{"id":365,"date":"2020-07-09T16:56:57","date_gmt":"2020-07-09T16:56:57","guid":{"rendered":"http:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/?page_id=365"},"modified":"2026-07-25T12:41:45","modified_gmt":"2026-07-25T12:41:45","slug":"research","status":"publish","type":"page","link":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/research\/","title":{"rendered":"Research"},"content":{"rendered":"<div class=\"reading-column\" style=\"max-width:760px;margin-left:auto;margin-right:auto\">\n<div class=\"research-toc\" style=\"margin:1.5em 0 2em;padding:1em 1.25em;border:1px solid #e2e2e2;border-radius:6px;background:#faf9f7\">\n<p style=\"margin:0 0 .6em;font-weight:600;font-size:.95em;letter-spacing:.02em;color:#cc0033\">Areas of Research Focus<\/p>\n<ul style=\"margin:0;padding-left:1.2em;line-height:1.9\">\n<li><a href=\"#advancing-the-no-u-turn-sampler\">Advancing the No-U-Turn Sampler<\/a><\/li>\n<li><a href=\"#couplings-for-kinetic-langevin-diffusions\">Couplings for Kinetic Langevin Diffusions<\/a><\/li>\n<li><a href=\"#the-no-underrun-sampler\">The No-Underrun Sampler<\/a><\/li>\n<li><a href=\"#mixing-time-of-the-no-u-turn-sampler\">Mixing Time of the No-U-Turn Sampler<\/a><\/li>\n<li><a href=\"#hamiltonian-monte-carlo-mixing-time\">Hamiltonian Monte Carlo Mixing Time<\/a><\/li>\n<li><a href=\"#randomized-time-integrators-for-hamiltonian-mcmc\">Randomized Time Integrators for Hamiltonian MCMC<\/a><\/li>\n<li><a href=\"#piecewise-deterministic-markov-processes\">Piecewise Deterministic Markov Processes<\/a><\/li>\n<li><a href=\"#path-integral-molecular-dynamics\">Path Integral Molecular Dynamics<\/a><\/li>\n<li><a href=\"#numerical-solution-of-stochastic-differential-equations\">Numerical Solution of Stochastic Differential Equations<\/a><\/li>\n<\/ul>\n<\/div>\n<h2 id=\"advancing-the-no-u-turn-sampler\">Advancing the No-U-Turn Sampler<\/h2>\n<p>The No-U-Turn Sampler (NUTS) employs a remarkably clever way to locally adapt the path length in Hamiltonian Monte Carlo (HMC), effectively avoiding U-turns during leapfrog integration while preserving detailed balance. This self-tuning capability has established NUTS as the default MCMC method for continuously differentiable densities in probabilistic programming languages such as STAN, PyMC3, NIMBLE, Turing, and NumPyro. Given its widespread success, a natural question arises: can this local adaptation strategy be extended to other HMC tuning parameters? In Bou-Rabee, Carpenter &amp; Marsden (2024), we address this question by developing a flexible framework for self-tuning HMC called GIST \u2013\u2013 Gibbs self-tuning for HMC. GIST not only generalizes the adaptation strategy of NUTS but also unifies a broad class of existing locally adaptive HMC samplers, including NUTS, the Apogee-to-Apogee Path Sampler, multinomial HMC and randomized HMC. What makes this unification particularly compelling is that it reveals an underlying universality: when reversibility is preserved, the transition step of any locally adaptive HMC sampler naturally aligns with the transition step of a GIST sampler. Beyond this unification, the GIST framework unlocks: (i) simpler alternatives to the No-U-Turn Sampler for path length tuning; but also (ii) local adaptation of NUTS\u2019s other parameters. This makes GIST a powerful and versatile extension of NUTS, with broad implications for the future development of sampling algorithms.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.jmlr.org\/papers\/v27\/25-1452.html\"><b>The Within-Orbit Adaptive Leapfrog No-U-Turn Sampler<\/b><\/a><br \/>\nN. Bou-Rabee, B. Carpenter, T. S. Kleppe &amp; S. Liu, Journal of Machine Learning Research 27(113), 1-64, 2026.