Kent-Andre Mardal

• Arbes, Florian Maurits; Mardal, Kent-Andre & Jensen, Øyvind (2026). Data regularization and deconvolution for non-intrusive reduced-order modeling of transport-dominated problems. Results in Applied Mathematics. ISSN 2590-0382. 32. doi: 10.1016/j.rinam.2026.100775. Fulltekst i vitenarkiv
• Girelli, Alberto; Solheim, Andreas; Lian, Sofie Lysholm; Storås, Tryggve Holck; Nordengen, Kaja & Mardal, Kent-Andre (2026). Multiphasic transport in the human cranial dura: how meningeal lymphatic vessels drive waste clearance from brain cerebrospinal fluid. Fluids and Barriers of the CNS. doi: 10.1186/s12987-026-00861-y. Fulltekst i vitenarkiv Vis sammendrag

  Background The discovery of meningeal lymphatic vessels (mLVs) has reshaped our understanding of brain waste clearance, positioning them as a very important clearance route. However, quantifying the precise hydrodynamics and solute transport mechanisms within the human dura mater in vivo remains highly challenging due to the multiscale nature of cranial fluid dynamics and imaging resolution limits. Methods We integrated patient-specific intrathecal contrast-enhanced MRI (gMRI) data from a single healthy subject with a novel anatomical segmentation of the cranial dura mater. By employing a hierarchy of inverse mathematical and micro-mechanical models, we estimated previously inaccessible physiological transport parameters. To ensure parameter identifiability, we deliberately modeled a scenario where the meningeal lymphatic vessels act as the sole fluid efflux route, thereby establishing an upper limit for their clearance capacity. These baseline estimates subsequently informed a spatially distributed, double-porosity 3D computational model of the human dura to simulate the fluid dynamics of cranial clearance. Results Our analysis suggests that purely diffusive macroscopic models fail to replicate in vivo tracer distribution. Conversely, a dual-porosity advection-diffusion framework successfully captures the tissue kinetics. The transport within the mLV compartment is characterized by highly convective clearance (with ), with macroscopic fluid velocities and effective lymphatic diffusion coefficients yielding an upper-bound advective capacity theoretically sufficient to clear the total albumin content in the Cerebrospinal Fluid (CSF). Conclusion Based on our findings on a single subject (Gonzo) solute clearance within the human cranial dura mater emerges as multiphasic, advection-dominated process. Rather than relying on passive Fickian diffusion, molecules appear to be actively and rapidly cleared through the meningeal lymphatic network via convective mechanisms. While our assumption of exclusively mLV-driven clearance provides a theoretical upper limit for their capacity, this hierarchical modeling approach successfully bridges clinical imaging and microscopic fluid dynamics, offering quantitative insights into the primary mechanical drivers of dural fluid clearance.

• Berre, Nanna; Mardal, Kent-Andre; Massing, André Jürgen & Yotov, Ivan (2026). A cut finite element method for the Biot system of poroelasticity. Computer Methods in Applied Mechanics and Engineering. ISSN 0045-7825. 458. doi: 10.1016/j.cma.2026.119048. Fulltekst i vitenarkiv
• Rognes, Marie; Mardal, Kent-Andre; Valnes, Lars & Vinje, Vegard (2026). From brain physiology to brain physics, Mathematical modeling of the human brain II: from glymphatics to deep learning. Springer Cham. ISSN 9783032006790. Fulltekst i vitenarkiv
• Hu, Xiaozhe; Kuchta, Miroslav; Mardal, Kent-Andre & Wang, Xue (2026). PARAMETER-ROBUST PRECONDITIONER FOR STOKES-DARCY COUPLED PROBLEM WITH LAGRANGE MULTIPLIER^\ast. SIAM Journal on Scientific Computing. ISSN 1064-8275. 48(2), s. A878–A904. doi: 10.1137/25M1744393. Fulltekst i vitenarkiv
• Solheim, Andreas; Ringstad, Geir; Eide, Per Kristian; Valnes, Lars Magnus & Mardal, Kent-Andre (2026). The impact of human brain geometry on the transport of an intrathecal tracer. Fluids and Barriers of the CNS. 23. doi: 10.1186/s12987-026-00764-y. Fulltekst i vitenarkiv Vis sammendrag

  Intrathecal contrast-enhanced magnetic resonance imaging (MRI), utilizing the contrast agent gadobutrol as cerebrospinal fluid (CSF) tracer is emerging as a useful method to study glymphatic function in the human brain. A consistent finding with this technique is large inter-individual variability regarding tracer propagation. In this study, we outline an approach which predicts the distribution of tracer in the parenchyma based only on geometric information from brain tissue as captured by MRI, addressing one possible explanation for this variability. Registrations are computed from pre-injection MRI, and used to map images at $$24$$ hours after tracer injection to perform predictions of tracer enrichment in the parenchyma in other patients. We apply the method to a dataset of human brain MRI of $$134$$ patients examined for different tentative diagnoses including idiopathic normal pressure hydrocephalus, spontaneous intracranial hypotension and idiopathic intracranial hypertension, as well as a group of reference patients. Tracer enrichment mapped between patients by image registration correlate strongly with actual observed enrichment. For patients in the reference group, the relative root mean squared error on our predictions is on average $$26\%$$ in the gray matter, and $$15\%$$ or less in other brain regions. Predictions are generally less reliable in the gray matter, and for patients with identified CSF leaks. We show that predictions made from purely geometrical considerations correlate strongly with actual MRI tracer enrichment for patients with similar diagnoses, thus quantifying the role of geometry in tracer enrichment.

• Poulain, Alexandre; Riseth, Jørgen Nilsen; Emblem, Kyrre E & Mardal, Kent-Andre (2026). Modeling glioma-induced impairments on the glymphatic system. Fluids and Barriers of the CNS. 23. doi: 10.1186/s12987-025-00736-8. Fulltekst i vitenarkiv Vis sammendrag

  Altered glymphatic function is observed for many neurological diseases. Glioma, one of the most common brain cancers, is known to have altered fluid dynamics in terms of edema and blood-brain barrier breakdown, both features potentially impacting the glymphatic function. To study glioma and its fluid dynamics, we propose a flexible mathematical model, including the tumor, the peri-tumoral edema and the healthy tissue. From a mechanical point of view, we consider the brain as a multicompartment porous medium and model both the fluid movement and the clearance of solutes within the brain. Our results indicate that the impairment of the glymphatic system due to glioma growth is two-fold. First, edema resulting from the leakage of fluid at the blood-brain barrier and/or the occlusion of the interstitial fluid exit routes (notably the perivascular spaces) due to migratory tumor cells result in a slight localized increase of pressure, consequently impairing negatively glymphatic clearance. Second, local changes of porosity (i.e. the volume fraction of certain compartments such as perivascular or extracellular spaces), result in a disruption of the transport of solutes in the brain. Our results indicate that an effect similar to the enhanced permeability and retention is obtained using biologically relevant changes of parameter values of our model. Our mathematical model is the first step towards a digital twin for drug or contrast product delivery within the cerebro-spinal fluid directly (e.g. from intrathecal injection) for patients suffering from gliomas.

• Mardal, Kent-Andre; Sogn, Jarle & Takacs, Stefan (2026). A robust and time-parallel preconditioner for parabolic reconstruction problems using Isogeometric analysis. Mathematical Models and Methods in Applied Sciences. ISSN 0218-2025. s. 26–26. doi: 10.1142/s0218202526500168. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we consider a PDE-constrained optimization problem of tracking type with parabolic state equation. The solution to the problem is characterized by the Karush–Kuhn–Tucker (KKT) system, which we formulate using a strong variational formulation of the state equation and a super weak formulation of the adjoined state equation. This allows us to propose a preconditioner that is robust both in the regularization and the diffusion parameter. In order to discretize the problem, we use Isogeometric Analysis since it allows the construction of sufficiently smooth basis functions effortlessly. To realize the preconditioner, one has to solve a problem over the whole space time cylinder that is elliptic with respect to certain non-standard norms. By using a fast diagonalization approach in time, we reformulate the problem as a collection of elliptic problems in space only. These problems are not only smaller, but our approach also allows to solve them in a time-parallel way. We show the efficiency of the preconditioner by rigorous analysis and illustrate it with numerical experiments.

• Riseth, Jørgen Nilsen; Koch, Timo; Lian, Sofie Lysholm; Storås, Tryggve Holck; T., Zikatanov, Ludmil & Valnes, Lars Magnus [Vis alle 8 forfattere av denne artikkelen] (2026). Human brain MRI data of intrathecally injected tracer evolution over 72 hours for data-integrated simulations. Scientific Data. 13. doi: 10.1038/s41597-026-06564-1. Fulltekst i vitenarkiv Vis sammendrag

  We present the Gonzo dataset: Brain MRI with processed and derivative data from one healthy male human volunteer ("Gonzo”) before and during the 72 hours after intrathecal injection of the contrast agent gadobutrol into the cerebrospinal fluid (CSF) of the spinal canal. The MRI data records include images highlighting the temporal and spatial evolution of the contrast agent in CSF, brain, and adjacent structures. In addition to raw MRI, we provide derivatives that enable numerical simulations of the transport process under study. Derivatives include T1 maps, tracer concentration maps, diffusion tensor maps, and unstructured triangulated volume meshes of the brain geometry. We also provide brain region markers obtained by image segmentation. A regional statistical analysis of the concentration data complements the image data. The presented data can be used to study the transport behavior and the underlying processes of a tracer in the brain. It is intended to contribute to and inspire new studies on the understanding of tracer transport, method development for image analysis, and simulation of brain fluid transport processes.

