Exponential integrators: tensor structured problems and applications

The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed

A second order directional split exponential integrator for systems of advection–diffusion–reaction equations

We propose a second order exponential scheme suitable for two-component coupled systems of stiff evolutionary advection–diffusion–reaction equations in two and three space dimensions. It is based on a directional splitting of the involved matrix functions, which allows for a simple yet efficient implementation through the computation of small sized exponential-like functions and tensor-matrix products. The procedure straightforwardly extends to the case of an arbitrary number of components and to any space dimension. Several numerical examples in 2D and 3D with physically relevant (advective) Schnakenberg, FitzHugh–Nagumo, DIB, and advective Brusselator models clearly show the advantage of the approach against state-of-the-art techniques

Direction splitting of φ\varphi-functions in exponential integrators for dd-dimensional problems in Kronecker form

In this manuscript, we propose an efficient, practical and easy-to-implement way to approximate actions of $\varphi$-functions for matrices with $d$-dimensional Kronecker sum structure in the context of exponential integrators up to second order. The method is based on a direction splitting of the involved matrix functions, which lets us exploit the highly efficient level 3 BLAS for the actual computation of the required actions in a $\mu$-mode fashion. The approach has been successfully tested on two- and three-dimensional problems with various exponential integrators, resulting in a consistent speedup with respect to a technique designed to compute actions of $\varphi$-functions for Kronecker sums

A second order directional split exponential integrator for systems of advection--diffusion--reaction equations

We propose a second order exponential scheme suitable for two-component coupled systems of stiff advection--diffusion--reaction equations in two and three space dimensions. It is based on a directional splitting of the involved matrix functions, which allows for a simple yet efficient implementation through the computation of small-sized exponential-like functions and tensor-matrix products. The procedure straightforwardly extends to the case of an arbitrary number of components and to any space dimension $d$. Several numerical experiments in 2D and 3D with physically relevant DIB, Schnakenberg, FitzHugh--Nagumo, and advective Brusselator models clearly show the advantage of the approach against state-of-the-art techniques

A μ-mode BLAS approach for multidimensional tensor-structured problems

In this manuscript, we present a common tensor framework which can be used to generalize one-dimensional numerical tasks to arbitrary dimension d by means of tensor product formulas. This is useful, for example, in the context of multivariate interpolation, multidimensional function approximation using pseudospectral expansions and solution of stiff differential equations on tensor product domains. The key point to obtain an efficient-to-implement BLAS formulation consists in the suitable usage of the mu-mode product (also known as tensor-matrix product or mode-n product) and related operations, such as the Tucker operator. Their MathWorks MATLAB (R)/GNU Octave implementations are discussed in the paper, and collected in the package KronPACK. We present numerical results on experiments up to dimension six from different fields of numerical analysis, which show the effectiveness of the approach

Accelerating exponential integrators to efficiently solve advection-diffusion-reaction equations

In this paper we consider an approach to improve the performance of exponential integrators/Lawson schemes in cases where the solution of a related, but usually much simpler, problem can be computed efficiently. While for implicit methods such an approach is common (e.g. by using preconditioners), for exponential integrators this has proven more challenging. Here we propose to extract a constant coefficient differential operator from advection-diffusion-reaction equations for which we are then able to compute the required matrix functions efficiently. Both a linear stability analysis and numerical experiments show that the resulting schemes can be unconditionally stable. In fact, we find that exponential integrators and Lawson schemes can have better stability properties than similarly constructed implicit-explicit schemes. We also propose new Lawson type integrators that further improve on these stability properties. The effectiveness of the approach is highlighted by a number of numerical examples in two and three space dimensions

A μ\mu-mode integrator for solving evolution equations in Kronecker form

In this paper, we propose a $\mu$-mode integrator for computing the solution of stiff evolution equations. The integrator is based on a d-dimensional splitting approach and uses exact (usually precomputed) one-dimensional matrix exponentials. We show that the action of the exponentials, i.e. the corresponding batched matrix-vector products, can be implemented efficiently on modern computer systems. We further explain how $\mu$-mode products can be used to compute spectral transformations efficiently even if no fast transform is available. We illustrate the performance of the new integrator by solving three-dimensional linear and nonlinear Schr\"odinger equations, and we show that the $\mu$-mode integrator can significantly outperform numerical methods well established in the field. We also discuss how to efficiently implement this integrator on both multi-core CPUs and GPUs. Finally, the numerical experiments show that using GPUs results in performance improvements between a factor of 10 and 20, depending on the problem

