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Publication# Certified Reduced Basis Method for Parametrized Partial Differential Equations: a Combination with ANOVA Expansion

Abstract

In many fields, the computation of an output depending on a field variable is of great interest. If the field variable depends on a high-dimensional parameter, the computational cost involved can be huge. Hence, it is necessary to find efficient and reliable methods to solve such a problem. In this report, we describe a method to solve in an efficient and reliable way elliptic coercive parametric partial differential equations which depend on high-dimensional parameters. The idea is to combine two methods already known, the Reduced Basis (RB) method and the ANOVA expansion. Since also the use of the Reduced Basis method for the approximation of high-dimensional parametric partial differential equations can be computationally expensive, it is important to find a method to approximate the solution in an efficient way. The method is divided in three steps, RB-ANOVA-RB. First, we use the Reduced Basis method with a big tolerance to have a coarse approximation of the output of interest. This allows us to use the ANOVA expansion in order to determine if there exists some less important parameters. If it is the case, we freeze them and then reapply the Reduced Basis method with a low tolerance to get a fine approximation of the output.

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Interest

In finance and economics, interest is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, the amount b

Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a d

Approximation

An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word approximation is derived from Latin approximatus, from prox

The multiquery solution of parametric partial differential equations (PDEs), that is, PDEs depending on a vector of parameters, is computationally challenging and appears in several engineering contexts, such as PDE-constrained optimization, uncertainty quantification or sensitivity analysis. When using the finite element (FE) method as approximation technique, an algebraic system must be solved for each instance of the parameter, leading to a critical bottleneck when we are in a multiquery context, a problem which is even more emphasized when dealing with nonlinear or time dependent PDEs. Several techniques have been proposed to deal with sequences of linear systems, such as truncated Krylov subspace recycling methods, deflated restarting techniques and approximate inverse preconditioners; however, these techniques do not satisfactorily exploit the parameter dependence. More recently, the reduced basis (RB) method, together with other reduced order modeling (ROM) techniques, emerged as an efficient tool to tackle parametrized PDEs.
In this thesis, we investigate a novel preconditioning strategy for parametrized systems which arise from the FE discretization of parametrized PDEs. Our preconditioner combines multiplicatively a RB coarse component, which is built upon the RB method, and a nonsingular fine grid preconditioner. The proposed technique hinges upon the construction of a new Multi Space Reduced Basis (MSRB) method, where a RB solver is built at each step of the chosen iterative method and trained to accurately solve the error equation.
The resulting preconditioner directly exploits the parameter dependence, since it is tailored to the class of problems at hand, and significantly speeds up the solution of the parametrized linear system.
We analyze the proposed preconditioner from a theoretical standpoint, providing assumptions which lead to its well-posedness and efficiency.
We apply our strategy to a broad range of problems described by parametrized PDEs:
(i) elliptic problems such as advection-diffusion-reaction equations, (ii) evolution problems such as time-dependent advection-diffusion-reaction equations or linear elastodynamics equations (iii) saddle-point problems such as Stokes equations, and, finally, (iv) Navier-Stokes equations.
Even though the structure of the preconditioner is similar for all these classes of problems, its fine and coarse components must be accurately chosen in order to provide the best possible results.
Several comparisons are made with respect to the current state-of-the-art preconditioning and ROM techniques.
Finally, we employ the proposed technique to speed up the solution of problems in the field of cardiovascular modeling.

Mathematical and numerical aspects of free surface flows are investigated. On one hand, the mathematical analysis of some free surface flows is considered. A model problem in one space dimension is first investigated. The Burgers equation with diffusion has to be solved on a space interval with one free extremity. This extremity is unknown and moves in time. An ordinary differential equation for the position of the free extremity of the interval is added in order to close the mathematical problem. Local existence in time and uniqueness results are proved for the problem with given domain, then for the free surface problem. A priori and a posteriori error estimates are obtained for the semi-discretization in space. The stability and the convergence of an Eulerian time splitting scheme are investigated. The same methodology is then used to study free surface flows in two space dimensions. The incompressible unsteady Navier-Stokes equations with Neumann boundary conditions on the whole boundary are considered. The whole boundary is assumed to be the free surface. An additional equation is used to describe the moving domain. Local existence in time and uniqueness results are obtained. On the other hand, a model for free surface flows in two and three space dimensions is investigated. The liquid is assumed to be surrounded by a compressible gas. The incompressible unsteady Navier-Stokes equations are assumed to hold in the liquid region. A volume-of-fluid method is used to describe the motion of the liquid domain. The velocity in the gas is disregarded and the pressure is computed by the ideal gas law in each gas bubble trapped by the liquid. A numbering algorithm is presented to recognize the bubbles of gas. Gas pressure is applied as a normal force on the liquid-gas interface. Surface tension effects are also taken into account for the simulation of bubbles or droplets flows. A method for the computation of the curvature is presented. Convergence and accuracy of the approximation of the curvature are discussed. A time splitting scheme is used to decouple the various physical phenomena. Numerical simulations are made in the frame of mould filling to show that the influence of gas on the free surface cannot be neglected. Curvature-driven flows are also considered.

Simone Deparis, Davide Forti, Alfio Quarteroni

We are interested in the approximation of partial differential equations on domains decomposed into two (or several) subdomains featuring non-conforming interfaces. The non-conformity may be due to different meshes and/or different polynomial degrees used from the two sides, or even to a geometrical mismatch. Across each interface, one subdomain is identified as master and the other as slave. We consider Galerkin methods for the discretization (such as finite element or spectral element methods) that make use of two interpolants for transferring information across the interface: one from master to slave and another one from slave to master. The former is used to ensure continuity of the primal variable (the problem solution), while the latter that of the dual variable (the normal flux). In particular, since the dual variable is expressed in weak form, we first compute a strong representation of the dual variable from the slave side, then interpolate it, transform the interpolated quantity back into weak form and eventually assign it to the master side. In case of slightly non-matching geometries, we use a radial-basis function interpolant instead of Lagrange interpolant. The proposed method is named INTERNODES (INTERpolation for NOnconforming DEcompositionS): it can be regarded as an alternative to the mortar element method and it is simpler to implement in a numerical code. We show on two dimension al problems that by using the Lagrange interpolation we obtain at least as good convergence results as with the mortar element method with any order of polynomials. When using low order polynomials, the radial-basis interpolant leads to the same convergence properties as the Lagrange interpolant. We conclude with a comparison between INTERNODES and a standard conforming approximation in a three dimensional case.