• Cai, Mingchao; Kuchta, Miroslav; Li, Jingzhi; Li, Ziliang & Mardal, Kent-Andre (2026). Parameter-Robust Preconditioners for a Four-Field Thermo-Poroelasticity Model. SIAM Journal on Scientific Computing. ISSN 1064-8275. 48(1), s. A49–A73. doi: 10.1137/25m1726923. Fulltekst i vitenarkiv Vis sammendrag

  We study a thermo-poroelasticity model which describes the interaction between the deformation of an elastic porous material and fluid flow under nonisothermal conditions. The model involves several parameters that can vary significantly in practical applications, posing a challenge for developing discretization techniques and solution algorithms that handle such variations effectively. We propose a four-field formulation and apply a conforming finite element discretization. The primary focus is on constructing and analyzing preconditioners for the resulting linear system. Two preconditioners are proposed: one involves regrouping variables and treating the 4-by-4 system as a 2-by-2 block form, while the other is directly constructed from the 4-by-4 coupled operator. Both preconditioners are demonstrated to be robust with respect to variations in parameters and mesh refinement. Numerical experiments are presented to demonstrate the effectiveness of the proposed preconditioners and validate their theoretical performance under varying parameter settings.

• Rieff, Melanie; Lapina, Oksana; Ringstad, Geir; Valnes, Lars; Warsza, Bogna & Eide, Per-Kristian [Vis alle 8 forfattere av denne artikkelen] (2025). U‐Net‐Based Prediction of Cerebrospinal Fluid Distribution and Ventricular Reflux Grading. NMR in Biomedicine. ISSN 0952-3480. Fulltekst i vitenarkiv
• Mardal, Kent-Andre; Sogn, Jarle & Takacs, Stefan (2025). A robust and time-parallel preconditioner for parabolic reconstruction problems using Isogeometric analysis. Mathematical Models and Methods in Applied Sciences. ISSN 0218-2025. Fulltekst i vitenarkiv
• Riseth, Jørgen N.; Koch, Timo & Mardal, Kent-Andre (2025). Two-compartment modeling of tracer transport in the brain. I dokken, jørgen; Mardal, Kent-Andre; Rognes, Marie Elisabeth; Valnes, Lars Magnus & Vinje, Vegard (Red.), Mathematical Modelling of the Human Brain II: From Glymphatics to Deep Learning | Springer Nature Link. Springer. s. 122–122. doi: 10.1007/978-3-032-00679-0_8. Fulltekst i vitenarkiv Vis sammendrag

  Open access briefs on neuroscience, mathematical modelling, deep learning, finite-element methods, open source software.

• Zapf, Bastian; Zeinhofer, Marius & Mardal, Kent-Andre (2025). Estimating molecular transport parameters using inverse PDE models. I dokken, jørgen; Mardal, Kent-Andre; Rognes, Marie Elisabeth; Valnes, Lars Magnus & Vinje, Vegard (Red.), Mathematical Modelling of the Human Brain II: From Glymphatics to Deep Learning | Springer Nature Link. Springer. s. 104–104. doi: 10.1007/978-3-032-00679-0_7. Fulltekst i vitenarkiv Vis sammendrag

  Open access briefs on neuroscience, mathematical modelling, deep learning, finite-element methods, open source software.

• Zeinhofer, Marius & Mardal, Kent-Andre (2025). Signal increase ratio prediction with CNNs. I dokken, jørgen; Mardal, Kent-Andre; Rognes, Marie Elisabeth; Valnes, Lars Magnus & Vinje, Vegard (Red.), Mathematical Modelling of the Human Brain II: From Glymphatics to Deep Learning | Springer Nature Link. Springer. s. 86–86. doi: 10.1007/978-3-032-00679-0_6. Fulltekst i vitenarkiv Vis sammendrag

  Open access briefs on neuroscience, mathematical modelling, deep learning, finite-element methods, open source software.

• Zapf, Bastian; Valnes, Lars Magnus; Mardal, Kent-Andre & Zikatanov, Ludmil (2025). Quantifying cerebrospinal fluid tracer concentration in the brain. I dokken, jørgen; Mardal, Kent-Andre; Rognes, Marie Elisabeth; Valnes, Lars Magnus & Vinje, Vegard (Red.), Mathematical Modelling of the Human Brain II: From Glymphatics to Deep Learning | Springer Nature Link. Springer. s. 76–76. doi: 10.1007/978-3-032-00679-0_5. Fulltekst i vitenarkiv Vis sammendrag

  Open access briefs on neuroscience, mathematical modelling, deep learning, finite-element methods, open source software.

• Valnes, Lars Magnus & Mardal, Kent-Andre (2025). Meshing the intracranial compartments: The cerebellum, cerebrum, brainstem, and cerebrospinal fluid, Mathematical modeling of the human brain II: from glymphatics to deep learning. Springer Cham. ISSN 9783032006790. s. 34–34. doi: 10.1007/978-3-032-00679-0_2. Fulltekst i vitenarkiv
• Rognes, Marie E.; Mardal, Kent-Andre; Valnes, Lars Magnus & Vinje, Vegard (2025). From brain physiology to brain physics. I dokken, jørgen; Mardal, Kent-Andre; Rognes, Marie Elisabeth; Valnes, Lars Magnus & Vinje, Vegard (Red.), Mathematical Modelling of the Human Brain II: From Glymphatics to Deep Learning | Springer Nature Link. Springer. s. 12–12. doi: 10.1007/978-3-032-00679-0_1. Fulltekst i vitenarkiv Vis sammendrag

  Open access briefs on neuroscience, mathematical modelling, deep learning, finite-element methods, open source software.

• Storås, Tryggve Holck; Lian, Sofie Lysholm; Mossige, Ingrid; Riseth, Jørgen Nilsen; Svensson, Siri Fløgstad & Løvland, Grethe [Vis alle 10 forfattere av denne artikkelen] (2025). T2-Weighted T1 Mapping and Automated Segmentation of CSF: Assessment of Solute Gradients in the Healthy Brain. Journal of Magnetic Resonance Imaging. ISSN 1053-1807. 0, s. 1–10. doi: 10.1002/jmri.70169. Fulltekst i vitenarkiv Vis sammendrag

  Background Cerebrospinal fluid (CSF) serves as a medium for nutrient delivery and waste clearance. The T1 relaxation rate, R1, can be used to measure the concentration of intrinsic solutes and extrinsic contrast agents. Purpose To implement a method for R1 mapping and segmentation of CSF and to use this method to explore how R1 of CSF relates to protein content and gadobutrol after intrathecal administration. Study Type Prospective cohort study, complemented by phantom analysis. Population Ten healthy control subjects (mean age 65.5 ± 4.4 years, range 57–72 years; five males and five females) and protein‐ and gadobutrol‐gradient phantom. Field Strength/Sequence 3 T Philips Ingenia scanner; 3D T2W mixed inversion recovery spin‐echo (T2W‐mixed IRSE) sequence and 3D T1W turbo field echo (3D T1W‐TFE). Assessment R1 maps were calculated by combining IR and SE data. An automated segmentation method derived from FreeSurfer employed SE data for CSF segmentation and T1W‐TFE for anatomical reference. CSF was collected by lumbar puncture for protein measurements, and 0.25 mmol gadobutrol was injected intrathecally. Post‐contrast assessments were performed at 3, 24, 48, and 72 h. Statistical Tests One‐way ANOVA, followed by a post hoc Tukey HSD test, and simple and multiple linear regression analysis; significance level of 0.05. Results R1 of ventricular CSF 0.216 ± 0.001 s −1 was significantly lower than that surrounding the cerebellum 0.225 ± 0.001 and cerebrum 0.228 ± 0.002 and correlated with lumbar protein concentration ( R 2  = 0.56). Peak gadobutrol concentrations were 101 ± 84 μM in ventricles, 185 ± 89 μM in cerebellar SAS and 166 ± 91 μM in cerebral SAS. Corresponding concentrations were 6 ± 4, 17 ± 8, and 37 ± 18 μM at 72 h. Data Conclusion Intrinsic R1 of CSF in the subarachnoid space correlated with protein content. Intracranial CSF enrichment after intrathecal administration of gadobutrol showed a large variation among healthy volunteers. Evidence Level 2. Technical Efficacy 3.

• Solheim, Andreas; Ringstad, Geir; Eide, Per Kristian & Mardal, Kent-Andre (2025). Geometry Reduced Order Modeling (GROM) with application to modeling of glymphatic function. Brain Research Bulletin. ISSN 0361-9230. 231. doi: 10.1016/j.brainresbull.2025.111558. Fulltekst i vitenarkiv Vis sammendrag

  Computational modeling of the brain has become a key part of understanding how the brain clears metabolic waste, but patient-specific modeling on a significant scale is still out of reach with current methods. We introduce a novel approach for leveraging model order reduction techniques in computational models of brain geometries to alleviate computational costs involved in numerical simulations. Using image registration methods based on magnetic resonance imaging, we compute inter-brain mappings which allow previously computed solutions on other geometries to be mapped on to a new geometry. We investigate this approach on two example problems typical of modeling of glymphatic function, applied to a dataset of 101 MRI of human patients. We discuss the applicability of the method when applied to a patient with no known neurological disease, as well as a patient diagnosed with idiopathic Normal Pressure Hydrocephalus displaying significantly enlarged ventricles. In each of our two example problems, we achieve a speedup of more 750 times compared to the full order problem, while introducing a comparably small additional system assembly overhead. The reduced solutions recover the full order solution with an error of less than 10% in most cases. Statement of significance: In many fields, model order reduction is a key technique in enabling high-throughput numerical simulations, but remains largely unexploited for biomedical modeling of the brain. In this work, we introduce a novel technique for building reduced representations integrating simulations performed on other brain geometries derived from MRI. Using this technique, we may leverage a dataset of previous solutions to accelerate simulations on new geometries, making patient-specific modeling more feasible.

