The majority of the most common physical phenomena can be described using partial differential equations (PDEs). However, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical methods to compute the solutions of systems of PDEs, even if the classical techniques are usually able to compute only a single solution for any value of a parameter when more branches exist. In this work, we implemented an elaborated deflated continuation method that relies on the spectral element method (SEM) and on the reduced basis (RB) one to efficiently compute bifurcation diagrams with more parameters and more bifurcation points. The deflated continuation method can be obtained combining the classical continuation method and the deflation one: the former is used to entirely track each known branch of the diagram, while the latter is exploited to discover the new ones. Finally, when more than one parameter is considered, the efficiency of the computation is ensured by the fact that the diagrams can be computed during the online phase while, during the offline one, one only has to compute one-dimensional diagrams. In this work, after a more detailed description of the method, we will show the results that can be obtained using it to compute a bifurcation diagram associated with a problem governed by the Navier-Stokes equations.

%B Advances in Computational Mathematics %V 47 %G eng %R 10.1007/s10444-020-09827-6 %0 Journal Article %J Advances in Computational Mathematics %D 2020 %T Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method %A Moreno Pintore %A Federico Pichi %A Martin Hess %A Gianluigi Rozza %A Claudio Canuto %XThe majority of the most common physical phenomena can be described using partial differential equations (PDEs). However, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical methods to compute the solutions of systems of PDEs, even if the classical techniques are usually able to compute only a single solution for any value of a parameter when more branches exist. In this work we implemented an elaborated deflated continuation method, that relies on the spectral element method (SEM) and on the reduced basis (RB) one, to efficiently compute bifurcation diagrams with more parameters and more bifurcation points. The deflated continuation method can be obtained combining the classical continuation method and the deflation one: the former is used to entirely track each known branch of the diagram, while the latter is exploited to discover the new ones. Finally, when more than one parameter is considered, the efficiency of the computation is ensured by the fact that the diagrams can be computed during the online phase while, during the offline one, one only has to compute one-dimensional diagrams. In this work, after a more detailed description of the method, we will show the results that can be obtained using it to compute a bifurcation diagram associated with a problem governed by the Navier-Stokes equations.

%B Advances in Computational Mathematics %G eng %U https://arxiv.org/abs/1912.06089 %0 Journal Article %J SIAM Journal on Scientific Computing %D 2020 %T A reduced order modeling technique to study bifurcating phenomena: Application to the gross-pitaevskii equation %A Federico Pichi %A Annalisa Quaini %A Gianluigi Rozza %XWe propose a computationally efficient framework to treat nonlinear partial differential equations having bifurcating solutions as one or more physical control parameters are varied. Our focus is on steady bifurcations. Plotting a bifurcation diagram entails computing multiple solutions of a parametrized, nonlinear problem, which can be extremely expensive in terms of computational time. In order to reduce these demanding computational costs, our approach combines a continuation technique and Newton's method with a reduced order modeling (ROM) technique, suitably supplemented with a hyperreduction method. To demonstrate the effectiveness of our ROM approach, we trace the steady solution branches of a nonlinear Schrödinger equation, called the Gross{Pitaevskii equation, as one or two physical parameters are varied. In the two-parameter study, we show that our approach is 60 times faster in constructing a bifurcation diagram than a standard full order method.

%B SIAM Journal on Scientific Computing %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85096768803&doi=10.1137%2f20M1313106&partnerID=40&md5=47d6012d10854c2f9a04b9737f870592 %R 10.1137/20M1313106 %0 Journal Article %J SIAM Journal on Scientific Computing %D 2020 %T A Reduced Order technique to study bifurcating phenomena: application to the Gross-Pitaevskii equation %A Federico Pichi %A Annalisa Quaini %A Gianluigi Rozza %XWe propose a computationally efficient framework to treat nonlinear partial differential equations having bifurcating solutions as one or more physical control parameters are varied. Our focus is on steady bifurcations. Plotting a bifurcation diagram entails computing multiple solutions of a parametrized, nonlinear problem, which can be extremely expensive in terms of computational time. In order to reduce these demanding computational costs, our approach combines a continuation technique and Newton's method with a Reduced Order Modeling (ROM) technique, suitably supplemented with a hyper-reduction method. To demonstrate the effectiveness of our ROM approach, we trace the steady solution branches of a nonlinear Schrödinger equation, called Gross-Pitaevskii equation, as one or two physical parameters are varied. In the two parameter study, we show that our approach is 60 times faster in constructing a bifurcation diagram than a standard Full Order Method.