<\/li>\n<li><a href=\"https:\/\/www.arxiv.org\/abs\/2408.08259\"><b>Incorporating Local Step-Size Adaptivity into the No-U-Turn Sampler using Gibbs Self Tuning<\/b><\/a><br \/>\nN. Bou-Rabee, B. Carpenter, T. S. Kleppe &amp; M. Marsden, Journal of Chemical Physics, Vol. 163, Issue 8, 084119, 2025.<\/span><\/li>\n<li><a href=\"https:\/\/doi.org\/10.1214\/26-SS156\"><b>GIST: Gibbs self-tuning for locally adaptive Hamiltonian Monte Carlo<\/b><\/a><br \/>\nN. Bou-Rabee, B. Carpenter, &amp; M. Marsden, Statist. Surv. 20: 135-179, 2026.<\/li>\n<li><a href=\"https:\/\/doi.org\/10.1090\/qam\/1748\"><b>From Continuous to Discrete: a No-Turn-Sampler for Permutations<\/b><\/a><br \/>\nN. Bou-Rabee &amp; Z. Wang, To appear, Quart. Appl. Math., 2026.<\/li>\n<\/ul>\n<div style=\"max-width:760px;margin:2em auto;background-color:#ede1c8;border:3px solid #5b4630;padding:9px\">\n<div style=\"border:1px solid #93795a;padding:22px 24px 16px\">\n<div style=\"border:2px solid #93795a;background:#fdfcf9;text-align:center;line-height:0\"><video style=\"max-width:100%;height:auto;vertical-align:top\" controls preload=\"metadata\" poster=\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/07\/nuts_vs_walnuts_page_poster.jpg\" src=\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/07\/nuts_vs_walnuts_page_1600x900.mp4\"><\/video><\/div>\n<p style=\"text-align:center;font-family:Georgia,serif;font-style:italic;color:#5b4630;font-size:1.02em;line-height:1.45;margin:14px 6px 2px\">NUTS vs WALNUTS in Neal\u2019s funnel: standard NUTS stalls at the neck and diverges (red), while WALNUTS adapts its step size within each trajectory and explores the full distribution with no divergences.<\/p>\n<\/div>\n<\/div>\n<h2 id=\"couplings-for-kinetic-langevin-diffusions\">Couplings for Kinetic Langevin Diffusions<\/h2>\n<p>Kinetic (underdamped) Langevin dynamics augments the overdamped Langevin diffusion with a velocity variable, and this momentum is what makes many modern gradient-based samplers fast. On ill-conditioned targets it is expected to reduce the mixing time from scaling linearly in the condition number to scaling with its square root, the sampling analogue of momentum acceleration in optimization. In practice the continuous dynamics are approximated by a splitting scheme, most commonly OBABO, which produces an implementable Markov chain. The same feature that yields the speedup also makes the chain hard to analyze: the noise enters only through the velocity and reaches the position coordinate indirectly, so the dynamics are hypoelliptic. This degeneracy obstructs coupling arguments, the main tool for sharp mixing-time bounds from a cold start, because two coupled copies must be driven together even though the position coordinate carries no noise of its own. Whether OBABO attains the square-root mixing time from a generic cold start remains open, and this question motivates the coupling constructions below. A longer and more accessible exposition appears in an <a href=\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/on-couplings-for-kinetic-langevin-diffusions\/\">accompanying blog post<\/a>.<\/p>\n<ul>\n<li><a href=\"https:\/\/arxiv.org\/abs\/2605.31088\"><b>On Couplings for Kinetic Langevin Diffusions<\/b><\/a><br \/>\nN. Bou-Rabee, S. Cox &amp; R. Schieven, Preprint, 2026.<\/li>\n<li><a href=\"https:\/\/arxiv.org\/abs\/2308.04634\"><b>Mixing of Metropolis-Adjusted Markov Chains via Couplings: The High Acceptance Regime<\/b><\/a><br \/>\nN. Bou-Rabee &amp; S. Oberd\u00f6rster, Electronic Journal of Probability, Vol. 29, paper no. 89, pp. 1-27, 2024.