5,621 research outputs found
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 -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
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