Mikael Mortensen

• Piterskaya, Anna & Mortensen, Mikael (2025). A study of the Orr–Sommerfeld and induction equations by Galerkin and Petrov–Galerkin spectral methods utilizing Chebyshev polynomials. Journal of Computational and Applied Mathematics. ISSN 0377-0427. 459. doi: 10.1016/j.cam.2024.116374. Full text in Research Archive
• Mortensen, Mikael (2024). A faster multipole Legendre-Chebyshev transform. SIAM Journal on Scientific Computing. ISSN 1064-8275. 46(6), p. A3803–A3826. doi: 10.1137/24M1659352. Full text in Research Archive Show summary

  This paper describes a fast algorithm for transforming Legendre coefficients into Chebyshev coefficients, and vice versa. The algorithm is based on the fast multipole method and is similar to the approach described by Alpert and Rokhlin [SIAM J. Sci. Comput., 12 (1991), pp. 158–179]. The main difference is that we utilize a modal Galerkin approach with Chebyshev basis functions instead of a nodal approach with a Lagrange basis. Part of the algorithm is a novel method that facilitates faster spreading of intermediate results through neighboring levels of hierarchical matrices. This enhancement leads to a method that is approximately 20 percent faster to execute, due to fewer floating point operations. We also describe an efficient initialization algorithm that for the Lagrange basis is roughly 5 times faster than the original method for large input arrays. The described method has both a planning and an execution stage that asymptotically require O(N) flops. The algorithm is simple enough that it can be implemented in 100 lines of vectorized Python code. Moreover, its efficiency is such that a single-threaded C implementation can transform 1,000,000 coefficients in approximately 20 milliseconds on a new MacBook Pro M3, representing about 3 times the execution time of a well-planned (single-threaded) type 2 discrete cosine transform from FFTW (www.fftw.org). Planning for these coefficients requires approximately 50 milliseconds.

• Vasanth, Joel; Rabault, Jean; Alcántara-Ávila, Francisco; Mortensen, Mikael & Vinuesa, Ricardo (2024). Multi-agent Reinforcement Learning for the Control of Three-Dimensional Rayleigh–Bénard Convection. Flow Turbulence and Combustion. ISSN 1386-6184. doi: 10.1007/s10494-024-00619-2. Full text in Research Archive
• Piterskaya, Anna; Miloch, Wojciech Jacek & Mortensen, Mikael (2023). A global spectral-Galerkin investigation of a Rayleigh-Taylor instability in plasma using an MHD-Boussinesq model. AIP Advances. 13(10). doi: 10.1063/5.0155976. Full text in Research Archive Show summary

  This paper presents a new efficient algorithm based on the spectral-Galerkin numerical approximations complemented by a magnetohydrodynamics–Boussinesq model and a new solver for studying the development of a Rayleigh–Taylor (RT) instability. We use the Shenfun computational framework in the Cartesian coordinates, which gives the spectral order and accuracy for the considered model based on the magnetohydrodynamics equations and the Boussinesq conjecture. The numerical simulations were conducted for each two- and three-dimensional case, both with and without an external static magnetic field. The validity of the numerical results was examined by comparing the calculated squared L2-norm of the density parameter with the linear stability analysis. We also examined the effects of a uniform tangential magnetic field on the onset and growth of an RT instability at different magnetic field strengths. The analysis of the effectiveness of the presented method suggests that it can be modified for further research on two-component plasma.

• Vignon, Colin; Rabault, Jean; Vasanth, Joel; Alcántara-Ávila, Francisco; Mortensen, Mikael & Vinuesa, Ricardo (2023). Effective control of two-dimensional Rayleigh-Bénard convection: Invariant multi-agent reinforcement learning is all you need. Physics of Fluids. ISSN 1070-6631. 35(6). doi: 10.1063/5.0153181. Full text in Research Archive Show summary