<\/li>\n<\/ul>\n<div style=\"text-align:center;margin:2em 0\"><img decoding=\"async\" src=\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/07\/coalescence-map.png\" alt=\"The coalescence map at work: two coupled copies of the process, started apart, are steered so their gap closes to zero and they meet exactly at a chosen time.\" style=\"max-width:100%;height:auto;border-radius:2px\"><\/div>\n<h2 id=\"the-no-underrun-sampler\">The No-Underrun Sampler<\/h2>\n<p>The No-Underrun Sampler (NURS) is a locally adaptive, gradient-free Markov Chain Monte Carlo method that approximates Hit-and-Run while incorporating orbit-based exploration, similar to the No-U-Turn Sampler, but without requiring gradients. Like Hit-and-Run, NURS selects a random direction uniformly from the unit sphere, avoiding artifacts from fixed coordinate systems. It then generates an orbit along this direction and dynamically adjusts the length of this orbit to match the local scale of the target distribution. State selection is performed using exact density evaluations along the orbit via categorical sampling. NURS has been rigorously analyzed, with formalized connections to Hit-and-Run, quantitative tuning guidelines, and an extension of coupling results for Hit-and-Run in the Gaussian case. Empirical tests on Neal\u2019s funnel, a challenging multiscale distribution, show that while NURS is diffusive in narrow regions, its ballistic movement in broader regions offsets this, enabling efficient sampling across the distribution\u2019s different scales.<\/p>\n<ul>\n<li><a href=\"https:\/\/arxiv.org\/abs\/2501.18548\"><b>The No-Underrun Sampler: A Locally-Adaptive, Gradient-Free MCMC Method<\/b><\/a><br \/>\nN. Bou-Rabee, B. Carpenter, S. Liu, &amp; S. Oberd\u00f6rster, arXiv:2501.18548 [math.ST], 2025.<\/li>\n<li><a href=\"https:\/\/doi.org\/10.1214\/25-EJP1418\"><b>Accelerated Convergence in Hit-and-Run Monte Carlo and a Coordinate-free Randomized Kaczmarz Algorithm<\/b><\/a><br \/>\nN. Bou-Rabee, A. Eberle, &amp; S. Oberd\u00f6rster, Electronic Journal of Probability, Vol. 30, paper no. 154, 1-28, 2025.<\/li>\n<\/ul>\n<div style=\"text-align:center;margin:2em 0\"><img decoding=\"async\" src=\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/07\/no-underrun-coupling-v2.png\" alt=\"Coupling two copies of the No-Underrun Sampler: by lining up the grids along each copy's search direction, the two copies are made to choose matching moves, so the distance between them contracts step by step.\" style=\"max-width:100%;height:auto;border-radius:2px\"><\/div>\n<h2 id=\"mixing-time-of-the-no-u-turn-sampler\">Mixing Time of the No-U-Turn Sampler<\/h2>\n<p>The No-U-Turn Sampler (NUTS) is the default sampler for continuously differentiable densities in probabilistic programming languages like Stan (with over 100K users), PyMC3, NIMBLE, Turing, and NumPyro. However, due to its recursive architecture, even proving its reversibility has been an ongoing challenge. For a concise proof based on interpreting NUTS as an auxiliary variable method, check out: <a href=\"https:\/\/arxiv.org\/abs\/2404.15253\">Bou-Rabee, Carpenter, and Marsden [2024]<\/a>. In Bou-Rabee &amp; Oberd\u00f6rster (2024), we prove that the mixing time of NUTS, when initialized in the concentration region of the canonical Gaussian measure, scales as d^{1\/4}, up to logarithmic factors, where d is the dimension. This is the first mixing time guarantee for NUTS and the scaling is expected to be sharp. The proof is based on a coupling argument that leverages the geometric structure of the target distribution. Specifically, concentration of measure results in a uniformity in NUTS&#8217; locally adapted transitions, which holds with high probability. This uniformity is formalized by interpreting NUTS as an accept\/reject Markov chain, where the mixing properties for the more uniform accept chain are analytically tractable. Additionally, our analysis uncovers a previously unnoticed issue with the path length adaptation procedure of NUTS, specifically related to looping behavior, which we address in detail.