• Rieff, Melanie; Holzberger, Fabian; Lapina, Oksana; Ringstad, Geir; Valnes, Lars Magnus & Warsza, Bogna [Vis alle 9 forfattere av denne artikkelen] (2025). U-Net-Based Prediction of Cerebrospinal Fluid Distribution and Ventricular Reflux Grading. NMR in Biomedicine. ISSN 0952-3480. 38(5). doi: 10.1002/nbm.70029. Fulltekst i vitenarkiv Vis sammendrag

  Previous work indicates evidence that cerebrospinal fluid (CSF) plays a crucial role in brain waste clearance processes and that altered flow patterns are associated with various diseases of the central nervous system. In this study, we investigate the potential of deep learning to predict the distribution in human brain of a gadolinium-based CSF contrast agent (tracer) administered intrathecal. For this, T1-weighted magnetic resonance imaging (MRI) scans taken at multiple time points before and after injection were utilized. We propose a U-net-based supervised learning model to predict pixel-wise signal increase at its peak after 24 h. Performance is evaluated based on different tracer distribution stages provided during training, including predictions from baseline scans taken before injection. Our findings show that training with imaging data from only the first 2-h postinjection yields tracer flow predictions comparable to models trained with additional later-stage scans. Validation against ventricular reflux gradings from neuroradiologists confirmed alignment with expert evaluations. These results demonstrate that deep learning-based methods for CSF flow prediction deserve more attention, as minimizing MR imaging without compromising clinical analysis could enhance efficiency, improve patient well-being and lower healthcare costs.

• Koch, Timo & Mardal, Kent-Andre (2025). Estimation of fluid flow velocities in cortical brain tissue driven by the microvasculature. Interface Focus. ISSN 2042-8898. 15(1). doi: 10.1098/rsfs.2024.0042. Fulltekst i vitenarkiv Vis sammendrag

  We present a modelling framework for describing bulk fluid flow in brain tissue. Within this framework, using computational simulation, we estimate bulk flow velocities in the grey matter parenchyma due to static or slowly varying water potential gradients—hydrostatic pressure gradients and osmotic pressure gradients. Working with the situation that experimental evidence and some model parameter estimates, as we point out, are presently insufficient to estimate velocities precisely, we explore feasible parameter ranges resulting in a range of estimates. We consider the effect of realistic microvascular architecture (extracted from mouse cortical grey matter). Although the estimated velocities are small in magnitude (e.g. in comparison to blood flow velocities), the passive transport of solutes with the bulk fluid can be a relevant process when considering larger molecules transported over larger distances. We compare velocity magnitudes resulting from filtration and pulsations. Filtration can lead to continuous directed fluid flow in the parenchyma, while pulsation-driven flow is (at least partly) reversible. For the first time, we consider the effect of the vascular architecture on the velocity distribution in a tissue sample of ca 1 mm3 cortical grey matter tissue. We conclude that both filtration and pulsations are potentially potent drivers for fluid flow.

• Schreiner, Jakob; Kessler, Ute; Ødegaard, Ketil; Mardal, Kent-Andre & Oltedal, Leif (2024). Exploring New Electroencephalogram Parameters in Electroconvulsive Therapy. The Journal of ECT. 40(1), s. 20–30. Fulltekst i vitenarkiv
• Zeinhofer, Marius; Masri, Rami & Mardal, Kent-Andre (2024). A unified framework for the error analysis of physics-informed neural networks. IMA Journal of Numerical Analysis. ISSN 0272-4979. doi: 10.1093/imanum/drae081. Fulltekst i vitenarkiv Vis sammendrag

  We prove a priori and a posteriori error estimates for physics-informed neural networks (PINNs) for linear PDEs. We analyze elliptic equations in primal and mixed form, elasticity, parabolic, hyperbolic and Stokes equations, and a PDE constrained optimization problem. For the analysis, we propose an abstract framework in the common language of bilinear forms, and we show that coercivity and continuity lead to error estimates. The obtained estimates are sharp and reveal that the penalty approach for initial and boundary conditions in the PINN formulation weakens the norm of the error decay. Finally, utilizing recent advances in PINN optimization, we present numerical examples that illustrate the ability of the method to achieve accurate solutions.

• Badia, Santiago; Hornkjøl, Martin; Khan, Arbaz; Mardal, Kent-Andre; Martín, Alberto F. & Ruiz-Baier, Ricardo (2024). Efficient and reliable divergence-conforming methods for an elasticity-poroelasticity interface problem. Computers and Mathematics with Applications. ISSN 0898-1221. 157, s. 173–194. doi: 10.1016/j.camwa.2023.12.038. Fulltekst i vitenarkiv Vis sammendrag

  We present a finite element discretization to model the interaction between a poroelastic structure and an elastic medium. The consolidation problem considers fully coupled deformations across an interface, ensuring continuity of displacement and total traction, as well as no-flux for the fluid phase. Our formulation of the poroelasticity equations incorporates displacement, fluid pressure, and total pressure, while the elasticity equations adopt a displacement-pressure formulation. Notably, the transmission conditions at the interface are enforced without the need for Lagrange multipliers. We demonstrate the stability and convergence of the divergence-conforming finite element method across various polynomial degrees. The a priori error bounds remain robust, even when considering large variations in intricate model parameters such as Lamé constants, permeability, and storativity coefficient. To enhance computational efficiency and reliability, we develop residual-based a posteriori error estimators that are independent of the aforementioned coefficients. Additionally, we devise parameter-robust and optimal block diagonal preconditioners. Through numerical examples, including adaptive scenarios, we illustrate the scheme's properties such as convergence and parameter robustness.

• Dimola, Nunzio; Kuchta, Miroslav; Mardal, Kent-Andre & Zunino, Paolo (2024). Robust Preconditioning of Mixed-Dimensional PDEs on 3d-1d Domains Coupled with Lagrange Multipliers. I Linninger, Andreas A.; Mardal, Kent-Andre & Zunino, Paolo (Red.), Quantitative Approaches to Microcirculation. Springer. ISSN 9783031585180. doi: 10.1007/978-3-031-58519-7_6. Fulltekst i vitenarkiv Vis sammendrag

  In the context of micro-circulation, the coexistence of two distinct length scales—the vascular radius and the tissue/organ scale—with a substantial difference in magnitude, poses significant challenges. To handle slender inclusions and simplify the geometry involved, a technique called topological dimensionality reduction is used, which suppresses the manifold dimensions associated with the smaller characteristic length. However, the algebraic structure of the resulting discretized system presents a challenge in constructing efficient solution algorithms. This chapter addresses this challenge by developing a robust preconditioner for the 3d-1d problem using the operator preconditioning technique. The robustness of the preconditioner is demonstrated with respect to the problem parameters, except for the vascular radius. The vascular radius, as demonstrated, plays a fundamental role in the mathematical well-posedness of the problem and the effectiveness of the preconditioner.

• Dreyer, Lars Willas; Eklund, Anders; Rognes, Marie Elisabeth; Malm, Jan; Qvarlander, Sara & Støverud, Karen-Helene [Vis alle 8 forfattere av denne artikkelen] (2024). Modeling CSF circulation and the glymphatic system during infusion using subject specific intracranial pressures and brain geometries. Fluids and Barriers of the CNS. 21(1). doi: 10.1186/s12987-024-00582-0. Fulltekst i vitenarkiv Vis sammendrag

  Background Infusion testing is an established method for assessing CSF resistance in patients with idiopathic normal pressure hydrocephalus (iNPH). To what extent the increased resistance is related to the glymphatic system is an open question. Here we introduce a computational model that includes the glymphatic system and enables us to determine the importance of (1) brain geometry, (2) intracranial pressure, and (3) physiological parameters on the outcome of and response to an infusion test. Methods We implemented a seven-compartment multiple network porous medium model with subject specific geometries from MR images using the finite element library FEniCS. The model consists of the arterial, capillary and venous blood vessels, their corresponding perivascular spaces, and the extracellular space (ECS). Both subject specific brain geometries and subject specific infusion tests were used in the modeling of both healthy adults and iNPH patients. Furthermore, we performed a systematic study of the effect of variations in model parameters. Results Both the iNPH group and the control group reached a similar steady state solution when subject specific geometries under identical boundary conditions was used in simulation. The difference in terms of average fluid pressure and velocity between the iNPH and control groups, was found to be less than 6% during all stages of infusion in all compartments. With subject specific boundary conditions, the largest computed difference was a 75% greater fluid speed in the arterial perivascular space (PVS) in the iNPH group compared to the control group. Changes to material parameters changed fluid speeds by several orders of magnitude in some scenarios. A considerable amount of the CSF pass through the glymphatic pathway in our models during infusion, i.e., 28% and 38% in the healthy and iNPH patients, respectively. Conclusions Using computational models, we have found the relative importance of subject specific geometries to be less important than individual differences in resistance as measured with infusion tests and model parameters such as permeability, in determining the computed pressure and flow during infusion. Model parameters are uncertain, but certain variations have large impact on the simulation results. The computations resulted in a considerable amount of the infused volume passing through the brain either through the perivascular spaces or the extracellular space.