  Rayleigh–Bénard convection (RBC) is a recurrent phenomenon in a number of industrial and geoscience flows and a well-studied system from a fundamental fluid-mechanics viewpoint. In the present work, we conduct numerical simulations to apply deep reinforcement learning (DRL) for controlling two-dimensional RBC using sensor-based feedback control. We show that effective RBC control can be obtained by leveraging invariant multi-agent reinforcement learning (MARL), which takes advantage of the locality and translational invariance inherent to RBC flows inside wide channels. MARL applied to RBC allows for an increase in the number of control segments without encountering the curse of dimensionality that would result from a naive increase in the DRL action-size dimension. This is made possible by the MARL ability for re-using the knowledge generated in different parts of the RBC domain. MARL is able to discover an advanced control strategy that destabilizes the spontaneous RBC double-cell pattern, changes the topology of RBC by coalescing adjacent convection cells, and actively controls the resulting coalesced cell to bring it to a new stable configuration. This modified flow configuration results in reduced convective heat transfer, which is beneficial in a number of industrial processes. We additionally draw comparisons with a conventional single-agent reinforcement learning (SARL) setup and report that in the same number of episodes, SARL is not able to learn an effective policy to control the cells. Thus, our work both shows the potential of MARL for controlling large RBC systems and demonstrates the possibility for DRL to discover strategies that move the RBC configuration between different topological configurations, yielding desirable heat-transfer characteristics.

• Mortensen, Mikael (2022). A Generic and Strictly Banded Spectral Petrov-Galerkin Method for Differential Equations with Polynomial Coefficients. SIAM Journal on Scientific Computing. ISSN 1064-8275. 45(1), p. A123–A146. doi: 10.1137/22M1492842. Full text in Research Archive Show summary

  In this paper we describe a generic spectral Petrov–Galerkin method that is sparse and strictly banded for any linear ordinary differential equation with polynomial coefficients. The method applies to all subdivisions of Jacobi polynomials (e.g., Chebyshev and Legendre), utilizes well-known recurrence relations of orthogonal polynomials, and leads to almost exactly the same discretized system of equations as the integration preconditioners [E. A. Coutsias, T. Hagstrom, and D. Torres, Math. Comp., 65 (1996), pp. 611–635] if this method was redesigned to make use of trial functions that satisfy a given problem’s boundary conditions. A link between the new Petrov–Galerkin method and IP is revealed through a new recursion relation for Jacobi polynomials. Because of the strictly banded nature of all coefficient matrices, the new method extends easily and efficiently to multiple dimensions though the use of tensor product methods.

• Friedemann, Christopher; Mortensen, Mikael & Nossen, Jan (2021). Two-phase co-current flow simulations using periodic boundary conditions in horizontal, 4, 10 and 90° inclined eccentric annulus, flow prediction using a modified interFoam solver and comparison with experimental results. International Journal of Heat and Fluid Flow. ISSN 0142-727X. 88, p. 1–20. doi: 10.1016/j.ijheatfluidflow.2020.108754. Full text in Research Archive Show summary

  Two-phase oil and gas flow were simulated in an entirely eccentric annulus and compared with experimental data at horizontal, 4, 10, and 90° inclination. The gas-phase was sulphur hexafluoride and the liquid phase a mixture of Exxsol D60 and Marcol 82 for the inclined cases (5–16), and pure Exxsol D60 for the horizontal cases (1–4). The diameter of the outer and inner cylinders was 0.1 and 0.04 m, respectively, for the inclined domains and 0.1 and 0.05 m for the horizontal domain. The cases studied consist of liquid phase fractions between 0.3 and 0.65 and mixture velocities from 1.2 to 4.25 m/s. The mean pressure gradient is within 33% of the expected experimental behavior for all inclined cases. In contrast, the low-velocity horizontal domains exhibit significant deviation, with a drastic over-prediction of the mean pressure gradient by as much as 200–335% for cases 1 and 2. The two remaining horizontal cases (3 and 4) are within 22% of the expected mean pressure gradient. Cases 13–16 are a replication of cases 5–8 at an increased inclination; the mean pressure gradient is within 6.5% of the expected increase due to the increase in hydrostatic pressure. By comparing cases 1–4 to previous published simulations at a lower eccentricity, we found a decrease of the mean pressure gradient by 30–40%, which is in line with existing literature, although for single-phase flows. The simulated and experimental liquid holdup profiles are in good agreement when comparing the fractional data; wave and slug frequencies match to within 0.5 Hz; however, at closer inspection, it is apparent that there is a decrease in the amount of phase-mixing of the simulations compared to the experiments. When increasing the mesh density from 115 k cells/m to 2 million cells/m, the simulations exhibit significantly more phase mixing, but are still unable to produce conventional slugs. In a simplified case, conventional slugs are observed at grid sizing of 1 × 1 × 1 mm, whereas the cells of the 2 million cells/m mesh are roughly 1.5 × 1.5 × 1.5 mm.