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.arxiv.org\/abs\/2410.06978\"><b>Mixing of the No-U-Turn Sampler and the Geometry of Gaussian Concentration<\/b><\/a><br \/>\nN. Bou-Rabee, &amp; S. Oberd\u00f6rster, arXiv:2410.06978 [math.PR], 2024.<\/li>\n<\/ul>\n<div style=\"text-align:center;margin:2em 0\"><img decoding=\"async\" src=\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/07\/nuts-u-turn-uniformity.png\" alt=\"In high dimension the No-U-Turn condition concentrates around a clean sine curve, so the length of each orbit has essentially the same distribution wherever the sampler currently sits.\" style=\"max-width:100%;height:auto;border-radius:2px\"><\/div>\n<h2 id=\"hamiltonian-monte-carlo-mixing-time\">Hamiltonian Monte Carlo Mixing Time<\/h2>\n<p>The Hamiltonian Monte Carlo (HMC) sampler combines Hamiltonian dynamics with momentum randomization. The Hamiltonian dynamics are discretized, and the resulting discretization bias is either accepted or corrected by a Metropolis-Hastings accept-reject step. Despite HMC&#8217;s empirical success, until recently, convergence bounds for HMC samplers were scarce. This has changed over the past few years with the development of new methods for quantifying convergence to equilibrium, including approaches based on coupling, conductance and hypocoercivity. In a series of works (referenced below), we introduced novel coupling techniques that establish mixing and complexity upper bounds for both uadjusted and adjusted HMC, particularly in models with high-dimensionality and non-convexity. These coupling techniques offer significant flexibility for further development, including potential applications as diagnostics to assess HMC convergence.<\/p>\n<ul>\n<li><a href=\"https:\/\/arxiv.org\/abs\/2601.09019\"><b>Tail-Sensitive KL and R&eacute;nyi Convergence of Unadjusted Hamiltonian Monte Carlo via One-Shot Couplings<\/b><\/a><br \/>\nN. Bou-Rabee, S. Mitra &amp; A. Wibisono, In revision, Ann. Appl. Probab., 2026.<\/li>\n<li><a href=\"https:\/\/arxiv.org\/abs\/2105.00887\"><b>Mixing Time Guarantees for Unadjusted Hamiltonian Monte Carlo<\/b><\/a><br \/>\nN. Bou-Rabee &amp; A. Eberle, Bernoulli, Vol. 29, pp. 75-104, 2023.<\/li>\n<li><a href=\"https:\/\/arxiv.org\/abs\/2009.08735\"><b>Convergence of Unadjusted Hamiltonian Monte Carlo for mean-field models<\/b><\/a><br \/>\nN. Bou-Rabee &amp; K. Schuh, Electronic Journal of Probability, Vol. 28, paper no. 91, pp. 1-40, 2023.<\/li>\n<li><b><\/b><a href=\"https:\/\/arxiv.org\/abs\/1909.07962\"><b>Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions<\/b><\/a><br \/>\nN. Bou-Rabee &amp; A. Eberle, Stoch PDE: Anal Comp, Vol. 9, pp. 207-242, 2021.<\/li>\n<li><a href=\"https:\/\/arxiv.org\/abs\/1805.00452\"><b>Coupling and Convergence for Hamiltonian Monte Carlo<\/b><\/a><br \/>\nN. Bou-Rabee, A. Eberle, &amp; R. Zimmer,\u00a0Annals of Applied Probability, Vol. 30, No. 3, pp. 1209-1250, 2020.<\/li>\n<\/ul>\n<div style=\"text-align:center;margin:2em 0\"><img decoding=\"async\" src=\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/07\/hmc-couplings-v2.png\" alt=\"A family of ways to couple two copies of Hamiltonian Monte Carlo, from synchronous to one-shot to two-scale, each yielding bounds on how quickly the sampler forgets where it started.