In this contribution we device and analyze improved variants of the non-conforming dual approach for trust-region reduced basis (TR-RB) approximation of PDE-constrained parameter optimization that has recently been introduced in [Keil et al.. A non-conforming dual approach for adaptive Trust-Region Reduced Basis approximation of PDE-constrained optimization. arXiv:2006.09297, 2020]. The proposed methods use model order reduction techniques for parametrized PDEs to significantly reduce the computational demand of parameter optimization with PDE constraints in the context of large-scale or multi-scale applications. The adaptive TR approach allows to localize the reduction with respect to the parameter space along the path of optimization without wasting unnecessary resources in an offline phase. The improved variants employ projected Newton methods to solve the local optimization problems within each TR step to benefit from high convergence rates. This implies new strategies in constructing the RB spaces, together with an estimate for the approximation of the hessian. Moreover, we present a new proof of convergence of the TR-RB method based on infinite-dimensional arguments, not restricted to the particular case of an RB approximation and provide an a posteriori error estimate for the approximation of the optimal parameter. Numerical experiments demonstrate the efficiency of the proposed methods.
In this contribution we propose and rigorously analyze new variants of adaptive Trust-Region methods for parameter optimization with PDE constraints and bilateral parameter constraints. The approach employs successively enriched Reduced Basis surrogate models that are constructed during the outer optimization loop and used as model function for the Trust-Region method. Each Trust-Region sub-problem is solved with the projected BFGS method. Moreover, we propose a non-conforming dual (NCD) approach to improve the standard RB approximation of the optimality system. Rigorous improved a posteriori …
This work focuses on the development of a non-conforming domain decomposition method for the approximation of PDEs based on weakly imposed transmission conditions: the continuity of the global solution is enforced by a discrete number of Lagrange multipliers defined over the interfaces of adjacent subdomains. The method falls into the class of primal hybrid methods, which also include the well-known mortar method. Differently from the mortar method, we discretize the space of basis functions on the interface by spectral approximation independently of the discretization of the two adjacent doma…
In the present paper non-convex multi-objective parameter optimization problems are considered which are governed by elliptic parametrized partial differential equations (PDEs). To solve these problems numerically the Pascoletti-Serafini scalarization is applied and the obtained scalar optimization problems are solved by an augmented Lagrangian method. However, due to the PDE constraints, the numerical solution is very expensive so that a model reduction is utilized by using the reduced basis (RB) method. The quality of the RB approximation is ensured by a trust-region strategy which does not …
Physics-informed machine learning and inverse modeling require the solution of ill-conditioned non-convex optimization problems. First-order methods, such as SGD and ADAM, and quasi-Newton methods, such as BFGS and L-BFGS, have been applied with some success to optimization problems involving deep neural networks in computational engineering inverse problems. However, empirical evidence shows that convergence and accuracy for these methods remain a challenge. Our study unveiled at least two intrinsic defects of these methods when applied to coupled systems of partial differential equations (PD…
We develop a general convergence theory for adaptive discontinuous Galerkin methods for elliptic PDEs covering the popular SIPG, NIPG and LDG schemes as well as all practically relevant marking strategies. Another key feature of the presented result is, that it holds for penalty parameters only necessary for the standard analysis of the respective scheme. The analysis is based on a quasi interpolation into a newly developed limit space of the adaptively created non-conforming discrete spaces, which enables to generalise the basic convergence result for conforming adaptive finite element method…
Coupled 3D-1D problems arise in many practical applications, in an attempt to reduce the computational burden in simulations where cylindrical inclusions with a small section are embedded in a much larger domain. Nonetheless the resolution of such problems can be non trivial, both from a mathematical and a geometrical standpoint. Indeed 3D-1D coupling requires to operate in non standard function spaces, and, also, simulation geometries can be complex for the presence of multiple intersecting domains. Recently, a PDE-constrained optimization based formulation has been proposed for such problems…
Reference-free logged energy-oracle recovery for neural approximations of symmetric coercive variational problems: conforming Riesz reconstruction and archive-level selection
Laboratory for Analysis and Modeling of Systems and Decision Support (LAMSAD), Hassan 1st University of Settat（塞塔特哈桑一世大学系统与决策支持分析建模实验室）；Al Akhawayn University in Ifrane（伊夫兰阿赫拉万大学）
This thesis presents recent advances in model order reduction methods with the primary aim to construct online-efficient reduced surrogate models for parameterized multiscale phenomena and accelerate large-scale PDE-constrained parameter optimization methods. In particular, we present several different adaptive RB approaches that can be used in an error-aware trust-region framework for progressive construction of a surrogate model used during a certified outer optimization loop. In addition, we elaborate on several different enhancements for the trust-region reduced basis (TR-RB) algorithm and…
It is quite common that a nonlinear partial differential equation (PDE) admits multiple distinct solutions and each solution may carry a unique physical meaning. One typical approach for finding multiple solutions is to use the Newton method with different initial guesses that ideally fall into the basins of attraction confining the solutions. In this paper, we propose a fast and accurate numerical method for multiple solutions comprised of three ingredients: (i) a well-designed spectral-Galerkin discretization of the underlying PDE leading to a nonlinear algebraic system (NLAS) with multiple …
An adaptive approach to using reduced-order models as surrogates in PDE-constrained optimization is introduced that breaks the traditional offline-online framework of model order reduction. A sequence of optimization problems constrained by a given Reduced-Order Model (ROM) is defined with the goal of converging to the solution of a given PDE-constrained optimization problem. For each reduced optimization problem, the constraining ROM is trained from sampling the High-Dimensional Model (HDM) at the solution of some of the previous problems in the sequence. The reduced optimization problems are…
Variationally correct operator learning: Reduced basis neural operator with a posteriori error estimation
School of Computational Science and Engineering（计算科学与工程学院）；Georgia Institute of Technology（佐治亚理工学院）；Department of Mathematics（数学系）；University of South Carolina（南卡罗来纳大学）
An efficient, globally convergent method for optimization under uncertainty using adaptive model reduction and sparse grids
Department of Mathematics, Lawrence Berkeley National Laboratory, University of California, Berkeley（加州大学伯克利分校劳伦斯伯克利国家实验室数学系）；University of California, Berkeley（加州大学伯克利分校）；Department of Aerospace and Mechanical Engineering, University of Notre Dame（圣母院大学航空航天与机械工程系）；University of Notre Dame（圣母院大学）；Sandia National Laboratories（桑迪亚国家实验室）