Minimally invasive vs. open segmental resection of the splenic flexure for cancer: a nationwide study of the Italian Society of Surgical Oncology-Colorectal Cancer Network (SICO-CNN)

Background Evidence on the efficacy of minimally invasive (MI) segmental resection of splenic flexure cancer (SFC) is not available, mostly due to the rarity of this tumor. This study aimed to determine the survival outcomes of MI and open treatment, and to investigate whether MI is noninferior to open procedure regarding short-term outcomes. Methods This nationwide retrospective cohort study included all consecutive SFC segmental resections performed in 30 referral centers between 2006 and 2016. The primary endpoint assessing efficacy was the overall survival (OS). The secondary endpoints included cancer-specific mortality (CSM), recurrence rate (RR), short-term clinical outcomes (a composite of Clavien-Dindo > 2 complications and 30-day mortality), and pathological outcomes (a composite of lymph nodes removed >= 12, and proximal and distal free resection margins length >= 5 cm). For these composites, a 6% noninferiority margin was chosen based on clinical relevance estimate. Results A total of 606 patients underwent either an open (208, 34.3%) or a MI (398, 65.7%) SFC segmental resection. At univariable analysis, OS and CSM were improved in the MI group (log-rank test p = 0.004 and Gray's tests p = 0.004, respectively), while recurrences were comparable (Gray's tests p = 0.434). Cox multivariable analysis did not support that OS and CSM were better in the MI group (p = 0.109 and p = 0.163, respectively). Successful pathological outcome, observed in 53.2% of open and 58.3% of MI resections, supported noninferiority (difference 5.1%; 1-sided 95%CI - 4.7% to infinity). Successful short-term clinical outcome was documented in 93.3% of Open and 93.0% of MI procedures, and supported noninferiority as well (difference - 0.3%; 1-sided 95%CI - 5.0% to infinity). Conclusions Among patients with SFC, the minimally invasive approach met the criterion for noninferiority for postoperative complications and pathological outcomes, and was found to provide results of OS, CSM, and RR comparable to those of open resection

Practice patterns and 90-day treatment-related morbidity in early-stage cervical cancer

To evaluate the impact of the Laparoscopic Approach to Cervical Cancer (LACC) Trial on patterns of care and surgery-related morbidity in early-stage cervical cancer

Bowel preparation for elective colorectal resection: multi-treatment machine learning analysis on 6241 cases from a prospective Italian cohort

background current evidence concerning bowel preparation before elective colorectal surgery is still controversial. this study aimed to compare the incidence of anastomotic leakage (AL), surgical site infections (SSIs), and overall morbidity (any adverse event, OM) after elective colorectal surgery using four different types of bowel preparation. methods a prospective database gathered among 78 Italian surgical centers in two prospective studies, including 6241 patients who underwent elective colorectal resection with anastomosis for malignant or benign disease, was re-analyzed through a multi-treatment machine-learning model considering no bowel preparation (NBP; No. = 3742; 60.0%) as the reference treatment arm, compared to oral antibiotics alone (oA; No. = 406; 6.5%), mechanical bowel preparation alone (MBP; No. = 1486; 23.8%), or in combination with oAB (MoABP; No. = 607; 9.7%). twenty covariates related to biometric data, surgical procedures, perioperative management, and hospital/center data potentially affecting outcomes were included and balanced into the model. the primary endpoints were AL, SSIs, and OM. all the results were reported as odds ratio (OR) with 95% confidence intervals (95% CI). results compared to NBP, MBP showed significantly higher AL risk (OR 1.82; 95% CI 1.23-2.71; p = .003) and OM risk (OR 1.38; 95% CI 1.10-1.72; p = .005), no significant differences for all the endpoints were recorded in the oA group, whereas MoABP showed a significantly reduced SSI risk (OR 0.45; 95% CI 0.25-0.79; p = .008). conclusions MoABP significantly reduced the SSI risk after elective colorectal surgery, therefore representing a valid alternative to NBP