\" style=\"max-width:100%;height:auto;border-radius:2px\"><\/div>\n<h2 id=\"randomized-time-integrators-for-hamiltonian-mcmc\">Randomized Time Integrators for Hamiltonian MCMC<\/h2>\n<p>Measure-preserving ODEs, SDEs, and PDMPs are key in constructing MCMC kernels. Among the time integrators for the corresponding flows, randomized time integrators have shown distinct advantages in terms of complexity. The core idea behind these randomized time integrators is to replace the deterministic quadrature rules typically used in each time integration step with a Monte Carlo integration rule. This approach not only reduces the regularity requirements on the underlying ODE, SDE or PDMP, but also significantly improves the asymptotic bias properties of the resulting Markov kernel, and thus, its complexity. Variants of these schemes that are Metropolis-adjustable were developed in Bou-Rabee &amp; Marsden (2022). Furthermore, the effectiveness of these randomized integrators for mean-field models was demonstrated in Bou-Rabee &amp; Schuh (2023). Additionally, under minimal regularity conditions for the drift coefficients, 2.5-order randomized Runge-Kutta-Nystr\u00f6m (rRKN) schemes were introduced in Bou-Rabee &amp; Kleppe (2023).<\/p>\n<ul>\n<li><a href=\"https:\/\/arxiv.org\/abs\/2310.07399\"><b>Randomized Runge-Kutta-Nystr\u00f6m Methods for Unadjusted Hamiltonian and Kinetic Langevin Monte Carlo <\/b><\/a><br \/>\nN. Bou-Rabee &amp; T. S. Kleppe, Mathematics of Computation 2025, Vol. 94, \u00a0No. 356, 2839-2865.<\/li>\n<li><a href=\"https:\/\/arxiv.org\/abs\/2308.11491\"><b>Nonlinear Hamiltonian Monte Carlo &amp; its Particle Approximation<\/b><\/a><br \/>\nN. Bou-Rabee &amp; K. Schuh, In revision, Stoch PDEs: Anal. Comp.<\/li>\n<li><a href=\"https:\/\/arxiv.org\/abs\/2211.11003\"><b>Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration<\/b><\/a><br \/>\nN. Bou-Rabee &amp; M. Marsden, Annals of Applied Probability 2025, Vol. 35, No. 1, 360-392.<\/li>\n<\/ul>\n<div style=\"text-align:center;margin:2em 0\"><img decoding=\"async\" src=\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/07\/randomized-integrator-accuracy.png\" alt=\"Accuracy of the randomized time integrator: as the step size is halved, its error falls at the predicted three-halves rate, faster than a standard integrator.\" style=\"max-width:100%;height:auto;border-radius:2px\"><\/div>\n<h2 id=\"piecewise-deterministic-markov-processes\">Piecewise Deterministic Markov Processes<\/h2>\n<p><a href=\"https:\/\/en.wikipedia.org\/wiki\/Andersen_thermostat\">Andersen dynamics<\/a> describes a piecewise deterministic Markov process that follows Hamiltonian dynamics, with the velocity of selected components randomized after exponentially distributed waiting times. It shares similarities with the kinetic Langevin diffusion but differs in that it lacks both diffusion and explicit dissipation, and instead involves a non-local jump operator. When the velocities of all components are randomized, it reduces to Randomized Hamiltonian Monte Carlo. In Bou-Rabee &amp; Sanz-Serna (2017), we verified geometric ergodicity for Randomized Hamiltonian Monte Carlo, and in Bou-Rabee &amp; Eberle (2022), we used a maximal coupling of the velocity randomizations to achieve optimal dimension dependence in Wasserstein convergence bounds for Andersen dynamics with weak interactions, without requiring global convexity of the potential energy.