A certified RB method for PDE-constrained parametric optimization problems

Abstract We present a certified reduced basis (RB) framework for the efficient solution of PDE-constrained parametric optimization problems. We consider optimization problems (such as optimal control and optimal design) governed by elliptic PDEs and involving possibly non-convex cost functionals, assuming that the control functions are described in terms of a parameter vector. At each optimization step, the high-fidelity approximation of state and adjoint problems is replaced by a certified RB approximation, thus yielding a very efficient solution through an "optimize-then-reduce" approach. We develop a posteriori error estimates for the solutions of state and adjoint problems, the cost functional, its gradient and the optimal solution. We confirm our theoretical results in the case of optimal control/design problems dealing with potential and thermal flows

Non equilibrium optical properties in semiconductors from first--principles: a combined theoretical and experimental study of bulk silicon

The calculation of the equilibrium optical properties of bulk silicon by using the Bethe--Salpeter equation solved in the Kohn--Sham basis represents a cornerstone in the development of an ab--initio approach to the optical and electronic properties of materials. Nevertheless calculations of the {\em transient} optical spectrum using the same efficient and successful scheme are scarce. We report, here, a joint theoretical and experimental study of the transient reflectivity spectrum of bulk silicon. Femtosecond transient reflectivity is compared to a parameter--free calculation based on the non--equilibrium Bethe--Salpeter equation. By providing an accurate description of the experimental results we disclose the different phenomena that determine the transient optical response of a semiconductor. We give a parameter--free interpretation of concepts like bleaching, photo--induced absorption and stimulated emission, beyond the Fermi golden rule. We also introduce the concept of optical gap renormalization, as a generalization of the known mechanism of band gap renormalization. The present scheme successfully describes the case of bulk silicon, showing its universality and accuracy.Comment: 14 pages, 13 figure

Computational Reduction for Parametrized PDEs: Strategies and Applications

In this paper we present a compact review on the mostly used techniques for computational reduction in numerical approximation of partial differential equations. We highlight the common features of these techniques and provide a detailed presentation of the reduced basis method, focusing on greedy algorithms for the construction of the reduced spaces. An alternative family of reduction techniques based on surrogate response surface models is briefly recalled too. Then, a simple example dealing with inviscid flows is presented, showing the reliability of the reduced basis method and a comparison between this technique and some surrogate model

Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order models

Deep learning-based reduced order models (DL-ROMs) have been recently proposed to overcome common limitations shared by conventional ROMs - built, e.g., exclusively through proper orthogonal decomposition (POD) - when applied to nonlinear time-dependent parametrized PDEs. In particular, POD-DL-ROMs can achieve extreme efficiency in the training stage and faster than real-time performances at testing, thanks to a prior dimensionality reduction through POD and a DL-based prediction framework. Nonetheless, they share with conventional ROMs poor performances regarding time extrapolation tasks. This work aims at taking a further step towards the use of DL algorithms for the efficient numerical approximation of parametrized PDEs by introducing the $\mu t$-POD-LSTM-ROM framework. This novel technique extends the POD-DL-ROM framework by adding a two-fold architecture taking advantage of long short-term memory (LSTM) cells, ultimately allowing long-term prediction of complex systems' evolution, with respect to the training window, for unseen input parameter values. Numerical results show that this recurrent architecture enables the extrapolation for time windows up to 15 times larger than the training time domain, and achieves better testing time performances with respect to the already lightning-fast POD-DL-ROMs.Comment: 28 page

A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs

Traditional reduced order modeling techniques such as the reduced basis (RB) method (relying, e.g., on proper orthogonal decomposition (POD)) suffer from severe limitations when dealing with nonlinear time-dependent parametrized PDEs, because of the fundamental assumption of linear superimposition of modes they are based on. For this reason, in the case of problems featuring coherent structures that propagate over time such as transport, wave, or convection-dominated phenomena, the RB method usually yields inefficient reduced order models (ROMs) if one aims at obtaining reduced order approximations sufficiently accurate compared to the high-fidelity, full order model (FOM) solution. To overcome these limitations, in this work, we propose a new nonlinear approach to set reduced order models by exploiting deep learning (DL) algorithms. In the resulting nonlinear ROM, which we refer to as DL-ROM, both the nonlinear trial manifold (corresponding to the set of basis functions in a linear ROM) as well as the nonlinear reduced dynamics (corresponding to the projection stage in a linear ROM) are learned in a non-intrusive way by relying on DL algorithms; the latter are trained on a set of FOM solutions obtained for different parameter values. In this paper, we show how to construct a DL-ROM for both linear and nonlinear time-dependent parametrized PDEs; moreover, we assess its accuracy on test cases featuring different parametrized PDE problems. Numerical results indicate that DL-ROMs whose dimension is equal to the intrinsic dimensionality of the PDE solutions manifold are able to approximate the solution of parametrized PDEs in situations where a huge number of POD modes would be necessary to achieve the same degree of accuracy.Comment: 28 page