• Budisa, Ana; Hu, Xiaozhe; Kuchta, Miroslav; Mardal, Kent-Andre & Zikatanov, Ludmil (2024). Algebraic multigrid methods for metric-perturbed coupled problems. SIAM Journal on Scientific Computing. ISSN 1064-8275. 46(3). doi: 10.1137/23M1572076. Fulltekst i vitenarkiv Vis sammendrag

  We develop multilevel methods for interface-driven multiphysics problems that can be coupled across dimensions and where complexity and strength of the interface coupling deteriorates the performance of standard methods. We focus on aggregation-based algebraic multigrid methods with custom smoothers that preserve the coupling information on each coarse level. We prove that, with the proper choice of subspace splitting, we obtain uniform convergence in discretization and physical parameters in the two-level setting. Additionally, we show parameter robustness and scalability with regard to the number of the degrees of freedom of the system on several numerical examples related to the biophysical processes in the brain, namely, the electric signaling in excitable tissue modeled by bidomain, the extracellular-membrane-intracellular (EMI) model, and reduced EMI equations.

• Vinje, Vegard; Zapf, Bastian; Ringstad, Geir; Eide, Per Kristian; Rognes, Marie Elisabeth & Mardal, Kent-Andre (2023). Human brain solute transport quantified by glymphatic MRI-informed biophysics during sleep and sleep deprivation. Fluids and Barriers of the CNS. 20. doi: 10.1186/s12987-023-00459-8. Fulltekst i vitenarkiv Vis sammendrag

  Whether you are reading, running or sleeping, your brain and its fluid environment continuously interacts to distribute nutrients and clear metabolic waste. Yet, the precise mechanisms for solute transport within the human brain have remained hard to quantify using imaging techniques alone. From multi-modal human brain MRI data sets in sleeping and sleep-deprived subjects, we identify and quantify CSF tracer transport parameters using forward and inverse subject-specific computational modelling. Our findings support the notion that extracellular diffusion alone is not sufficient as a brain-wide tracer transport mechanism. Instead, we show that human MRI observations align well with transport by either by an effective diffusion coefficent 3.5 $$\times $$ × that of extracellular diffusion in combination with local clearance rates corresponding to a tracer half-life of up to 5 h, or by extracellular diffusion augmented by advection with brain-wide average flow speeds on the order of 1–9 $$\mu $$ μ m/min. Reduced advection fully explains reduced tracer clearance after sleep-deprivation, supporting the role of sleep and sleep deprivation on human brain clearance.

• Budisa, Ana; Hu, Xiaozhe; Kuchta, Miroslav; Mardal, Kent-Andre & Zikatanov, Ludmil (2023). Rational Approximation Preconditioners for Multiphysics Problems. I Georgiev, Ivan; Datcheva, Maria; Georgiev, Krassimir & Nikolov, Geno (Red.), Numerical Methods and Applications 10th International Conference, NMA 2022, Borovets, Bulgaria, August 22–26, 2022, Proceedings. Springer. ISSN 9783031324123. s. 100–113. Fulltekst i vitenarkiv
• Koch, Timo; Vinje, Vegard & Mardal, Kent-Andre (2023). Estimates of the permeability of extra-cellular pathways through the astrocyte endfoot sheath. Fluids and Barriers of the CNS. 20(1). doi: 10.1186/s12987-023-00421-8. Fulltekst i vitenarkiv Vis sammendrag

  Background Astrocyte endfoot processes are believed to cover all micro-vessels in the brain cortex and may play a significant role in fluid and substance transport into and out of the brain parenchyma. Detailed fluid mechanical models of diffusive and advective transport in the brain are promising tools to investigate theories of transport. Methods We derive theoretical estimates of astrocyte endfoot sheath permeability for advective and diffusive transport and its variation in microvascular networks from mouse brain cortex. The networks are based on recently published experimental data and generated endfoot patterns are based on Voronoi tessellations of the perivascular surface. We estimate corrections for projection errors in previously published data. Results We provide structural-functional relationships between vessel radius and resistance that can be directly used in flow and transport simulations. We estimate endfoot sheath filtration coefficients in the range Lp = 2 × 10−11 mPa−1 s−1 to 3 × 10−11 mPa−1 s−1 , diffusion membrane coefficients for small solutes in the range CM = 5 × 102 m−1 to 6 × 102 m−1 , and gap area fractions in the range 0.2–0.6%, based on a inter-endfoot gap width of 20 nm. Conclusions The astrocyte endfoot sheath surrounding microvessels forms a secondary barrier to extra-cellular transport, separating the extra-cellular space of the parenchyma and the perivascular space outside the endothelial layer. The filtration and membrane diffusion coefficients of the endfoot sheath are estimated to be an order of magnitude lower than those of the extra-cellular matrix while being two orders of magnitude higher than those of the vessel wall.

• Gómez-Vargas, Bryan; Mardal, Kent-Andre; Ruiz-Baier, Ricardo & Vinje, Vegard (2023). Twofold Saddle-Point Formulation of Biot Poroelasticity with Stress-Dependent Diffusion. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 61(3), s. 1449–1481. doi: 10.1137/21m1449695. Fulltekst i vitenarkiv Vis sammendrag

  We present a new stress/total-pressure formulation for poroelasticity that incorporates the coupling with steady nonlinear diffusion modified by stress. This nonlinear problem is written in mixed-primal form, which combines a perturbed twofold saddle-point system with an elliptic problem. We analyze the continuous formulation within the framework of abstract fixed-point theory and Fredholm alternative for compact operators. A mixed finite element method is proposed, and its stability and convergence are rigorously analyzed. We also provide a few representative numerical examples to illustrate the effectiveness of the proposed formulation. The resulting model can be used to study the steady case of waste removal in the brain, providing insight into the transport of solutes in poroelastic structures under the influence of stress.

• Schreiner, Jakob Emil; Kessler, Ute; Ødegaard, Ketil Joachim; Mardal, Kent-Andre & Oltedal, Leif (2023). Exploring New Electroencephalogram Parameters in Electroconvulsive Therapy. Journal of ECT. ISSN 1095-0680. 40(1), s. 20–30. doi: 10.1097/YCT.0000000000000930. Fulltekst i vitenarkiv Vis sammendrag

  Background Electroconvulsive therapy (ECT) is an effective treatment against severe depressive episodes, which has been shown to induce volume changes in the hippocampus. The power spectrum of the electroencephalogram (EEG) follows a characteristic power-law relation, but its utility as a metric of ECT-induced seizures has not been explored. Objective This pilot study aims to evaluate a novel metric based on the power spectrum of the EEG recordings from ECT-induced seizures—its association to volume changes in the hippocampus after ECT and improvement in depression rating scores. Methods Depressed patients treated with ECT underwent brain magnetic resonance imaging before and after treatment and the EEG from each seizure was recorded (N = 29). Hippocampal volume changes and EEG parameters were recorded in addition to clinician-rated and self-reported measures of depressive symptoms. The slope of the power law in the power spectral density of the EEG was calculated. Multivariate linear models relating seizure parameters to volume change or clinical outcome were systematically and successively simplified. The best models were selected according to Akaike information criterion. Results The slope of the power law was steeper in the right than the left hemisphere (P < 0.001). Electroencephalogram measures were included in the best models of volume change for both hippocampi as well as in the models explaining clinical outcome (P = 0.014, P = 0.004). Conclusions In this pilot study, novel EEG measures were explored and contributed in models explaining the variation in volume change in the hippocampus and in clinical outcome after ECT.

• Budisa, Ana; Hu, Xiaozhe; Kuchta, Miroslav; Mardal, Kent-Andre & Zikatanov, Ludmil (2023). HAZniCS – Software Components for Multiphysics Problems. ACM Transactions on Mathematical Software. ISSN 0098-3500. 49(4). doi: 10.1145/3625561. Fulltekst i vitenarkiv Vis sammendrag

  We introduce the software toolbox HAZniCS for solving interface-coupled multiphysics problems. HAZniCS is a suite of modules that combines the well-known FEniCS framework for finite element discretization with solver and graph library HAZmath. The focus of this article is on the design and implementation of robust and efficient solver algorithms which tackle issues related to the complex interfacial coupling of the physical problems often encountered in applications in brain biomechanics. The robustness and efficiency of the numerical algorithms and methods is shown in several numerical examples, namely the Darcy-Stokes equations that model the flow of cerebrospinal fluid in the human brain and the mixed-dimensional model of electrodiffusion in the brain tissue.

• Bojarskaite, Laura; Vallet, Alexandra Honorine Emilie; Bjørnstad, Daniel Marelius; Binder, Kristin Maria Gullestad; Cunen, Celine & Heuser, Kjell [Vis alle 9 forfattere av denne artikkelen] (2023). Sleep cycle-dependent vascular dynamics in male mice and the predicted effects on perivascular cerebrospinal fluid flow and solute transport. Nature Communications. 14. doi: 10.1038/s41467-023-36643-5. Fulltekst i vitenarkiv Vis sammendrag

  Perivascular spaces are important highways for fluid and solute transport in the brain enabling efficient waste clearance during sleep. However, the underlying mechanisms augmenting perivascular flow in sleep are unknown. Using two-photon imaging of naturally sleeping male mice we demonstrate sleep cycle-dependent vascular dynamics of pial arteries and penetrating arterioles: slow, large-amplitude oscillations in NREM sleep, a vasodilation in REM sleep, and a vasoconstriction upon awakening at the end of a sleep cycle and microarousals in NREM and intermediate sleep. These vascular dynamics are mirrored by changes in the size of the perivascular spaces of the penetrating arterioles: slow fluctuations in NREM sleep, reduction in REM sleep and an enlargement upon awakening after REM sleep and during microarousals in NREM and intermediate sleep. By biomechanical modeling we demonstrate that these sleep cycle-dependent perivascular dynamics likely enhance fluid flow and solute transport in perivascular spaces to levels comparable to cardiac pulsation-driven oscillations.