<\/p>\n<ul>\n<li><a href=\"https:\/\/arxiv.org\/abs\/2009.14239\"><b>Couplings for Andersen Dynamics<\/b><\/a><br \/>\nN. Bou-Rabee &amp; A. Eberle, Annales de l&#8217;Institut Henri Poincar\u00e9 (B) Probabilit\u00e9s et Statistiques, Vol. 58, No. 2, pp. 916-944, 2022.<\/li>\n<li><a href=\"https:\/\/arxiv.org\/abs\/1511.09382\"><b>Randomized Hamiltonian Monte Carlo<\/b><\/a><br \/>\nN. Bou-Rabee &amp; J. M. Sanz-Serna, \u00a0Annals of Applied Probability, Vol. 27, No. 4, pp. 2159-2194, 2017.<\/li>\n<\/ul>\n<div style=\"text-align:center;margin:2em 0\"><img decoding=\"async\" src=\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/07\/andersen-velocity-coupling.png\" alt=\"Coupling the velocity randomization step of Andersen dynamics: each copy\u2019s new velocity is chosen, nudged together or mirror-reflected, so two copies of the system steadily draw closer.\" style=\"max-width:100%;height:auto;border-radius:2px\"><\/div>\n<h2 id=\"path-integral-molecular-dynamics\">Path Integral Molecular Dynamics<\/h2>\n<p>Path Integral Molecular Dynamics (PIMD) leverages Feynmann&#8217;s path-integral formulation of quantum statistical mechanics to compute static and dynamic quantum statistics, especially when quantum nuclear effects are significant. Applications include calculating chemical reaction rates, diffusion coefficients, and absorption spectra. PIMD&#8217;s appeal lies in its use of a classical ring-polymer-bead system to approximate quantum properties. Central to PIMD simulations is the free ring-polymer update, but fast harmonic motions present within the ring-polymer, necessitate a strongly stable approximation. Additionally, the integration scheme must remain accurate as the number of beads approaches infinity. Unfortunately, existing methods &#8212; including most ring-polymer molecular dynamics (RPMD) and thermostatted RPMD (T-RPMD) schemes &#8212; fail in this regard, as the overlap between the exact and sampled ring-polymer Boltzmann distributions tends to zero in the infinite bead limit. A key result of the papers referenced below is the development of strongly stable, &#8220;dimension-free&#8221; numerical integration schemes that preserve non-zero overlap with the exact distribution in the infinite bead limit. These methods, valued numerically for quantum liquid water, enable highly accurate and stable ergodic time-averaging. The approach involves a simple &#8220;Cayley&#8221; modification of standard &#8220;BAOAB&#8221; splitting schemes from molecular dynamics, leading to new &#8220;BCOCB&#8221; methods which are expected to be widely adopted for future PIMD simulations.<\/p>\n<ul>\n<li><a href=\"https:\/\/arxiv.org\/abs\/2011.01601\"><b>A generalized class of strongly stable and dimension-free T-RPMD integrators<\/b><\/a><br \/>\nJ. L. Rosa-Ra\u00edces, J. Sun, N. Bou-Rabee, &amp; T. F. Miller III, J. Chem. Phys. 154, 024106, 2021.<\/li>\n<li><a href=\"https:\/\/arxiv.org\/abs\/1909.07962\"><b>Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions<\/b><\/a><br \/>\nN. Bou-Rabee &amp; A. Eberle, Stoch PDE: Anal Comp, Vol. 9, pp. 207-242, 2021.<\/li>\n<li><a href=\"https:\/\/doi.org\/10.1063\/1.5134810\"><b>Dimension-free path-integral molecular dynamics without preconditioning<\/b><\/a> [Editor&#8217;s Pick]<br \/>\nR. Korol, J. L. Rosa-Ra\u00edces, N. Bou-Rabee &amp; T. F. Miller III, J. Chem. Phys. 152, 104102, 2020.<\/li>\n<li><a href=\"https:\/\/arxiv.org\/abs\/1907.07941\"><b>Cayley modification for strongly stable path-integral and ring-polymer molecular dynamics<\/b><\/a> [Editor&#8217;s Pick]<br \/>\nR. Korol, N. Bou-Rabee &amp; T. F. Miller III, J. Chem. Phys. 151, 124103, 2019.