The reduced basis method (RBM) empowers repeated and rapid evaluation of parametrized partial differential equations through an offline–online decomposition, a.k.a. a learning‐execution process. A key feature of the method is a greedy algorithm repeatedly scanning the training set, a fine discretization of the parameter domain, to identify the next dimension of the parameter‐induced solution manifold along which we expand the surrogate solution space. Although successfully applied to problems with fairly high parametric dimensions, the challenge is that this scanning cost dominates the offline cost due to it being proportional to the cardinality of the training set which is exponential with respect to the parameter dimension. In this work, we review three recent attempts in effectively delaying this curse of dimensionality, and propose two new hybrid strategies through successive refinement and multilevel maximization of the error estimate over the training set. All five offline‐enhanced methods and the original greedy algorithm are tested and compared on two types of problems: the thermal block problem and the geometrically parameterized Helmholtz problem.
Reduced Basis Greedy Selection Using Random Training Sets
Reduced bases have been introduced for the approximation of parametrized PDEs in applications where many online queries are required. Their numerical efficiency for such problems has been theoretically confirmed in Binev et al. ( SIAM J. Math. Anal. 43 (2011) 1457–1472) and DeVore et al. ( Constructive Approximation 37 (2013) 455–466), where it is shown that the reduced basis space V n of dimension n , constructed by a certain greedy strategy, has approximation error similar to that of the optimal space associated to the Kolmogorov n -width of the solution manifold. The greedy construction of the reduced basis space is performed in an offline stage which requires at each step a maximization of the current error over the parameter space. For the purpose of numerical computation, this maximization is performed over a finite training set obtained through a discretization of the parameter domain. To guarantee a final approximation error ε for the space generated by the greedy algorithm requires in principle that the snapshots associated to this training set constitute an approximation net for the solution manifold with accuracy of order ε . Hence, the size of the training set is the ε covering number for M and this covering number typically behaves like exp( Cε −1/s ) for some C > 0 when the solution manifold has n -width decay O ( n −s ). Thus, the shear size of the training set prohibits implementation of the algorithm when ε is small. The main result of this paper shows that, if one is willing to accept results which hold with high probability, rather than with certainty, then for a large class of relevant problems one may replace the fine discretization by a random training set of size polynomial in ε −1 . Our proof of this fact is established by using inverse inequalities for polynomials in high dimensions.
more »
« less
- NSF-PAR ID:
- 10181903
- Date Published:
- Journal Name:
- ESAIM: Mathematical Modelling and Numerical Analysis
- Volume:
- 54
- Issue:
- 5
- ISSN:
- 0764-583X
- Page Range / eLocation ID:
- 1509 to 1524
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
more » « less
Abstract -
more » « less
Abstract We consider the problem of covering multiple submodular constraints. Given a finite ground set
N , a weight function$$w: N \rightarrow \mathbb {R}_+$$ r monotone submodular functions$$f_1,f_2,\ldots ,f_r$$ N and requirements$$k_1,k_2,\ldots ,k_r$$ $$S \subseteq N$$ $$f_i(S) \ge k_i$$ $$1 \le i \le r$$ Multi-Submod-Cover and it was recently considered by Har-Peled and Jones (Few cuts meet many point sets. CoRR.arxiv:abs1808.03260 Har-Peled and Jones 2018) who were motivated by an application in geometry. Even with$$r=1$$ Multi-Submod-Cover generalizes the well-known Submodular Set Cover problem (Submod-SC ), and it can also be easily reduced toSubmod-SC . A simple greedy algorithm gives an$$O(\log (kr))$$ $$k = \sum _i k_i$$ Multi-Submod-Cover that covers each constraint to within a factor of$$(1-1/e-\varepsilon )$$ $$O(\frac{1}{\epsilon }\log r)$$ $$f_i$$ Partial-SC ), covering integer programs (CIPs ) and multiple vertex cover constraints Bera et al. (Theoret Comput Sci 555:2–8 Bera et al. 2014). Both these algorithms are based on mathematical programming relaxations that avoid the limitations of the greedy algorithm. We demonstrate the implications of our algorithms and related ideas to several applications ranging from geometric covering problems to clustering with outliers. Our work highlights the utility of the high-level model and the lens of submodularity in addressing this class of covering problems. -
