survey of concepts in localized model order reduction methods for parameterized partial differential equations. The key concept of localized model order reduction is to construct local reduced spaces that have only support on part of the domain and compute a global approximation by a suitable coupling of the local spaces. In detail, we show how optimal local approximation spaces can be constructed and approximated by random sampling. An overview of possible conforming and non-conforming couplings of the local spaces is provided and corresponding localized a posteriori error estimates are derived. We introduce concepts of local basis enrichment, which includes a discussion of adaptivity. Implementational aspects of localized model reduction methods are addressed. Finally, we illustrate the presented concepts for multiscale, linear elasticity and fluid-flow problems, providing several numerical experiments. This work has been accepted as a chapter in P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Sileira. Handbook on Model Order Reduction. Walter De Gruyter GmbH, Berlin, 2019+

Stabilized reduced order models for low speed flows

This thesis presents the a stabilized projection-based Reduced Order Model (ROM) formulation in low speed fluid flows using a Variational Multi-Scale (VMS) approach. To develop this formulation we use a Finite Element (FE) method for the Full Order Model (FOM) and a Proper Orthogonal Decomposition (POD) to construct the basis. Additional to the ROM formulation, we introduce two techniques that became possible using this approach: a mesh-based hyper-reduction that uses an Adaptive Mesh Refinement (AMR) approach, and a domain decomposition scheme for ROMs. To illustrate and test the proposed formulation we use five different models: a convection–diffusion–reaction, the incompressible Navier–Stokes, a Boussinesq approximation, a low Mach number model, and a three-field incompressible Navier–Stokes.Esta tesis presenta un modelo de orden reducido estabilizado paran fluidos a baja velocidad utilizando un enfoque de multiescala variacional. Para desarrollar esta formulación utilizamos el método de elementos finitos para el modelo no reducido y una descomposición en autovalores del mismo para construir la base. Adicional a la formulación del modelo reducido, presentamos dos técnicas que podemos formular al utilizar este enfoque: una reducción adicional del dominio, basada en la reducción de la malla, donde usamos una técnica de refinamiento adaptativa y un esquema de descomposición de dominio para el modelo reducido. Para ilustrar y probar la formulación propuesta, utilizamos cuatro diferentes modelos fisicos: una ecuación de convección-difusión-reacción, la ecuación de Navier-Stokes para fluidos incompresibles, una aproximación de Boussinesq para la ecuación de Navier-Stokes, y una aproximación para números de Mach bajos de la ecuación de Navier-Stokes

Stabilized reduced order models for low speed flows

This thesis presents the a stabilized projection-based Reduced Order Model (ROM) formulation in low speed fluid flows using a Variational Multi-Scale (VMS) approach. To develop this formulation we use a Finite Element (FE) method for the Full Order Model (FOM) and a Proper Orthogonal Decomposition (POD) to construct the basis. Additional to the ROM formulation, we introduce two techniques that became possible using this approach: a mesh-based hyper-reduction that uses an Adaptive Mesh Refinement (AMR) approach, and a domain decomposition scheme for ROMs. To illustrate and test the proposed formulation we use five different models: a convection–diffusion–reaction, the incompressible Navier–Stokes, a Boussinesq approximation, a low Mach number model, and a three-field incompressible Navier–Stokes.Esta tesis presenta un modelo de orden reducido estabilizado paran fluidos a baja velocidad utilizando un enfoque de multiescala variacional. Para desarrollar esta formulación utilizamos el método de elementos finitos para el modelo no reducido y una descomposición en autovalores del mismo para construir la base. Adicional a la formulación del modelo reducido, presentamos dos técnicas que podemos formular al utilizar este enfoque: una reducción adicional del dominio, basada en la reducción de la malla, donde usamos una técnica de refinamiento adaptativa y un esquema de descomposición de dominio para el modelo reducido. Para ilustrar y probar la formulación propuesta, utilizamos cuatro diferentes modelos fisicos: una ecuación de convección-difusión-reacción, la ecuación de Navier-Stokes para fluidos incompresibles, una aproximación de Boussinesq para la ecuación de Navier-Stokes, y una aproximación para números de Mach bajos de la ecuación de Navier-Stokes.Postprint (published version

