We present a novel active learning algorithm, termed as iterative surrogate model optimization (ISMO), for robust and efficient numerical approximation of PDE constrained problems. This algorithm is based on deep neural networks its key feature the selection training data through feedback loop between any underlying standard algorithm. Under suitable hypotheses, we show that resulting optimizers converge exponentially fast (and with decaying variance), respect to increasing number samples....

The quantum approximate optimization algorithm/quantum alternating operator ansatz (QAOA) is a heuristic to find solutions of combinatorial problems. Most the literature limited quadratic problems without constraints. However, many practically relevant do have (hard) constraints that need be fulfilled. In this article, we present framework for constructing mixing operators restrict evolution subspace full Hilbert space given by these We generalize “XY”-mixer designed preserve “one-hot”...

Pseudo-Hamiltonian neural networks (PHNN) were recently introduced for learning dynamical systems that can be modelled by ordinary differential equations. In this paper, we extend the method to partial The resulting model is comprised of up three networks, modelling terms representing conservation, dissipation and external forces, discrete convolution operators either learned or given as input. We demonstrate numerically superior performance PHNN compared a baseline models full dynamics...

Abstract We develop auto-tuned GPU implementations of ILU(0) and DILU preconditioners using mixed precision compare their performance against various GPU-based implementations, integrated into the open-source OPM Flow reservoir simulator. The are parallelized conservatively to ensure that results match those from serial computation up instruction set equivalence. Our implementation incorporates techniques such as graph coloring, row reordering, matrix splitting, mixed-precision schemes,...

We propose a multi-level method to increase the accuracy of machine learning algorithms for approximating observables in scientific computing, particularly those that arise systems modelled by differential equations. The algorithm relies on judiciously combining large number computationally cheap training data coarse resolutions with few expensive samples fine grid resolutions. Theoretical arguments lowering generalisation error, based reducing variance underlying maps, are provided and...

Statistical solutions are time-parameterized probability measures on spaces of integrable functions, which have been proposed recently as a framework for global and uncertainty quantification multi-dimensional hyperbolic system conservation laws. By combining high-resolution finite volume methods with Monte Carlo sampling procedure, we present numerical algorithm to approximate statistical solutions. Under verifiable assumptions the method, prove that approximations, generated by algorithm,...

We propose efficient numerical algorithms for approximating statistical solutions of scalar conservation laws. The proposed combine finite volume spatio-temporal approximations with Monte Carlo and multilevel discretizations the probability space. Both sets methods are proved to converge entropy solution. also prove that there is a considerable gain in efficiency resulting from method over standard method. Numerical experiments illustrating ability both accurately compute multipoint...

We propose a Multilevel Monte-Carlo (MLMC) method for computing entropy measure valued solutions of hyperbolic conservation laws. Sharp bounds the narrow convergence MLMC are proposed. An optimal work-vs-error bound is derived assuming only an abstract decay criterion on variance. Finally, we display numerical experiments cases where is, and not, efficient when compared to Monte-Carlo.

We present a software framework that facilitates the development of OpenGL applications utilizing limited GPU capacities portable client in combination with high-end rendering hardware on server. The resulting web-application uses standard technologies and can be run wide variety devices, such as smart phones, tablets laptops. is designed to make it simple changing an existing application into web-application, gradually adding client-side rendering. Furthermore, provides automatic network...

We present the Alsvinn simulator, a fast multi general purpose graphical processing unit (GPGPU) finite volume solver for hyperbolic conservation laws in multiple space dimensions. has native support uncertainty quantifications, and exhibits excellent scaling on top tier compute clusters.

Abstract. Multi-level Monte Carlo methods have established as a tool in uncertainty quantification for decreasing the computational costs while maintaining same statistical accuracy single-level Carlo. Lately, there also been theoretical efforts to use similar ideas facilitate multi-level data assimilation. By applying ensemble Kalman filter assimilating sparse observations of ocean currents into simplified model based on shallow-water equations, we study practical challenges these method...

Highlights•OPM Flow is an open-source simulator for subsurface flow in porous media.•Quasi-Monte Carlo with the Sobol sequence generated input values OPM evaluation time domain.•Sensitivity analysis identified key parameters efficient CO2 sequestration domain.•The optimal convergence rate quasi-Monte observed.•For studied model ensemble, 'Trapped CO2' depends dominantly on permeabilities of shale between sand layers.•Machine learning: Symbolic regression provided approximations using...

Pseudo-Hamiltonian neural networks (PHNN) were recently introduced for learning dynamical systems that can be modelled by ordinary differential equations. In this paper, we extend the method to partial The resulting model is comprised of up three networks, modelling terms representing conservation, dissipation and external forces, discrete convolution operators either learned or given as input. We demonstrate numerically superior performance PHNN compared a baseline models full dynamics...

Abstract We prove convergence rates of monotone schemes for conservation laws Hölder continuous initial data with unbounded total variation, provided that the exponent is greater than $$\nicefrac {1}{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> . For strictly $${{\,\mathrm{Lip}\,}}^+$$ <mml:msup> <mml:mspace /> <mml:mi>Lip</mml:mi> <mml:mo>+</mml:mo> </mml:msup> stable schemes,...