In many mechanistic medical, biological, physical, and engineered spatiotemporal dynamic models the numerical solution of partial differential equations (PDEs), especially for diffusion, fluid flow and mechanical relaxation, can make simulations impractically slow. Biological models of tissues and organs often require the simultaneous calculation of the spatial variation of concentration of dozens of diffusing chemical species. One clinical example where rapid calculation of a diffusing field is of use is the estimation of oxygen gradients in the retina, based on imaging of the retinal vasculature, to guide surgical interventions in diabetic retinopathy. Furthermore, the ability to predict blood perfusion and oxygenation may one day guide clinical interventions in diverse settings, i.e., from stent placement in treating heart disease to BOLD fMRI interpretation in evaluating cognitive function (Xie et al., 2019 ; Lee et al., 2020 ). Since the quasi-steady-state solutions required for fast-diffusing chemical species like oxygen are particularly computationally costly, we consider the use of a neural network to provide an approximate solution to the steady-state diffusion equation. Machine learning surrogates, neural networks trained to provide approximate solutions to such complicated numerical problems, can often provide speed-ups of several orders of magnitude compared to direct calculation. Surrogates of PDEs could enable use of larger and more detailed models than are possible with direct calculation and can make including such simulations in real-time or near-real time workflows practical. Creating a surrogate requires running the direct calculation tens of thousands of times to generate training data and then training the neural network, both of which are computationally expensive. Often the practical applications of such models require thousands to millions of replica simulations, for example for parameter identification and uncertainty quantification, each of which gains speed from surrogate use and rapidly recovers the up-front costs of surrogate generation. We use a Convolutional Neural Network to approximate the stationary solution to the diffusion equation in the case of two equal-diameter, circular, constant-value sources located at random positions in a two-dimensional square domain with absorbing boundary conditions. Such a configuration caricatures the chemical concentration field of a fast-diffusing species like oxygen in a tissue with two parallel blood vessels in a cross section perpendicular to the two blood vessels. To improve convergence during training, we apply a training approach that uses roll-back to reject stochastic changes to the network that increase the loss function. The trained neural network approximation is about 1000 times faster than the direct calculation for individual replicas. Because different applications will have different criteria for acceptable approximation accuracy, we discuss a variety of loss functions and accuracy estimators that can help select the best network for a particular application. We briefly discuss some of the issues we encountered with overfitting, mismapping of the field values and the geometrical conditions that lead to large absolute and relative errors in the approximate solution.more » « less
-
null (Ed.)Typical model reduction methods for parametric partial differential equations construct a linear space V n which approximates well the solution manifold M consisting of all solutions u ( y ) with y the vector of parameters. In many problems of numerical computation, nonlinear methods such as adaptive approximation, n -term approximation, and certain tree-based methods may provide improved numerical efficiency over linear methods. Nonlinear model reduction methods replace the linear space V n by a nonlinear space Σ n . Little is known in terms of their performance guarantees, and most existing numerical experiments use a parameter dimension of at most two. In this work, we make a step towards a more cohesive theory for nonlinear model reduction. Framing these methods in the general setting of library approximation, we give a first comparison of their performance with the performance of standard linear approximation for any compact set. We then study these methods for solution manifolds of parametrized elliptic PDEs. We study a specific example of library approximation where the parameter domain is split into a finite number N of rectangular cells, with affine spaces of dimension m assigned to each cell, and give performance guarantees with respect to accuracy of approximation versus m and N .more » « less
-
We consider the problem of estimating the number of distinct elements in a large data set (or, equivalently, the support size of the distribution induced by the data set) from a random sample of its elements. The problem occurs in many applications, including biology, genomics, computer systems and linguistics. A line of research spanning the last decade resulted in algorithms that estimate the support up to ±εn from a sample of size O(log2(1/ε)·n/logn), where n is the data set size. Unfortunately, this bound is known to be tight, limiting further improvements to the complexity of this problem. In this paper we consider estimation algorithms augmented with a machine-learning-based predictor that, given any element, returns an estimation of its frequency. We show that if the predictor is correct up to a constant approximation factor, then the sample complexity can be reduced significantly, to log(1/ε)·n1−Θ(1/log(1/ε)).We evaluate the proposed algorithms on a collection of data sets, using the neural-network based estimators from Hsu et al, ICLR’19 as predictors. Our experiments demonstrate substantial (up to 3x) improvements in the estimation accuracy com-pared to the state of the art algorithm.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1051/m2an/2020004