• Kuchta, Miroslav; Wubshet, Sileshi; Afseth, Nils; Mardal, Kent-Andre & Liland, Kristian (2022). Encoder–decoder neural networks for predicting future. Journal of Biophotonics. ISSN 1864-063X. 15(9). doi: 10.1002/jbio.v15.910.1002/jbio.202200097. Fulltekst i vitenarkiv
• Mardal, Kent-Andre; Sogn, Jarle & Takacs, Stefan (2022). Robust Preconditioning and Error Estimates for Optimal Control of the Convection--Diffusion--Reaction Equation with Limited Observation in Isogeometric Analysis. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 60(1), s. 195–221. doi: 10.1137/21M139147X. Fulltekst i vitenarkiv
• Hornkjøl, Martin; Valnes, Lars Magnus; Ringstad, Geir; Rognes, Marie Elisabeth; Eide, Per Kristian & Mardal, Kent-Andre [Vis alle 7 forfattere av denne artikkelen] (2022). CSF circulation and dispersion yield rapid clearance from intracranial compartments. Frontiers in Bioengineering and Biotechnology. 10, s. 1–14. doi: 10.3389/fbioe.2022.932469. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we used a computational model to estimate the clearance of a tracer driven by the circulation of cerebrospinal fluid (CSF) produced in the choroid plexus (CP) located within the lateral ventricles. CSF was assumed to exit the subarachnoid space (SAS) via different outflow routes such as the parasagittal dura, cribriform plate, and/or meningeal lymphatics. We also modelled a reverse case where fluid was produced within the spinal canal and absorbed in the choroid plexus in line with observations on certain iNPH patients. No directional interstitial fluid flow was assumed within the brain parenchyma. Tracers were injected into the foramen magnum. The models demonstrate that convection in the subarachnoid space yields rapid clearance from both the SAS and the brain interstitial fluid and can speed up intracranial clearance from years, as would be the case for purely diffusive transport, to days.

• Zapf, Bastian; Haubner, Johannes; Kuchta, Miroslav; Ringstad, Geir; Eide, Per Kristian & Mardal, Kent-Andre (2022). Investigating molecular transport in the human brain from MRI with physics-informed neural networks. Scientific Reports. 12. doi: 10.1038/s41598-022-19157-w. Fulltekst i vitenarkiv Vis sammendrag

  In recent years, a plethora of methods combining neural networks and partial differential equations have been developed. A widely known example are physics-informed neural networks, which solve problems involving partial differential equations by training a neural network. We apply physics-informed neural networks and the finite element method to estimate the diffusion coefficient governing the long term spread of molecules in the human brain from magnetic resonance images. Synthetic testcases are created to demonstrate that the standard formulation of the physics-informed neural network faces challenges with noisy measurements in our application. Our numerical results demonstrate that the residual of the partial differential equation after training needs to be small for accurate parameter recovery. To achieve this, we tune the weights and the norms used in the loss function and use residual based adaptive refinement of training points. We find that the diffusion coefficient estimated from magnetic resonance images with physics-informed neural networks becomes consistent with results from a finite element based approach when the residuum after training becomes small. The observations presented here are an important first step towards solving inverse problems on cohorts of patients in a semi-automated fashion with physics-informed neural networks.

• Kuchta, Miroslav; Wubshet, Sileshi Gizachew; Afseth, Nils Kristian; Mardal, Kent-Andre & Liland, Kristian Hovde (2022). Encoder–decoder neural networks for predicting future FTIR spectra – application to enzymatic protein hydrolysis. Journal of Biophotonics. ISSN 1864-063X. 15(9), s. 1–18. doi: 10.1002/jbio.202200097. Fulltekst i vitenarkiv Vis sammendrag

  In the process of converting food-processing by-products to value-addedingredients, fine grained control of the rawmaterials, enzymes and process conditionsensures the best possible yield and eco-nomic return. However, when raw mate-rial batches lack good characterization andcontain high batch variation, online or at-line monitoring of the enzymatic reac-tions would be beneficial. We investigate the potential of deep neural networks inpredicting the future state of enzymatic hydrolysis as described by Fourier-trans-form infrared spectra of the hydrolysates. Combined with predictions of averagemolecular weight, this provides a flexible and transparent tool for process moni-toring and control, enabling proactive adaption of process parameters.

• Boon, Wietse; Hornkjøl, Martin; Kuchta, Miroslav; Mardal, Kent-Andre & Ruiz-Baier, Ricardo (2022). Parameter-robust methods for the Biot–Stokes interfacial coupling without Lagrange multipliers. Journal of Computational Physics. ISSN 0021-9991. 467. doi: 10.1016/j.jcp.2022.111464. Fulltekst i vitenarkiv
• Mardal, Kent-Andre; Sogn, Jarle & Takacs, Stefan (2022). ROBUST PRECONDITIONING AND ERROR ESTIMATES FOR OPTIMAL CONTROL OF THE CONVECTION-DIFFUSION-REACTION EQUATION WITH LIMITED OBSERVATION IN ISOGEOMETRIC ANALYSIS. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 60(1), s. 195–221. doi: 10.1137/21M139147X. Fulltekst i vitenarkiv
• Schreiner, Jakob Emil & Mardal, Kent-Andre (2022). Simulating epileptic seizures using the bidomain model. Scientific Reports. 12(1). doi: 10.1038/s41598-022-12101-y. Fulltekst i vitenarkiv Vis sammendrag

  Abstract Epileptic seizures are due to excessive and synchronous neural activity. Extensive modelling of seizures has been done on the neuronal level, but it remains a challenge to scale these models up to whole brain models. Measurements of the brain’s activity over several spatiotemporal scales follow a power-law distribution in terms of frequency. During normal brain activity, the power-law exponent is often found to be around 2 for frequencies between a few Hz and up to 150 Hz, but is higher during seizures and for higher frequencies. The Bidomain model has been used with success in modelling the electrical activity of the heart, but has been explored far less in the context of the brain. This study extends previous models of epileptic seizures on the neuronal level to the whole brain using the Bidomain model. Our approach is evaluated in terms of power-law distributions. The electric potentials were simulated in 7 idealized two-dimensional models and 3 three-dimensional patient-specific models derived from magnetic resonance images (MRI). Computed electric potentials were found to follow power-law distributions with slopes ranging from 2 to 5 for frequencies greater than 10–30 Hz.

• Hong, Qingguo; Kraus, Johannes; Kuchta, Miroslav; Lymbery, Maria; Mardal, Kent-Andre & Rognes, Marie Elisabeth (2022). Robust Approximation of Generalized Biot-Brinkman Problems. Journal of Scientific Computing. ISSN 0885-7474. 93(3). doi: 10.1007/s10915-022-02029-w. Fulltekst i vitenarkiv
• Arbes, Florian Maurits; Jensen, Øyvind; Mardal, Kent-Andre & Dokken, Jorgen S. (2022). Model order reduction of solidification problems, 8th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2022). European Community on Computional Methods in Applied Sciences (ECCOMAS). ISSN 9788412322286. doi: 10.23967/eccomas.2022.179. Fulltekst i vitenarkiv Vis sammendrag

  Advection driven problems are known to be difficult to model with a reduced basis because of a slow decay of the Kolmogorov N -width. This paper investigates how this challenge transfers to the context of solidification problems and tries to answer when and to what extend reduced order models (ROMs) work for solidification problems. In solidification problems, the challenge is not the advection per se, but rather a moving solidification front. This paper studies reduced spaces for 1D step functions that move in time, which can either be seen as advection of a quantity or as a moving solidification front. Furthermore, the reduced space of a 2D solidification test case is compared with the reduced space of an alloy solidification featuring a mushy zone. The results show that not only the PDE itself, but the smoothness of the solution is crucial for the decay of the singular values and thus the quality of a reduced space representation.

• Bohr, Tomas; Hjorth, Poul G.; Holst, Sebastian C.; Hrabětová, Sabina; Kiviniemi, Vesa & Lilius, Tuomas [Vis alle 19 forfattere av denne artikkelen] (2022). The glymphatic system: Current understanding and modeling. iScience. 25(9). doi: 10.1016/j.isci.2022.104987. Fulltekst i vitenarkiv Vis sammendrag

  We review theoretical and numerical models of the glymphatic system, which circulates cerebrospinal fluid and interstitial fluid around the brain, facilitating solute transport. Models enable hypothesis development and predictions of transport, with clinical applications including drug delivery, stroke, cardiac arrest, and neurodegenerative disorders like Alzheimer’s disease. We sort existing models into broad categories by anatomical function: Perivascular flow, transport in brain parenchyma, interfaces to perivascular spaces, efflux routes, and links to neuronal activity. Needs and opportunities for future work are highlighted wherever possible; new models, expanded models, and novel experiments to inform models could all have tremendous value for advancing the field.

