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Johan Ulander

Publications and source records attributed to Johan Ulander.

8 recordsLinked to original sources

Boundary-preserving Lamperti--It\^o--Taylor approximations for some stochastic differential equations

In this work, we propose high-order boundary-preserving numerical schemes for the strong approximation for some scalar stochastic differential equations with invariant domains being open and bounded intervals. The proposed methods involve using the Lamperti transform to map the SDE to another SDE with additive noise with a trivial invariant domain. Then, by imposing regularity assumptions on the original coefficient functions, we can guarantee that the drift coefficient function of the transformed SDE is regular, and known high-order schemes can be used to achieve the desired convergence order. We confirm the theoretical results with numerical experiments.

math.NA

Analysis of an exponential integrator for stochastic PDEs driven by Riesz noise

We present and study an explicit exponential integrator for parabolic SPDEs in any dimension driven by a Gaussian noise which is white in time and with spatial correlation given by a Riesz kernel. Under assumptions on the coefficients of the SPDE, we prove strong error bounds and exhibit how the rate of convergence depends on the exponent in the Riesz kernel. Finally, numerical experiments in spatial dimensions $1$ and $2$ are provided in order to confirm our convergence results.

math.NA

Prediction of Retention Time in Larger Antisense Oligonucleotide Datasets using Machine Learning

Antisense oligonucleotides (ASOs) are nucleic acid molecules with transformative therapeutic potential, especially for diseases that are untreatable by traditional drugs. However, the production and purification of ASOs remain challenging due to the presence of unwanted impurities. One tool successfully used to separate an ASO compound from the impurities is ion pair liquid chromatography (IPC). It is a critical step in separation, where each compound is identified by its retention time (tR) in the IPC. Due to the complex sequence-dependent behavior of ASOs and variability in chromatographic conditions, the accurate prediction of tR is a difficult task. This study addresses this challenge by applying machine learning (ML) to predict tR based on the sequence characteristics of ASOs. Four ML models Gradient Boosting, Random Forest, Decision Tree, and Support Vector Regression were evaluated on three large ASO datasets with different gradient times. Through feature engineering and grid search optimization, key predictors were identified and compared for model accuracy using root mean square error, coefficient of determination R-squared, and run time. The results showed that Gradient Boost performance competes with the Support Vector Machine in two of the three datasets, but is 3.94 times faster to tune. Additionally, newly proposed features representing the sulfur count and the nucleotides residing at the first and last positions of a sequence were found to improve the predictive power of the models. This study demonstrates the advantages of ML-based tR prediction at scale and provides insights into interpretable and efficient utilization of ML in chromatographic applications.

q-bio.OT

Boundary-preserving weak approximation for some semilinear stochastic partial differential equations

We propose and analyse a boundary-preserving numerical scheme for the weak approximation for some stochastic partial differential equations (SPDEs) with bounded state-space. We impose regularity assumptions on the drift and diffusion coefficients only locally on the state-space. In particular, the drift and diffusion coefficients may be non-globally Lipschitz continuous and superlinearly growing. The scheme consists of a finite difference discretisation in space and a Lie--Trotter time splitting followed by exact simulation and exact integration in time. The proposed scheme converges in the weak sense of order $1/4$ in time and of order $1/2$ in space, for globally Lipschitz continuous test functions. We prove the weak convergence order in time by proving strong convergence towards a strong solution driven by a different noise process. The convergence order in space follows from known results. The boundary-preserving property is ensured by the use of Lie--Trotter time splitting followed by exact simulation and exact integration. Numerical experiments confirm the theoretical results and demonstrate the practical advantages of the proposed Lie--Trotter-Exact (LTE) scheme compared to existing schemes for SPDEs.

math.NA

Artificial Barriers for stochastic differential equations and for construction of boundary-preserving schemes

We develop the novel method of artificial barriers for scalar stochastic differential equations (SDEs) and use it to construct boundary-preserving numerical schemes for strong approximation of scalar SDEs, possibly with non-globally Lipschitz drift and diffusion coefficients, whose state-space is either bounded or half-bounded. The idea of artificial barriers is to augment the SDE with artificial barriers outside the state-space to not change the solution process, and then apply a boundary-preserving numerical scheme to the resulting reflected SDE (RSDE). This enables us to construct boundary-preserving numerical schemes that achieve the same strong convergence rate as the corresponding RSDE scheme. Based on the method of artificial barriers, we construct two boundary-preserving schemes that we call the Artificial Barriers Euler--Maruyama (ABEM) scheme and the Artificial Barriers Euler--Peano (ABEP) scheme, respectively. We provide numerical experiments for the ABEM scheme and the numerical results agree with the obtained theoretical results.

math.NA

Boundary-preserving Lamperti-splitting schemes for some Stochastic Differential Equations

We propose and analyse boundary-preserving schemes for the strong approximations of some scalar SDEs with non-globally Lipschitz drift and diffusion coefficients whose state-space is bounded. The schemes consists of a Lamperti transform followed by a Lie--Trotter splitting. We prove $L^{p}(\Omega)$-convergence of order $1$, for every $p \geq 1$, of the schemes and exploit the Lamperti transform to confine the numerical approximations to the state-space of the considered SDE. We provide numerical experiments that confirm the theoretical results and compare the proposed Lamperti-splitting schemes to other numerical schemes for SDEs.

math.NA

Positivity-preserving schemes for some nonlinear stochastic PDEs

We introduce a positivity-preserving numerical scheme for a class of nonlinear stochastic heat equations driven by a purely time-dependent Brownian motion. The construction is inspired by a recent preprint by the authors where one-dimensional equations driven by space-time white noise are considered. The objective of this paper is to illustrate the properties of the proposed integrators in a different framework, by numerical experiments and by giving convergence results.

math.NA

Analysis of a positivity-preserving splitting scheme for some nonlinear stochastic heat equations

We construct a positivity-preserving Lie--Trotter splitting scheme with finite difference discretization in space for approximating the solutions to a class of nonlinear stochastic heat equations with multiplicative space-time white noise. We prove that this explicit numerical scheme converges in the mean-square sense, with rate $1/4$ in time and rate $1/2$ in space, under appropriate CFL conditions. Numerical experiments illustrate the superiority of the proposed numerical scheme compared with standard numerical methods which do not preserve positivity.

math.NA