• Landet, Tormod; Mardal, Kent-Andre & Mortensen, Mikael (2020). Slope limiting the velocity field in a discontinuous Galerkin divergence-free two-phase flow solver. Computers & Fluids. ISSN 0045-7930. 196. doi: 10.1016/j.compfluid.2019.104322. Full text in Research Archive Show summary

  Solving the Navier–Stokes equations when the density field contains a large and sharp discontinuity—such as a water/air free surface—is numerically challenging. Convective instabilities cause Gibbs oscillations which quickly destroy the solution. We investigate the use of slope limiters for the velocity field to overcome this problem in a way that does not compromise on the mass-conservation properties. The equations are discretised using a symmetric interior-penalty discontinuous Galerkin finite element method that is divergence-free to machine precision. A slope limiter made specifically for exactly divergence-free (solenoidal) fields is presented and used to illustrate the difficulties in obtaining convectively stable fields that are also exactly solenoidal. The lessons learned from this are applied in constructing a simpler method based on the use of an existing scalar slope limiter applied to each velocity component. We show by numerical examples how both presented slope limiting methods are vastly superior to the naive non-limited method. The methods can solve difficult two-phase problems with high density-ratios and high Reynolds numbers—typical for marine and offshore water/air simulations—in a way that conserves mass and stops unbounded energy growth caused by the Gibbs phenomenon.

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• Miloch, Wojciech Jacek; Usui, Hideyuki; Miyake, Yohei; Mortensen, Mikael; Chernyshov, A.A. & Baddeley, Lisa [Show all 11 contributors for this article] (2021). Long-term and sustainable research- based student mobility projects within space weather. Full text in Research Archive
• Kjeldsberg, Henrik Aasen; Valen-Senstad, Kristian; Sundnes, Joakim & Mortensen, Mikael (2025). The Clot Thickens: Investigating Thrombus Formation Through Left Atrial Flow Simulations. Universitetet i Oslo. Full text in Research Archive Show summary

  Atrial fibrillation (AF) is a common cardiac arrhythmia characterized by an irregular and often rapid heartbeat, which significantly increases the risk of thrombus formation and subsequent stroke. While there is no definitive cure for AF, patients are typically managed With medications, such as anticoagulants, to reduce stroke risk. The decision to undergo such treatment is often determined by clinical risk scores, such as the CHA2DS2–VASc score, which assesses stroke risk based on factors like age, hypertension, and prior stroke history. However, these scores do not account for patient-specific cardiac geometry or blood flow patterns. This dissertation addresses this limitation by using high-resolution computational fluid dynamics (CFD) simulations to quantify patient-specific cardiac anatomy and blood flow. We investigate how these factors contribute to thrombus formation and, by extension, stroke, and assess their potential as complementary markers to existing clinical risk scores. In pursuit of this goal, we developed an open-source, moving-domain CFD solver (OasisMove) and an automated modeling pipeline (VaMPy) to enable high-resolution, high-performance simulations of left atrial (LA) hemodynamics in patients with AF. We first verify and validate these tools using benchmark problems and experimental data. We then systematically compare the impact of rigid, generic, and patient-specific wall motion on LA hemodynamics, demonstrating that incorporating patient-specific motion notably alters flow patterns and proposed indicators of thrombus formation. To address the Challenge of prescribing inflow conditions in patient-specific simulations, we introduce an anatomically based and physiologically constrained method for estimating inlet boundary conditions. This approach enables physiologically plausible flow without the need for invasive measurements. Finally, we apply our workflow to a larger cohort of AF patients and show that combining CFD-derived hemodynamic metrics and morphological parameters with traditional clinical scores results in stronger correlations with thrombus presence and stroke history than clinical scores alone. Overall, our framework provides deeper Insight into AF-related flow phenomena, and results from its application to a patient cohort suggests that CFD-derived metrics and morphological parameters could be valuable additions to stroke risk assessment in AF patients.

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