Stabilized reduced order models for low speed flows

This thesis presents the a stabilized projection-based Reduced Order Model (ROM) formulation in low speed fluid flows using a Variational Multi-Scale (VMS) approach. To develop this formulation we use a Finite Element (FE) method for the Full Order Model (FOM) and a Proper Orthogonal Decomposition (POD) to construct the basis. Additional to the ROM formulation, we introduce two techniques that became possible using this approach: a mesh-based hyper-reduction that uses an Adaptive Mesh Refinement (AMR) approach, and a domain decomposition scheme for ROMs. To illustrate and test the proposed formulation we use five different models: a convection–diffusion–reaction, the incompressible Navier–Stokes, a Boussinesq approximation, a low Mach number model, and a three-field incompressible Navier–Stokes.Esta tesis presenta un modelo de orden reducido estabilizado paran fluidos a baja velocidad utilizando un enfoque de multiescala variacional. Para desarrollar esta formulación utilizamos el método de elementos finitos para el modelo no reducido y una descomposición en autovalores del mismo para construir la base. Adicional a la formulación del modelo reducido, presentamos dos técnicas que podemos formular al utilizar este enfoque: una reducción adicional del dominio, basada en la reducción de la malla, donde usamos una técnica de refinamiento adaptativa y un esquema de descomposición de dominio para el modelo reducido. Para ilustrar y probar la formulación propuesta, utilizamos cuatro diferentes modelos fisicos: una ecuación de convección-difusión-reacción, la ecuación de Navier-Stokes para fluidos incompresibles, una aproximación de Boussinesq para la ecuación de Navier-Stokes, y una aproximación para números de Mach bajos de la ecuación de Navier-Stokes

MATHICSE Technical Report: Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries

The aim of this work is to solve parametrized partial differential equations in computational domains represented by networks of repetitive geometries by combining reduced basis and domain decomposition techniques. The main idea behind this approach is to compute once, locally and for few reference shapes, some representative finite element solutions for different values of the parameters and with a set of different suitable boundary conditions on the boundaries: these functions will represent the basis of a reduced space where the global solution is sought for. The continuity of the latter is assured by a classical domain decomposition approach. Test results on Poisson problem show the flexibility of the proposed method in which accuracy and computational time may be tuned by varying the number of reduced basis functions employed, or the set of boundary conditions used for defining locally the basis functions. The proposed approach simplifies the pre-computation of the reduced basis space by splitting the global problem into smaller local subproblems.Thanks to this feature, it allows dealing with arbitrarily complex network and features more flexibility than a classical global reduced basis approximation where the topology of the geometry is fixed

Proper Generalized Decomposition solutions within a Domain Decomposition strategy

"This is the peer reviewed version of the following article: Huerta, Antonio, Enrique Nadal, and Francisco Chinesta. 2018. Proper Generalized Decomposition Solutions within a Domain Decomposition Strategy. International Journal for Numerical Methods in Engineering 113 (13). Wiley: 1972 94. doi:10.1002/nme.5729, which has been published in final form at https://doi.org/10.1002/nme.5729. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving."[EN] Domain decomposition strategies and proper generalized decomposition are efficiently combined to obtain a fast evaluation of the solution approximation in parameterized elliptic problems with complex geometries. The classical difficulties associated to the combination of layered domains with arbitrarily oriented midsurfaces, which may require in-plane-out-of-plane techniques, are now dismissed. More generally, solutions on large domains can now be confronted within a domain decomposition approach. This is done with a reduced cost in the offline phase because the proper generalized decomposition gives an explicit description of the solution in each subdomain in terms of the solution at the interface. Thus, the evaluation of the approximation in each subdomain is a simple function evaluation given the interface values (and the other problem parameters). The interface solution can be characterized by any a priori user-defined approximation. Here, for illustration purposes, hierarchical polynomials are used. The repetitiveness of the subdomains is exploited to reduce drastically the offline computational effort. The online phase requires solving a nonlinear problem to determine all the interface solutions. However, this problem only has degrees of freedom on the interfaces and the Jacobian matrix is explicitly determined. Obviously, other parameters characterizing the solution (material constants, external loads, and geometry) can also be incorporated in the explicit description of the solution.European Commission, Grant/Award Number: MSCA ITN-ETN 675919; ESI group, Grant/Award Number: ENSAM ESI Chair; Spanish Ministry of Economy and Competitiveness, Grant/Award Number: DPI2017-85139-C2-2-R; Generalitat de Catalunya, Grant/Award Number: 2014SGR1471Huerta, A.; Nadal, E.; Chinesta Soria, FJ. (2018). Proper Generalized Decomposition solutions within a Domain Decomposition strategy. International Journal for Numerical Methods in Engineering. 113(13):1972-1994. https://doi.org/10.1002/nme.5729S1972199411313Ammar, A., Mokdad, B., Chinesta, F., & Keunings, R. (2006). A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. 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