arXiv subjects

Sascha Portaro

Publications and source records attributed to Sascha Portaro.

4 recordsLinked to original sources

arXiv2026

Numerical methods for Langevin-type SPDE: an implicit Milstein approach and multilevel Monte Carlo techniques

In this work, we investigate the numerical approximation of degenerate Langevin-type stochastic partial differential equations (SPDEs) in two spatial dimensions. These SPDEs arise in stochastic dynamics and mathematical finance, among other applications. In order to handle the mixed deterministic-stochastic structure of the equation and the degeneracy of the differential operator, we propose a semi-implicit Milstein finite difference scheme for the numerical solution. Through the Fourier analysis of the mean-square stability and convergence, we derive explicit conditions on the coefficients under which the scheme is stable, jointly with explicit convergence rates in terms of the discretization parameters. We further embed the proposed scheme within a Multilevel Monte Carlo (MLMC) framework to reduce the computational cost associated with SPDE simulations, and we derive its theoretical computational complexity. Numerical experiments confirm theoretical convergence rates and show that the MLMC strategy achieves an accuracy comparable to standard Monte Carlo at a fraction of the computational cost, reducing the complexity from $\mathcal{O}(\varepsilon^{-5})$ to $\mathcal{O}(\varepsilon^{-3})$ for a target root-mean-square error $\varepsilon$. These results show that combining semi-implicit Milstein schemes with MLMC techniques provides an effective approach for the numerical simulation of Langevin-type SPDEs.

math.NA↗

arXiv2026

A Practical Mode-parallel Implementation of the (H-)Tucker Decomposition via Randomization

In the last decades, tensors have emerged as the right tool to represent multidimensional data in a compact yet informative manner. Moreover, it is well-known that by performing low-rank factorizations of such tensors one is often able to effectively unveil possible hidden structure in data, mainly due to unexpected dependencies among the different variables encoded in the given tensor. However, computing these factorizations is extremely energy-consuming and memory-demanding, especially for high-dimensional tensors, namely those with a large number of modes. In this paper we focus on two state-of-the-art tensor decompositions: the Tucker and H-Tucker decompositions. We propose novel numerical strategies able to perform these factorizations in a mode-parallel fashion, that is the operations required by the algorithm along all modes are performed in parallel. This is in contrast to what is achieved by many procedures available in the literature that parallelize some of the operations along each mode, e.g., tensor-times-matrix steps, while still visiting one mode at the time in a sequential manner. Our strategies make use of cutting-edge randomization techniques comprising fiber sampling and randomized range-finding steps. In case of Tucker decomposition, we provide upper bounds on the expected value of the error provided sufficiently large sampling parameters have been adopted. A panel of numerical results showcases the potential of our approach in reducing both the running time and the storage demand of computing Tucker and H-Tucker decompositions. Moreover, experiments carried out in HPC environments illustrate the good scaling of our mode-parallel approach.

math.NA↗

arXiv2025

Fractional Sobolev spaces related to an ultraparabolic operator

We propose a functional framework of fractional Sobolev spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak H\"ormander condition. We characterize these spaces as real interpolation of natural order intrinic Sobolev spaces recently introduced in [27], and prove continuous embeddings into $L^p$ and intrinsic H\"older spaces from [24]. These embeddings naturally extend the standard Euclidean ones, coherently with the homogeneous structure of the associated Kolmogorov group. Our approach to interpolation is based on approximation of intrinsically regular functions, the latter heavily relying on integral estimates of the intrinsic Taylor remainder. The embeddings exploit the aforementioned interpolation property and the corresponding embeddings of natural order intrinsic spaces.

math.AP↗

arXiv2024

Row-aware Randomized SVD with applications

The randomized singular value decomposition proposed in [27] has certainly become one of the most well-established randomization-based algorithms in numerical linear algebra. The key ingredient of the entire procedure is the computation of a subspace which is close to the column space of the target matrix $\mathbf{A}$ up to a certain probabilistic confidence. In this paper we employ a modification to the standard randomized SVD procedure which leads, in general, to better approximations to $\text{Range}(\mathbf{A})$ at the same computational cost. To this end, we explicitly construct information from the row space of $\mathbf{A}$ enhancing the quality of the approximation. We derive novel error bounds which improve over existing results for $\mathbf{A}$ having important gaps in its singular values. We also observe that very few pieces of information from $\text{Range}(\mathbf{A}^T)$ may be necessary. We thus design a variant of this algorithm equipped with a subsampling step which largely increases the efficiency of the procedure while often attaining competitive accuracy records. Our findings are supported by both theoretical analysis and numerical results.

math.NA↗