• Boon, Wietse; Koch, Timo; Kuchta, Miroslav & Mardal, Kent-Andre (2022). Robust Monolithic Solvers for the Stokes--Darcy Problem with the Darcy Equation in Primal Form. SIAM Journal on Scientific Computing. ISSN 1064-8275. 44(4), s. B1148–B1174. doi: 10.1137/21M1452974. Fulltekst i vitenarkiv Vis sammendrag

  We construct mesh-independent and parameter-robust monolithic solvers for the coupled primal Stokes--Darcy problem. Three different formulations and their discretizations in terms of conforming and nonconforming finite element methods and finite volume methods are considered. In each case, robust preconditioners are derived using a unified theoretical framework. In particular, the suggested preconditioners utilize operators in fractional Sobolev spaces. Numerical experiments demonstrate the parameter-robustness of the proposed solvers.

• Kuchta, Miroslav; Mardal, Kent-Andre; Rognes, Marie & Tveito, Aslak (2021). Solving the EMI Equations using Finite Element Methods. doi: 10.1007/978-3-030-61157-6_5. Fulltekst i vitenarkiv Vis sammendrag

  This chapter discusses 2 X 2 symmetric variational formulations and associated finite element methods for the EMI equations. We demonstrate that the presented methods converge at expected rates, and compare the approaches in terms of approximation of the transmembrane potential. Overall, the choice of which formulation to employ for solving EMI models becomes largely a matter of desired accuracy and available computational resources.

• Buccino, Alessio; Kuchta, Miroslav; Schreiner, Jakob; Mardal, Kent-Andre; Tveito, Aslak & Rognes, Marie (2021). Improving Neural Simulations with the EMI Model. doi: 10.1007/978-3-030-61157-6_7. Fulltekst i vitenarkiv Vis sammendrag

  Mathematical modeling of neurons is an essential tool to investigate neuronal activity alongside with experimental approaches. However, the conventional modeling framework to simulate neuronal dynamics and extracellular potentials makes several assumptions that might need to be revisited for some applications. In this chapter we apply the EMI model to investigate the ephaptic effect and the effect of the extracellular probes on the measured potential. Finally, we introduce reduced EMI models, which provide a more computationally efficient framework for simulating neurons with complex morphologies.

• Kuchta, Miroslav; Mardal, Kent-Andre; Tveito, Aslak & Rognes, Marie (2021). Iterative Solvers for EMI Models. doi: 10.1007/978-3-030-61157-6_6. Fulltekst i vitenarkiv Vis sammendrag

  This chapter deals with iterative solution algorithms for the four EMI formulations derived in (17, Chapter 5). Order optimal monolithic solvers robust with respect to material parameters, the number of degrees of freedom of discretization as well as the time-stepping parameter are presented and compared in terms of computational cost. Domain decomposition solver for the single-dimensional primal formulation is discussed.

• Telle, Åshild; Wall, Samuel; Sundnes, Joakim; Tveito, Aslak; Rognes, Marie & Mardal, Kent-Andre (2021). Modeling Cardiac Mechanics on a Sub-Cellular Scale. I Tveito, Aslak; Mardal, Kent-Andre & Rognes, Marie (Red.), Modeling Excitable Tissue -- The EMI Framework. Springer. ISSN 9783030611569. s. 28–43. Fulltekst i vitenarkiv Vis sammendrag

  We aim to extend existing models of single-cell mechanics to the EMI framework, to define spatially resolved mechanical models of cardiac myocytes embedded in a passive extracellular space. The models introduced here will be pure mechanics models employing fairly simple constitutive laws for active and passive mechanics. Future extensions of the models may include a coupling to the electrophysiology and electro-diffusion models described in the other chapters, to study the impact of spatially heterogeneous ion concentrations on the cell and tissue mechanics.

• Mardal, Kent-Andre; Rognes, Marie & Thompson, Travis (2021). Accurate discretization of poroelasticity without Darcy stability – Stokes-Biot revisited. BIT Numerical Mathematics. ISSN 0006-3835. 61, s. 941–976. doi: 10.1007/s10543-021-00849-0. Fulltekst i vitenarkiv
• Ellingsrud, Ada; Daversin-Catty, Cécile; Rognes, Marie; Mardal, Kent-Andre & Tveito, Aslak (2021). A cell-based model for ionic electrodiffusion in excitable tissue. I Tveito, Aslak; Mardal, Kent-Andre & Rognes, Marie (Red.), Modeling Excitable Tissue -- The EMI Framework. Springer. ISSN 9783030611569. s. 14–27. Fulltekst i vitenarkiv
• Young, Bruce A.; Adams, James; Beary, Jonathan M.; Mardal, Kent-Andre; Schneider, Robert & Kondrashova, Tatyana (2021). Variations in the cerebrospinal fluid dynamics of the American alligator (Alligator mississippiensis). Fluids and Barriers of the CNS. 18. doi: 10.1186/s12987-021-00248-1. Fulltekst i vitenarkiv Vis sammendrag

  Background Studies of mammalian CSF dynamics have been focused on three things: paravascular flow, pressure and pulsatility, and “bulk” flow; and three (respective) potential motive forces have been identified: vasomotor, cardiac, and ventilatory. There are unresolved questions in each area, and few links between the different areas. The American alligator (Alligator mississippiensis) has pronounced plasticity in its ventilatory and cardiovascular systems. This study was designed to test the hypothesis that the greater cardiovascular and ventilatory plasticity of A. mississippiensis would result in more variation within the CSF dynamics of this species. Methods Pressure transducers were surgically implanted into the cranial subarachnoid space of 12 sub-adult alligators; CSF pressure and pulsatility were monitored along with EKG and the exhalatory gases. In four of the alligators a second pressure transducer was implanted into the spinal subarachnoid space. In five of the alligators the CSF was labeled with artificial microspheres and Doppler ultrasonography used to quantify aspects of the spinal CSF flow. Results Both temporal and frequency analyses of the CSF pulsations showed highly variable contributions of both the cardiac and ventilatory cycles. Unlike the mammalian condition, the CSF pressure pulsations in the alligator are often of long (~ 3 s) duration, and similar duration CSF unidirectional flow pulses were recorded along the spinal cord. Reduction of the duration of the CSF pulsations, as during tachycardia, can lead to a “summation” of the pulsations. There appears to be a minimum duration (~ 1 s) of isolated CSF pulsations. Simultaneous recordings of cranial and spinal CSF pressures reveal a 200 ms delay in the propagation of the pressure pulse from the cranium to the vertebral canal. Conclusions Most of the CSF flow dynamics recorded from the alligators, are similar to what has been reported from studies of the human CSF. It is hypothesized that the link between ventilatory mechanics and CSF pulsations in the alligator is mediated by displacement of the spinal dura. The results of the study suggest that understanding the CSF dynamics of Alligator may provide unique insights into the evolutionary origins and functional regulation of the human CSF dynamics.

• Kuchta, Miroslav; Laurino, Federica; Mardal, Kent-Andre & Zunino, Paolo (2021). Analysis and approximation of mixed-dimensional pdes on 3d-1d domains coupled with lagrange multipliers. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 59(1), s. 558–582. doi: 10.1137/20M1329664. Fulltekst i vitenarkiv Vis sammendrag

  Coupled partial differential equations (PDEs) defined on domains with different dimensionality are usually called mixed-dimensional PDEs. We address mixed-dimensional PDEs on three-dimensional (3D) and one-dimensional (1D) domains, which gives rise to a 3D-1D coupled problem. Such a problem poses several challenges from the standpoint of existence of solutions and numerical approximation. For the coupling conditions across dimensions, we consider the combination of essential and natural conditions, which are basically the combination of Dirichlet and Neumann conditions. To ensure a meaningful formulation of such conditions, we use the Lagrange multiplier method suitably adapted to the mixed-dimensional case. The well-posedness of the resulting saddle-point problem is analyzed. Then, we address the numerical approximation of the problem in the framework of the finite element method. The discretization of the Lagrange multiplier space is the main challenge. Several options are proposed, analyzed, and compared, with the purpose of determining a good balance between the mathematical properties of the discrete problem and flexibility of implementation of the numerical scheme. The results are supported by evidence based on numerical experiments.

• Piersanti, Eleonora; Lee, Jeonghun; Thompson, Travis; Mardal, Kent-Andre & Rognes, Marie (2021). Parameter robust preconditioning by congruence for multiple-network poroelasticity. SIAM Journal on Scientific Computing. ISSN 1064-8275. 43(4), s. B984–B1007. doi: 10.1137/20M1326751. Fulltekst i vitenarkiv Vis sammendrag

  The mechanical behavior of a poroelastic medium permeated by multiple interacting fluid networks can be described by a system of time-dependent partial differential equations known as the multiple-network poroelasticity (MPET) equations or multiporosity/multipermeability systems. These equations generalize Biot's equations, which describe the mechanics of the one network case. The efficient numerical solution of the MPET equations is challenging, in part due to the complexity of the system and in part due to the presence of interacting parameter regimes. In this paper, we present a new strategy for efficiently and robustly solving the MPET equations numerically. In particular, we discuss an approach to formulating finite element methods and associated preconditioners for the MPET equations based on simultaneous diagonalization of the element matrices. We demonstrate the technique for the multicompartment Darcy equations, with large exchange variability, and the MPET equations for a nearly incompressible medium with large exchange variability. The approach is based on designing transformations of variables that simultaneously diagonalize (by congruence) the equations' key operators and subsequently constructing parameter-robust block diagonal preconditioners for the transformed system. The proposed approach is supported by theoretical considerations as well as by numerical results.

