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.

1 aPintore, Moreno1 aPichi, Federico1 aHess, Martin, W.1 aRozza, Gianluigi1 aCanuto, Claudio uhttps://www.math.sissa.it/publication/efficient-computation-bifurcation-diagrams-deflated-approach-reduced-basis-spectral-000496nas a2200109 4500008004100000245009200041210006900133100002000202700002000222700002100242856012300263 2021 eng d00aModel order reduction for bifurcating phenomena in Fluid-Structure Interaction problems0 aModel order reduction for bifurcating phenomena in FluidStructur1 aKhamlich, Moaad1 aPichi, Federico1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/model-order-reduction-bifurcating-phenomena-fluid-structure-interaction-problems00597nas a2200121 4500008004100000245014900041210006900190100002000259700002200279700002400301700002100325856012900346 2020 eng d00aDriving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier-Stokes equations with model order reduction0 aDriving bifurcating parametrized nonlinear PDEs by optimal contr1 aPichi, Federico1 aStrazzullo, Maria1 aBallarin, Francesco1 aRozza, Gianluigi uhttps://www.math.sissa.it/publication/driving-bifurcating-parametrized-nonlinear-pdes-optimal-control-strategies-application02133nas a2200145 4500008004100000245011600041210006900157520162200226100002001848700002001868700002101888700002101909700002001930856003701950 2020 eng d00aEfficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method0 aEfficient computation of bifurcation diagrams with a deflated ap3 aThe 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.

1 aPintore, Moreno1 aPichi, Federico1 aHess, Martin, W.1 aRozza, Gianluigi1 aCanuto, Claudio uhttps://arxiv.org/abs/1912.0608901537nas a2200121 4500008004100000245011600041210006900157520098400226100002001210700002101230700002101251856014301272 2020 eng d00aA reduced order modeling technique to study bifurcating phenomena: Application to the gross-pitaevskii equation0 areduced order modeling technique to study bifurcating phenomena 3 aWe 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.

1 aPichi, Federico1 aQuaini, Annalisa1 aRozza, Gianluigi uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85096768803&doi=10.1137%2f20M1313106&partnerID=40&md5=47d6012d10854c2f9a04b9737f87059201419nas a2200121 4500008004100000245010700041210006900148520098100217100002001198700002101218700002101239856003701260 2020 eng d00aA Reduced Order technique to study bifurcating phenomena: application to the Gross-Pitaevskii equation0 aReduced Order technique to study bifurcating phenomena applicati3 aWe 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.

1 aPichi, Federico1 aQuaini, Annalisa1 aRozza, Gianluigi uhttps://arxiv.org/abs/1907.0708201440nas a2200133 4500008004100000245010900041210006900150300001200219490000700231520087600238100002001114700002101134856015101155 2019 eng d00aReduced Basis Approaches for Parametrized Bifurcation Problems held by Non-linear Von Kármán Equations0 aReduced Basis Approaches for Parametrized Bifurcation Problems h a112-1350 v813 aThis 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.

1 aPichi, Federico1 aRozza, Gianluigi uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85068973907&doi=10.1007%2fs10915-019-01003-3&partnerID=40&md5=a09af83ce45183d6965cdb79d87a919b01370nas a2200133 4500008004100000245010900041210006900150300001400219490000700233520091800240100002001158700002101178856003701199 2019 eng d00aReduced basis approaches for parametrized bifurcation problems held by non-linear Von Kármán equations0 aReduced basis approaches for parametrized bifurcation problems h a112–1350 v813 aThis 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

1 aPichi, Federico1 aRozza, Gianluigi uhttps://arxiv.org/abs/1804.0201400501nas a2200121 4500008004100000245013400041210006900175490000700244100002100251700002000272700002100292856006600313 2018 eng d00aReduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings0 aReduced Basis Approximation and A Posteriori Error Estimation Ap0 v151 aHuynh, D., B. P.1 aPichi, Federico1 aRozza, Gianluigi uhttps://link.springer.com/chapter/10.1007/978-3-319-94676-4_802258nas a2200145 4500008004100000245013400041210006900175300001200244490000700256520163800263100001801901700002001919700002101939856015201960 2018 eng d00aReduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings0 aReduced Basis Approximation and A Posteriori Error Estimation Ap a203-2470 v153 aIn 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.

1 aHuynh, D.B.P.1 aPichi, Federico1 aRozza, Gianluigi uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85055036627&doi=10.1007%2f978-3-319-94676-4_8&partnerID=40&md5=e9c07038e7bcc6668ec702c0653410dc