A staggered-in-time and non-conforming-in-space numerical framework for realistic cardiac electrophysiology outputs

Computer-based simulations of non-invasive cardiac electrical outputs, such as electrocardiograms and body surface potential maps, usually entail severe computational costs due to the need of capturing fine-scale processes and to the complexity of the heart-torso morphology. In this work, we model cardiac electrical outputs by employing a coupled model consisting of a reaction-diffusion model - either the bidomain model or the most efficient pseudo-bidomain model - on the heart, and an elliptic model in the torso. We then solve the coupled problem with a segregated and staggered in-time numerical scheme, that allows for independent and infrequent solution in the torso region. To further reduce the computational load, main novelty of this work is in introduction of an interpolation method at the interface between the heart and torso domains, enabling the use of non-conforming meshes, and the numerical framework application to realistic cardiac and torso geometries. The reliability and efficiency of the proposed scheme is tested against the corresponding state-of-the-art bidomain-torso model. Furthermore, we explore the impact of torso spatial discretization and geometrical non-conformity on the model solution and the corresponding clinical outputs. The investigation of the interface interpolation method provides insights into the influence of torso spatial discretization and of the geometrical non-conformity on the simulation results and their clinical relevance.Comment: 26 pages,11 figures, 3 table

Promjene urbane strukture i svakodnevice u Puntu na otoku Krku od sredine XX. stoljeća

Punat je naselje smješteno na jugozapadnom dijelu otoka Krka, na istočnoj obali Puntarske uvale – Drage. Najstariju jezgru naselja čine ulice Kolušin, Galija, Javorika i Guvnić, koncentrirani oko Male place, a takva struktura naselja vidljiva je na karti iz 1821. godine. Sredinom XIX. stoljeća, dolazi do naglog porasta broja stanovnika, što je popraćeno širenjem naselja na sjever i zapad, prema obali, vidljivo na karti iz 1879. godine. Krajem XIX. i početkom XX. stoljeća, u Puntu se počinje razvijati turizam koji potiče nove izgradnje, posebice ugostiteljskih objekata. Prvi i Drugi svjetski rat privremeno zaustavljaju napredak, no sredinom XX. stoljeća, turizam doživljava novi uzlet. Otada, posebice nakon 1960-ih, u Punat dolazi veliki broj domaćih turista koji kupuju i adaptiraju stare kuće. Počinje izgradnja potpuno novoga dijela Punta, Buke, čime se naselje širi prema jugu. Promjena urbane strukture popraćena je promjenom svakodnevice gdje dolazi do prelaska s poljoprivrede, stočarstva i ribarstva na novu granu gospodarstva – turizam. Dok kuće u staroj jezgri naselja imaju elemente tipične puntarske kuće,kuće na Buki su izgrađene novim materijalima i prilagođene novom obliku privređivanja. S obzirom da je stara jezgra naselja zaštićena kao kulturno dobro 1968. godine, uklanjanja i adaptacije starih arhitektonskih elemenata kojima se potpuno mijenja urbana slika naselja, trebala bi se adekvatno kontrolirati.Punat is a small town situated on the south-west part of the island Krk, on the east coast of Punat cove named Draga. The earliest district consists of streets Kolušin, Galija, Javorika and Guvnić, concentrated around the square which can be seen on the map from 1821. In the mid 19th century, the population started to grow, followed by town expansion to the west which we can see on the map from 1879. In the late 19th century, tourism started to expand, along with new buildings, but it was hindered by the two World wars. After the 1960s, tourism starts to develop again and many old houses were sold and adapted. In these years, thanks to the new constructions, a new part of town named Buka was created. Along with urban expansion, everyday life changed as well. People were no longer farmers, cattlemen or fishermen. Now they were working in tourism. The main difference between the houses in the old part of town and Buka is that the new ones no longer contained traditional architectural elements of Punat houses. They were built from new materials and adjusted to a new economy market. Considering that the old town center was officially recognized as cultural heritage in 1968, every removal or adaption of old architecture must be closely and adequately monitored