Greedy Kernel Approximation for Sparse Surrogate Modeling

Haasdonk, Bernard; Santin, Gabriele

doi:10.1007/978-3-319-75319-5_2

Cited by 7 publications

(13 citation statements)

References 16 publications

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“…Finally, we remark that the partial surrogates can be efficiently updated when adding a new point, i.e.,

{\overset{f}{true}}_{n}

can be obtained from

{\overset{f}{true}}_{n - 1}

by computing only a new coefficient in the expansion, while the already computed ones are not modified. We point to the paper for a more in‐depth explanation of this efficient computational process.…”

Section: Kernel‐based Surrogate Modelsmentioning

confidence: 99%

“…We remark that the values of this test set are not contained in the training sets (except for R s = 10 −6 and R s = 1), so the results are reliable assessments of the models' accuracy. For every value ( R s ) i in the test set, we consider the absolute and relative errors

e_{A}^{(i)} : = ‖ f ({(R_{s})}_{i}) - \hat{f} ({(R_{s})}_{i}) {false‖}_{2}, e_{R}^{(i)} : = \frac{‖ f ({(R_{s})}_{i}) - \hat{f} ({(R_{s})}_{i}) {false‖}_{2}}{‖ f ({(R_{s})}_{i}) {false‖}_{2}},

and, to measure the overall error over the test set, we compute both the maximum absolute and relative errors, i.e.,

E_{A} : = \max_{1 \leq i \leq 1000} e_{A}^{(i)}, E_{R} : = \max_{1 \leq i \leq 1000} e_{R}^{(i)} .

We use the Gaussian kernel, and the f ‐VKOGA is stopped using a tolerance 5·10 −8 on the regularised Power Function , which controls the model stability …”

Section: Numerical Testsmentioning

confidence: 99%

“…() Contrary to the RB method, the surrogate model is in this case represented by a linear combination of kernel functions like the Gaussian or the Wendland kernel. () The coefficients in the linear combination and the parameters for the kernel functions are obtained from a training and validation process, which is performed in an offline or training phase . These methods have the advantage of constructing non‐linear, data‐dependent surrogates that can reach significant degrees of accuracy, while not needing an excessively large amount of data, as for different machine learning techniques.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods

Köppl

Santin

Haasdonk

et al. 2018

Numer Methods Biomed Eng

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In this work, we consider 2 kinds of model reduction techniques to simulate blood flow through the largest systemic arteries, where a stenosis is located in a peripheral artery, i.e., in an artery that is located far away from the heart. For our simulations, we place the stenosis in one of the tibial arteries belonging to the right lower leg (right posterior tibial artery). The model reduction techniques that are used are on the one hand dimensionally reduced models (1-D and 0-D models, the so-called mixed-dimension model) and on the other hand surrogate models produced by kernel methods. Both methods are combined in such a way that the mixed-dimension models yield training data for the surrogate model, where the surrogate model is parametrised by the degree of narrowing of the peripheral stenosis. By means of a well-trained surrogate model, we show that simulation data can be reproduced with a satisfactory accuracy and that parameter optimisation or state estimation problems can be solved in a very efficient way. Furthermore, it is demonstrated that a surrogate model enables us to present after a very short simulation time the impact of a varying degree of stenosis on blood flow, obtaining a speedup of several orders over the full model.

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“…Finally, we remark that the partial surrogates can be efficiently updated when adding a new point, i.e.,

{\overset{f}{true}}_{n}

can be obtained from

{\overset{f}{true}}_{n - 1}

Section: Kernel‐based Surrogate Modelsmentioning

confidence: 99%

e_{A}^{(i)} : = ‖ f ({(R_{s})}_{i}) - \hat{f} ({(R_{s})}_{i}) {false‖}_{2}, e_{R}^{(i)} : = \frac{‖ f ({(R_{s})}_{i}) - \hat{f} ({(R_{s})}_{i}) {false‖}_{2}}{‖ f ({(R_{s})}_{i}) {false‖}_{2}},

and, to measure the overall error over the test set, we compute both the maximum absolute and relative errors, i.e.,

E_{A} : = \max_{1 \leq i \leq 1000} e_{A}^{(i)}, E_{R} : = \max_{1 \leq i \leq 1000} e_{R}^{(i)} .

We use the Gaussian kernel, and the f ‐VKOGA is stopped using a tolerance 5·10 −8 on the regularised Power Function , which controls the model stability …”

Section: Numerical Testsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods

Köppl

Santin

Haasdonk

et al. 2018

Numer Methods Biomed Eng

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“…We briefly outline here the fundamentals of interpolation with kernels and of the VKOGA algorithm, and we refer to [10] and to [11,4] for the respective details.…”

Section: Kernel Based Surrogates and The Vkogamentioning

confidence: 99%

Greedy Kernel Methods for Accelerating Implicit Integrators for Parametric ODEs

Brünnette

Santin

Haasdonk

2019

Lecture Notes in Computational Science and Engineering

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We present a novel acceleration method for the solution of parametric ODEs by single-step implicit solvers by means of greedy kernelbased surrogate models. In an offline phase, a set of trajectories is precomputed with a high-accuracy ODE solver for a selected set of parameter samples, and used to train a kernel model which predicts the next point in the trajectory as a function of the last one. This model is cheap to evaluate, and it is used in an online phase for new parameter samples to provide a good initialization point for the nonlinear solver of the implicit integrator. The accuracy of the surrogate reflects into a reduction of the number of iterations until convergence of the solver, thus providing an overall speedup of the full simulation. Interestingly, in addition to providing an acceleration, the accuracy of the solution is maintained, since the ODE solver is still used to guarantee the required precision. Although the method can be applied to a large variety of solvers and different ODEs, we will present in details its use with the Implicit Euler method for the solution of the Burgers equation, which results to be a meaningful test case to demonstrate the method's features.

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“…for kernel methods (see e.g. [18,20,23,29]) and lead to sparse models which turn out to be helpful in many applications, see e.g. [16].…”

Section: Introductionmentioning

confidence: 99%

Greedy algorithms for learning via exponential-polynomial splines

Campagna¹,

Marchi²,

Perracchione³

et al. 2021

Preprint

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Kernel-based schemes are state-of-the-art techniques for learning by data. In this work we extend some ideas about kernel-based greedy algorithms to exponential-polynomial splines, whose main drawback consists in possible overfitting and consequent oscillations of the approximant. To partially overcome this issue, we introduce two algorithms which perform an adaptive selection of the spline interpolation points based on the minimization either of the sample residuals (f -greedy), or of an upper bound for the approximation error based on the spline Lebesgue function (λgreedy). Both methods allow us to obtain an adaptive selection of the sampling points, i.e. the spline nodes. However, while the f -greedy selection is tailored to one specific target function, the λ-greedy algorithm is independent of the function values and enables us to define a priori optimal interpolation nodes.

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Greedy Kernel Approximation for Sparse Surrogate Modeling

Cited by 7 publications

References 16 publications

Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods

Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods

Greedy Kernel Methods for Accelerating Implicit Integrators for Parametric ODEs

Greedy algorithms for learning via exponential-polynomial splines

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