• Boon, Wietse; Kuchta, Miroslav; Mardal, Kent-Andre & Ruiz-Baier, Ricardo (2021). Robust preconditioners for perturbed saddle-point problems and conservative discretizations of Biot's equations utilizing total pressure. SIAM Journal on Scientific Computing. ISSN 1064-8275. 43(4), s. B961–B983. doi: 10.1137/20M1379708. Fulltekst i vitenarkiv Vis sammendrag

  We develop robust solvers for a class of perturbed saddle-point problems arising in the study of a second-order elliptic equation in mixed form (in terms of flux and potential), and of the four-field formulation of Biot's consolidation problem for linear poroelasticity (using displacement, filtration flux, total pressure, and fluid pressure). The stability of the continuous variational mixed problems, which hinges upon using adequately weighted spaces, is addressed in detail; and the efficacy of the proposed preconditioners, as well as their robustness with respect to relevant material properties, is demonstrated through several numerical experiments.

• Mardal, Kent-Andre; Rognes, Marie E. & Thompson, Travis (2021). Accurate discretization of poroelasticity without Darcy stability -- Stokes–Biot stability revisited. BIT Numerical Mathematics. ISSN 0006-3835. 61, s. 941–976. doi: 10.1007/s10543-021-00849-0. Fulltekst i vitenarkiv Vis sammendrag

  Abstract In this manuscript we focus on the question: what is the correct notion of Stokes–Biot stability? Stokes–Biot stable discretizations have been introduced, independently by several authors, as a means of discretizing Biot’s equations of poroelasticity; such schemes retain their stability and convergence properties, with respect to appropriately defined norms, in the context of a vanishing storage coefficient and a vanishing hydraulic conductivity. The basic premise of a Stokes–Biot stable discretization is: one part Stokes stability and one part mixed Darcy stability. In this manuscript we remark on the observation that the latter condition can be generalized to a wider class of discrete spaces. In particular: a parameter-uniform inf-sup condition for a mixed Darcy sub-problem is not strictly necessary to retain the practical advantages currently enjoyed by the class of Stokes–Biot stable Euler–Galerkin discretization schemes.

• Eide, Per Kristian; Vinje, Vegard; Pripp, Are Hugo; Mardal, Kent-Andre & Ringstad, Geir (2021). Sleep deprivation impairs molecular clearance from the human brain. Brain. ISSN 0006-8950. 144(3), s. 863–874. doi: 10.1093/brain/awaa443. Fulltekst i vitenarkiv
• Eide, Per Kristian; Valnes, Lars Magnus; Lindstrøm, Erika Kristina; Mardal, Kent-Andre & Ringstad, Geir (2021). Direction and magnitude of cerebrospinal fluid flow vary substantially across central nervous system diseases. Fluids and Barriers of the CNS. 18(1). doi: 10.1186/s12987-021-00251-6. Fulltekst i vitenarkiv Vis sammendrag

  Background Several central nervous system diseases are associated with disturbed cerebrospinal fluid (CSF) flow patterns and have typically been characterized in vivo by phase-contrast magnetic resonance imaging (MRI). This technique is, however, limited by its applicability in space and time. Phase-contrast MRI has yet to be compared directly with CSF tracer enhanced imaging, which can be considered gold standard for assessing long-term CSF flow dynamics within the intracranial compartment. Methods Here, we studied patients with various CSF disorders and compared MRI biomarkers of CSF space anatomy and phase-contrast MRI at level of the aqueduct and cranio-cervical junction with dynamic intrathecal contrast-enhanced MRI using the contrast agent gadobutrol as CSF tracer. Tracer enrichment of cerebral ventricles was graded 0–4 by visual assessment. An intracranial pressure (ICP) score was used as surrogate marker of intracranial compliance. Results The study included 94 patients and disclosed marked variation of CSF flow measures across disease categories. The grade of supra-aqueductal reflux of tracer varied, with strong reflux (grades 3–4) in half of patients. Ventricular tracer reflux correlated with stroke volume and aqueductal CSF pressure gradient. CSF flow in the cerebral aqueduct was retrograde (from 4th to 3rd ventricle) in one third of patients, with estimated CSF net flow volume about 1.0 L/24 h. In the cranio-cervical junction, net flow was cranially directed in 78% patients, with estimated CSF net flow volume about 4.7 L/24 h. Conclusions The present observations provide in vivo quantitative evidence for substantial variation in direction and magnitude of CSF flow, with re-direction of aqueductal flow in communicating hydrocephalus, and significant extra-cranial CSF production. The grading of ventricular reflux of tracer shows promise as a clinical useful method to assess CSF flow pattern disturbances in patients.

• Erhardt, André; Mardal, Kent-Andre & Schreiner, Jakob (2020). Dynamics of a neuron–glia system: the occurrence of seizures and the influence of electroconvulsive stimuli: A mathematical and numerical study. Journal of Computational Neuroscience. ISSN 0929-5313. 48, s. 229–251. Fulltekst i vitenarkiv

Se alle arbeider i NVA

• Dokken, Jørgen; Mardal, Kent-Andre; Rognes, Marie; Valnes, Lars & Vinje, Vegard (2026). Mathematical modeling of the human brain II: from glymphatics to deep learning. Springer Cham. ISBN 9783032006790. Fulltekst i vitenarkiv
• dokken, jørgen; Mardal, Kent-Andre; Rognes, Marie Elisabeth; Valnes, Lars Magnus & Vinje, Vegard (2026). Mathematical Modelling of the Human Brain II: From Glymphatics to Deep Learning | Springer Nature Link. Springer. doi: 10.1007/978-3-032-00679-0. Fulltekst i vitenarkiv Vis sammendrag

  Open access briefs on neuroscience, mathematical modelling, deep learning, finite-element methods, open source software.

• linninger, andreas; Mardal, Kent-Andre & zunino, paolo (2024). Quantitative Approaches to Microcirculation: Mathematical Models, Computational Methods and Data Analysis | Springer Nature Link. Springer. doi: 10.1007/978-3-031-58519-7. Fulltekst i vitenarkiv Vis sammendrag

  This volume brings together a series of insightful chapters that highlight the latest trends in modeling microcirculation.

• Linninger, Andreas A.; Mardal, Kent-Andre & Zunino, Paolo (2024). Quantitative Approaches to Microcirculation. Springer. ISBN 9783031585180. 220 s. Fulltekst i vitenarkiv
• Mardal, Kent-Andre; Rognes, Marie; Thompson, Travis & Valnes, Lars (2022). Mathematical modeling of the human brain: from magnetic resonance images to finite element simulation. Springer. 118 s. Fulltekst i vitenarkiv
• Mardal, Kent-Andre; Rognes, Marie Elisabeth; Thompson, Travis & Valnes, Lars Magnus (2022). Mathematical Modeling of the Human Brain - From Magnetic Resonance Images to Finite Element Simulation. Springer Nature. ISBN 9783030951351. 118 s. Fulltekst i vitenarkiv
• Tveito, Aslak; Mardal, Kent-Andre & Rognes, Marie (2021). Modeling Excitable Tissue: The EMI Framework. Springer. Fulltekst i vitenarkiv

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• Solheim, Andreas & Mardal, Kent-Andre (2026). The importance of barriers and interfaces in brain fluid modeling​. Fulltekst i vitenarkiv
• Solheim, Andreas & Mardal, Kent-Andre (2026). Data-integrated simulations of solute transport in the rodent brain. Fulltekst i vitenarkiv
• Solheim, Andreas; Ringstad, Geir; Eide, Per Kristian; Acar, Evrim & Mardal, Kent-Andre (2026). Data-driven insights from glymphatic MRI​. Fulltekst i vitenarkiv
• Solheim, Andreas; Ringstad, Geir; Eide, Per Kristian; Nordengen, Kaja & Mardal, Kent-Andre (2026). Image registration as a tool for modeling and analysis of brain solute transport​. Fulltekst i vitenarkiv
• Girelli, Alberto; Solheim, Andreas; Lian, Sofie Lysholm; Storås, Tryggve Holck; Nordengen, Kaja & Mardal, Kent-Andre (2026). Exploring dura hydrodynamics: The Role of the Meningeal Lymphatic Vessels in Brain Clearance. Fulltekst i vitenarkiv
• Solheim, Andreas; Girelli, Alberto; Lian, Sofie Lysholm; Storås, Tryggve Holck; Nordengen, Kaja & Mardal, Kent-Andre (2026). Exploring the brain dura: Estimating meningeal lymphatic clearance through mathematical models. Fulltekst i vitenarkiv
• Girelli, Alberto; Solheim, Andreas; Lian, Sofie Lysholm; Nordengen, Kaja; Storås, Tryggve Holck & Mardal, Kent-Andre (2026). Exploring the brain dura: a multiscale framework for dural lymphatic transport. Fulltekst i vitenarkiv
• Mardal, Kent-Andre (2026). Computational modeling of brain clearance during sleep. Fulltekst i vitenarkiv
• Girelli, Alberto; Solheim, Andreas; Lian, Sofie Lysholm; Storås, Tryggve Holck; Nordengen, Kaja & Mardal, Kent-Andre (2026). Computational Modeling of transport in the human brain dura. Fulltekst i vitenarkiv
• Mardal, Kent-Andre (2026). Why do we sleep? -- Computational models of brain clearance. Fulltekst i vitenarkiv
• Mardal, Kent-Andre (2026). Computational modeling of brain clearance: why do we sleep? Fulltekst i vitenarkiv
• Causemann, Marius & Mardal, Kent-Andre (2026). Inferring Human Intracranial CSF Absorption Sites via Inverse Modelling of Protein Transport. Fulltekst i vitenarkiv Vis sammendrag