%B SIAM Journal on Scientific Computing %G eng %U https://arxiv.org/abs/1907.07082 %R https://doi.org/10.1137/20M1313106 %0 Journal Article %J Journal of Scientific Computing %D 2019 %T Reduced Basis Approaches for Parametrized Bifurcation Problems held by Non-linear Von Kármán Equations %A Federico Pichi %A Gianluigi Rozza %XThis work focuses on the computationally efficient detection of the buckling phenomena and bifurcation analysis of the parametric Von Kármán plate equations based on reduced order methods and spectral analysis. The computational complexity—due to the fourth order derivative terms, the non-linearity and the parameter dependence—provides an interesting benchmark to test the importance of the reduction strategies, during the construction of the bifurcation diagram by varying the parameter(s). To this end, together the state equations, we carry out also an analysis of the linearized eigenvalue problem, that allows us to better understand the physical behaviour near the bifurcation points, where we lose the uniqueness of solution. We test this automatic methodology also in the two parameter case, understanding the evolution of the first buckling mode.

%B Journal of Scientific Computing %V 81 %P 112-135 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85068973907&doi=10.1007%2fs10915-019-01003-3&partnerID=40&md5=a09af83ce45183d6965cdb79d87a919b %R 10.1007/s10915-019-01003-3 %0 Journal Article %D 2019 %T Reduced basis approaches for parametrized bifurcation problems held by non-linear Von Kármán equations %A Federico Pichi %A Gianluigi Rozza %XThis work focuses on the computationally efficient detection of the buckling phenomena and bifurcation analysis of the parametric Von Kármán plate equations based on reduced order methods and spectral analysis. The computational complexity - due to the fourth order derivative terms, the non-linearity and the parameter dependence - provides an interesting benchmark to test the importance of the reduction strategies, during the construction of the bifurcation diagram by varying the parameter(s). To this end, together the state equations, we carry out also an analysis of the linearized eigenvalue problem, that allows us to better understand the physical behaviour near the bifurcation points, where we lose the uniqueness of solution. We test this automatic methodology also in the two parameter case, understanding the evolution of the first buckling mode. journal = Journal of Scientific Computing

%V 81 %P 112–135 %G eng %U https://arxiv.org/abs/1804.02014 %R 10.1007/s10915-019-01003-3 %0 Journal Article %J SEMA SIMAI Springer Series %D 2018 %T Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings %A D.B.P. Huynh %A Federico Pichi %A Gianluigi Rozza %XIn this work we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for elasticity problems in affinely parametrized geometries. The essential ingredients of the methodology are: a Galerkin projection onto a low-dimensional space associated with a smooth “parametric manifold”—dimension reduction; an efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations—rapid convergence; an a posteriori error estimation procedures—rigorous and sharp bounds for the functional outputs related with the underlying solution or related quantities of interest, like stress intensity factor; and Offline-Online computational decomposition strategies—minimum marginal cost for high performance in the real-time and many-query (e.g., design and optimization) contexts. We present several illustrative results for linear elasticity problem in parametrized geometries representing 2D Cartesian or 3D axisymmetric configurations like an arc-cantilever beam, a center crack problem, a composite unit cell or a woven composite beam, a multi-material plate, and a closed vessel. We consider different parametrization for the systems: either physical quantities—to model the materials and loads—and geometrical parameters—to model different geometrical configurations—with isotropic and orthotropic materials working in plane stress and plane strain approximation. We would like to underline the versatility of the methodology in very different problems. As last example we provide a nonlinear setting with increased complexity.

%B SEMA SIMAI Springer Series %V 15 %P 203-247 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85055036627&doi=10.1007%2f978-3-319-94676-4_8&partnerID=40&md5=e9c07038e7bcc6668ec702c0653410dc %R 10.1007/978-3-319-94676-4_8 %0 Book Section %B Numerical Methods for PDEs %D 2018 %T Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings %A Huynh, D. B. P. %A Federico Pichi %A Gianluigi Rozza %B Numerical Methods for PDEs %V 15 %G eng %U https://link.springer.com/chapter/10.1007/978-3-319-94676-4_8 %R https://doi.org/10.1007/978-3-319-94676-4_8