Single Review on past events

The velocity of a car or a plane is in large part dependent on its shape. But how can its ideal shape be defined exactly? This is one of the questions Laura Iapichino deals with in her research that she presented in her talk on “Reduced Basis Method and Optimization Strategies for the Solution of complex Systems in Real Applications”.

She works with mathematical models: a set of equations that interpret a phenomenon in the abstract terms of mathematics. A physical model often is represented by a set of partial differential equations (PDEs) in which a set of parameters characterizes the system of interest and describes physical quantities (like source terms, boundary conditions, material properties) and/or geometrical configuration, so that the system solution is parameter dependent. The original physical problem (e.g. optimize the shape of a car) can be seen as a “problem of infinite dimension” that requires an infinite number of information to exactly represent it. Computers can only deal with a finite number of information. Hence an important role of the Numerical Analysis is to project an infinite dimensional problem to a finite dimensional problem. Model order reduction further reduces the dimension of the problem by reducing the number of information needed to solve the problem but keeping the same level of accuracy of a method that requires a significantly larger number and requiring very low computational time.

The general idea of Laura Iapichino´s project is 1. to replace the full-order, expensive finite element (FE) approximation with a reduced-order inexpensive solution obtained by the Reduced Basis (RB) method and to apply this procedure among several kind of applications and methodology; and 2. to efficiently use the RB method to define the prediction of the system solution required for each different value of the parameters by providing an efficient and rigorous error estimation.

“Our research goal is not only to use the Reduced Basis (RB) method for solving parametric complex systems, but also trying to improve the method itself, in order to improve results and to enlarge the range of applications”, explains the mathematician. “Moreover we are interested in optimal control problems where suitable functions of state variables have to be minimized. These problems can be efficiently solved by applying the RB method since state variables are solutions of the parametric PDEs representing a constraint for the minimization problem.”

In the end Laura Iapichino summarized the possibilities of the RB method: “With a classical technique, as the FE method, 3-D models of large structures can be solved to simulate how to design under stress, vibrations, heat, and other real-world conditions. These simulations require intensive computation done by powerful computers over many hours. The RB method, based on the FE one, can use pre-calculated supercomputer data for few specific conditions to solve the same model in few seconds, thus simulation that could take hours with conventional FE method, could rapidly be done on an ordinary smartphone.”