  The clearance of metabolic waste products and the distribution of therapeutic agents within the central nervous system are critically dependent on the circulation of cerebrospinal fluid (CSF). However, the precise anatomical locations of CSF production and absorption in the human brain remain subjects of ongoing investigation. Recent clinical research [Storås et al. 2025] and open tracer evolution datasets [Riseth et al. 2026] utilized high-resolution T1 mapping (R1) to demonstrate significant regional solute gradients in the CSF of healthy brains, where R1 in the subarachnoid space (SAS) correlates strongly with intrinsic protein concentration. These findings suggest that quantitative R1 maps can serve as a reliable, non-invasive proxy for protein distribution within the human intracranial space. Building on this physiological insight, we present a computational framework to infer CSF absorption pathways by integrating mathematical modeling with clinically acquired R1 imaging data. We model the transport of proteins within the CSF-filled spaces using a coupled fluid flow and mass transport model. First, we generate a set of basis velocity fields by solving the steady-state Stokes equations, where each field represents a distinct hypothesis for a specific absorption site. Second, we model the transport of proteins using a stationary advection-diffusion equation driven by a linear combination of these basis fields. To identify the active absorption sites, we formulate the inverse problem as a PDE-constrained optimization. We define a objective function that minimizes the mismatch between the computed equilibrium protein concentration and the measured R1 data field. By solving for the optimal combination of basis flows that best reproduces the observed protein gradients, this approach provides a mathematical tool for non-invasively characterizing intracranial fluid dynamics and validating physiological transport hypotheses.

• Causemann, Marius & Mardal, Kent-Andre (2026). Inferring Human Intracranial CSF Absorption Sites via Inverse Modelling of Protein Transport. Fulltekst i vitenarkiv
• Valseth, Kristen Joy; Valnes, Lars Magnus; Silantyeva, Olga; Lappegard, Gaute & Mardal, Kent-Andre (2025). CAMELS-Nordic, a large-scale hydrometeorological and catchment properties dataset for Norway and Sweden. doi: https:/www.hydrologiraadet.no/wp-content/uploads/2025/08/AbstractsLillehammer2025_210825.pdf. Fulltekst i vitenarkiv Vis sammendrag

  Historically, hydrologic studies have focused on one or a small number of basins. In many cases, these studies were limited by data availability and computational resources. In the last 20 years, the availability of large hydrological datasets, such as gridded meteorological data sets and streamflow timeseries, and increased computing resources have empowered large-sample hydrology studies. Having accessible and high-quality large datasets available to the science community facilitates the evaluation of hydrologic processes and prediction questions. To support modeling and climate research efforts in the Nordics the CAMELS (Catchment Attributes and Meteorology for Large-sample Studies)-Nordic was collected and processed into a coherent dataset for the entirety of Norway and Sweden. CAMELS-Nordic combines not only meteorological and hydrological, but also topography, climate, streamflow, land cover, and soil data with python package to update time series automatically where possible. The development of the data package takes advantage of high-quality and freely available data from various Norwegian, Swedish, and European agencies. It includes: (1) daily forcing data (e.g. observations, interpolations, and modeled data) for catchments located in Norway and Sweden; (2) daily streamflow data; (3) digital elevation model; (4) catchment properties (size, location, elevation, and catchment files); (5) landcover; and (6) soil type data. Dataset time series span 1980 to 2022.

• Valseth, Kristen Joy; Valnes, Lars Magnus; Lappegard, Gaute; Silantyeva, Olga & Mardal, Kent-Andre (2025). Development of CAMELS-Nordic, a large-scale hydrometeorological and catchment properties dataset for Norway and Sweden. doi: 10.5194/egusphere-egu25-10411. Fulltekst i vitenarkiv Vis sammendrag

  Historically, hydrologic studies have focused on one or a small number of basins, in many cases these studies were limited by data availability and computational resources. The increased availability of large hydrological datasets in the last 20 years, such as gridded meteorological data sets and streamflow timeseries, and increased computing resources have empowered large-sample hydrology studies having accessible and high-quality large datasets available to the science community to facilitate the evaluation of hydrologic processes and prediction questions. To support modeling and climate research efforts in the Nordics the CAMELS (Catchment Attributes and Meteorology for Large-sample Studies)-Nordic was collected and processed from multiple sources and databases into a coherent dataset for the entirety of Norway and Sweden. CAMELS-Nordic combines not only meteorological and hydrological, but also topography, climate, streamflow, land cover, soil, and geology data with python tools to update time series automatically. The development of the data package takes advantage of high-quality and freely available data from various Norwegian, Swedish, and European agencies. &#160;It includes: (1) daily forcing data (e.g. observations, interpolations, and modeled data) for catchments located in Norway and Sweden; (2) daily streamflow data; (3) digital elevation model; (4) catchment properties (size, location, elevation, and catchment files); (5) landcover; and (6) soil type data. Dataset time series span 1980 to 2022.

• Solheim, Andreas; Ringstad, Geir; Valnes, Lars Magnus; Eide, Per Kristian & Mardal, Kent-Andre (2025). The role of brain geometry in glymphatic modeling. Fulltekst i vitenarkiv
• Solheim, Andreas; Ringstad, Geir; Eide, Per Kristian & Mardal, Kent-Andre (2025). An image registration-based approach for model order reduction in brain geometries. Fulltekst i vitenarkiv
• Mardal, Kent-Andre (2025). Mathematical Challenges of Modeling the Fluid Flow of the Brain - Brain Clearance, Sleep and Dementia. Fulltekst i vitenarkiv
• Mardal, Kent-Andre (2025). On the inf-sup condition in slender geometries – mixed dimensions and application to brain fluid dynamics. Fulltekst i vitenarkiv
• Mardal, Kent-Andre (2025). Mathematical challenges of modeling the fluid flow of the brain - brain clearance, sleep and dementia. Fulltekst i vitenarkiv
• Koch, Timo; Vinje, Vegard & Mardal, Kent-Andre (2023). Correction to: Estimates of the permeability of extra-cellular pathways through the astrocyte endfoot sheath (Fluids and Barriers of the CNS, (2023), 20, 1, (20), 10.1186/s12987-023-00421-8). Fluids and Barriers of the CNS. 20(1). doi: 10.1186/s12987-023-00456-x. Fulltekst i vitenarkiv
• Koch, Timo; Mardal, Kent-Andre & Valnes, Lars Magnus (2023). The Complex Pore Spaces of Our Brain: A Challenge and Opportunity for Computer Simulations. I Médici, Ezequiel F. & Otero, Alejandro D. (Red.), Album of Porous Media. Springer. ISSN 9783031238000. s. 29–29. doi: 10.1007/978-3-031-23800-0_20. Fulltekst i vitenarkiv
• Kuchta, Miroslav; Budisa, Ana; Hu, Xiaozhe; Zikatanov, Ludmil & Mardal, Kent-Andre (2022). Fractional Laplacians and Hybridized DG. Fulltekst i vitenarkiv
• Kuchta, Miroslav; Boon, Wietse; Hornkjøl, Martin; Koch, Timo; Mardal, Kent-Andre & Ruiz-Baier, Ricardo (2022). Parameter-robust Methods for Biot-Stokes and Darcy-Stokes Interfacial Coupling without Lagrange Multipliers. Fulltekst i vitenarkiv
• Kuchta, Miroslav; Boon, Wietse; Koch, Timo; Mardal, Kent-Andre & Ruiz-Baier, Ricardo (2022). Parameter-robust monolithic solvers for coupled Biot/Darcy-Stokes models. Fulltekst i vitenarkiv
• Kuchta, Miroslav; Boon, Wietse; Holter, Karl; Hornkjøl, Martin; Koch, Timo & Mardal, Kent-Andre [Vis alle 7 forfattere av denne artikkelen] (2022). Fractional operators in coupled multiphysics problems with implicit coupling. Fulltekst i vitenarkiv
• Kuchta, Miroslav; Boon, Wietse; Holter, Karl; Hornkjøl, Martin; Koch, Timo & Mardal, Kent-Andre [Vis alle 7 forfattere av denne artikkelen] (2022). Parameter-robust monolithic solvers for Stokes-Darcy/Biot systems. Fulltekst i vitenarkiv
• Daversin-Catty, Cécile; Gjerde, Ingeborg; Mardal, Kent-Andre; Rognes, Marie & Vinje, Vegard (2021). Modeling fluid flow in perivascular networks. doi: 10.5281/zenodo.7848945. Fulltekst i vitenarkiv
• Piersanti, Eleonora; Rognes, Marie & Mardal, Kent-Andre (2021). Parameter robust preconditioning for multi-compartmental Darcy equations. doi: 10.1007/978-3-030-55874-1. Fulltekst i vitenarkiv
• Arbes, Florian Maurits; Mardal, Kent-Andre; Jensen, Øyvind & Dokken, Jorgen S. (2021). Modelling of solidification problems with FEniCS. Fulltekst i vitenarkiv
• Mardal, Kent-Andre (2020). Of mice and men, etc. Fulltekst i vitenarkiv
• Sande, Espen; Koch, Timo; Kuchta, Miroslav & Mardal, Kent-Andre (2025). On a robust inf-sup condition for the Stokes problem in slender domains - with application to preconditioning. arxiv.org. doi: https:/arxiv.org/abs/2510.24590. Fulltekst i vitenarkiv

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