<\/li>\n<\/ul>\n<div style=\"text-align:center;margin:2em 0\"><img decoding=\"async\" src=\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/07\/cayley-exponential-eigenvalues-v2.png\" alt=\"Keeping fast vibrations stable: the Cayley update (right) keeps every mode on the unit circle, while the exact exponential (left) loses strong stability when overlapping frequencies resonate.\" style=\"max-width:100%;height:auto;border-radius:2px\"><\/div>\n<h2 id=\"numerical-solution-of-stochastic-differential-equations\">Numerical Solution of Stochastic Differential Equations<\/h2>\n<p>Sticky diffusion processes frequently arise in models of physical and natural phenomena where the system&#8217;s variables can change dimension. However, until recently, there were no effective methods to simulate these processes. This changed with our work with Eric Vanden-Eijnden (NYU), where we developed a method that discretizes SDEs in space rather than time, resulting in a Markov jump process (MJP) approximation. Because the increments of the MJP can be uniformly bounded, the algorithm is numerically stable by construction. Additionally, MJPs naturally handle boundary conditions, such as positivity constraints in population or interest rate models, which are challenging with standard methods. In collaboration with Miranda Holmes-Cerfon (UBC), we extended this approach to include &#8220;sticky&#8221; boundary conditions, demonstrating that MJPs can dramatically speed up simulations, sometimes by orders of magnitude. Beyond sticky diffusions, MJPs offer a practical approach for approximating SDEs and SPDEs with complex conditions, such as interface conditions, and can be applied to processes where the underlying differential operators are perhaps not fully understood, making them a versatile tool for simulating various stochastic processes.<\/p>\n<ul>\n<li><a href=\"https:\/\/arxiv.org\/abs\/1906.06803?context=math.PR\"><b>Sticky Brownian Motion and its Numerical Solution<\/b><\/a><br \/>\nN. Bou-Rabee &amp; M. Holmes-Cerfon, SIAM Review, Vol. 62, No. 1, pp. 164-195, 2020.<\/li>\n<li><b><a href=\"https:\/\/bookstore.ams.org\/memo-256-1228\">Continuous-Time Random Walks for the Numerical Solution of Stochastic Differential Equations<\/a><\/b><br \/>\nN. Bou-Rabee &amp; E. Vanden-Eijnden, Memoirs of the American Mathematical Society, Vol. 256, 2018.<\/li>\n<li><b><a href=\"https:\/\/bookstore.ams.org\/memo-256-1228\">SPECTRWM: spectral random walk method for the numerical solution of stochastic partial differential equations<\/a><\/b><br \/>\nN. Bou-Rabee, SIAM Review, Vol. 60, No. 2, pp. 386-406, 2018.<\/li>\n<\/ul>\n<div style=\"text-align:center;margin:2em 0\"><img decoding=\"async\" src=\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/wp-content\/uploads\/sites\/908\/2026\/07\/sticky-brownian-motion-paths-v2.png\" alt=\"A random walk that captures 'sticky' diffusion: as the stickiness grows, the approximate paths linger longer at zero, the sticky behavior that standard schemes fail to reproduce.\" style=\"max-width:100%;height:auto;border-radius:2px\"><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Areas of Research Focus Advancing the No-U-Turn Sampler Couplings for Kinetic Langevin Diffusions The No-Underrun Sampler Mixing Time of the No-U-Turn Sampler Hamiltonian Monte Carlo Mixing Time Randomized Time Integrators &hellip; <a href=\"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/research\/\" class=\"\">Read More<\/a><\/p>\n","protected":false},"author":21,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-custom.php","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-365","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v23.5 - 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