| 1. |
- Sabahno, Hamed, et al.
(författare)
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Variable parameters memory-type control charts for simultaneous monitoring of the mean and variability of multivariate multiple linear regression profiles
- 2024
-
Ingår i: Scientific Reports. - : Nature Publishing Group. - 2045-2322. ; 14:1
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Tidskriftsartikel (refereegranskat)abstract
- Variable parameters (VP) schemes are the most effective adaptive schemes in increasing control charts' sensitivity to detect small to moderate shift sizes. In this paper, we develop four VP adaptive memory-type control charts to monitor multivariate multiple linear regression profiles. All the proposed control charts are single-chart (single-statistic) control charts, two use a Max operator and two use an SS (squared sum) operator to create the final statistic. Moreover, two of the charts monitor the regression parameters, and the other two monitor the residuals. After developing the VP control charts, we developed a computer algorithm with which the charts' time-to-signal and run-length-based performances can be measured. Then, we perform extensive numerical analysis and simulation studies to evaluate the charts’ performance and the result shows significant improvements by using the VP schemes. Finally, we use real data from the national quality register for stroke care in Sweden, Riksstroke, to illustrate how the proposed control charts can be implemented in practice.
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| 2. |
- Mejia, A. F., et al.
(författare)
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A Bayesian General Linear Modeling Approach to Cortical Surface fMRI Data Analysis
- 2020
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Ingår i: Journal of the American Statistical Association. - : Informa UK Limited. - 0162-1459 .- 1537-274X.
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Tidskriftsartikel (refereegranskat)abstract
- Cortical surface functional magnetic resonance imaging (cs-fMRI) has recently grown in popularity versus traditional volumetric fMRI. In addition to offering better whole-brain visualization, dimension reduction, removal of extraneous tissue types, and improved alignment of cortical areas across subjects, it is also more compatible with common assumptions of Bayesian spatial models. However, as no spatial Bayesian model has been proposed for cs-fMRI data, most analyses continue to employ the classical general linear model (GLM), a "massive univariate" approach. Here, we propose a spatial Bayesian GLM for cs-fMRI, which employs a class of sophisticated spatial processes to model latent activation fields. We make several advances compared with existing spatial Bayesian models for volumetric fMRI. First, we use integrated nested Laplacian approximations, a highly accurate and efficient Bayesian computation technique, rather than variational Bayes. To identify regions of activation, we utilize an excursions set method based on the joint posterior distribution of the latent fields, rather than the marginal distribution at each location. Finally, we propose the first multi-subject spatial Bayesian modeling approach, which addresses a major gap in the existing literature. The methods are very computationally advantageous and are validated through simulation studies and two task fMRI studies from the Human Connectome Project. for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.
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| 3. |
- Sabahno, Hamed, et al.
(författare)
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New statistical and machine learning based control charts with variable parameters for monitoring generalized linear model profiles
- 2023
-
Ingår i: Computers & industrial engineering. - : Elsevier. - 0360-8352 .- 1879-0550. ; 184
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Tidskriftsartikel (refereegranskat)abstract
- In this research, we develop three statistical based control charts: the Hotelling’s T2, MEWMA (multivariate exponentially weighted moving average), and LRT (likelihood ratio test) as well as three machine learning (ML) based control charts: the ANN (artificial neural network), SVR (support vector regression), and RFR (random forest regression), for monitoring generalized linear model (GLM) profiles. We train these ML models with two different training methods to get a linear (regression) output and then apply our classification technique to see if the process is in-control or out-of-control, at each sampling time. In addition to developing the FP (fixed parameters) schemes, for the first time in GLM profiles, we design an adaptive VP (variable parameters) scheme for each control chart as well to increase the charts’ sensitivity in detecting shifts. We develop some algorithms with which the values of the control chart parameters in both FP and VP schemes can be obtained. Then, we develop two algorithms to measure the charts’ performance in both FP and VP schemes, by using the run-length and time-to-signal based performance measures. This is also the first control chart-related research that develops an algorithm to compute the performance measures that applies to any VP adaptive control scheme. After designing the control charts as well as performance measures, we perform extensive simulation studies and evaluate and compare all our control charts under different shift sizes and scenarios, and in three different simulation environments. Finally, we present a numerical example regarding a drug dose-response study to show how the proposed control charts can be implemented in real practice.
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| 4. |
- Casarino, V., et al.
(författare)
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On the variation operator for the Ornstein-Uhlenbeck semigroup in dimension one
- 2024
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Ingår i: Annali Di Matematica Pura Ed Applicata. - 0373-3114 .- 1618-1891. ; 203:1, s. 205-219
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Tidskriftsartikel (refereegranskat)abstract
- Consider the variation seminorm of the Ornstein-Uhlenbeck semigroup Ht in dimension one, taken with respect to t. We show that this seminorm defines an operator of weak type (1, 1) for the relevant Gaussian measure. The analogous Lp estimates for 1 < p < 8 were already known.
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| 5. |
- Bayisa, Fekadu, et al.
(författare)
-
Large-scale modelling and forecasting of ambulance calls in northern Sweden using spatio-temporal log-Gaussian Cox processes
- 2020
-
Ingår i: Spatial Statistics. - : Elsevier BV. - 2211-6753. ; 39
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Tidskriftsartikel (refereegranskat)abstract
- Although ambulance call data typically come in the form of spatio-temporal point patterns, point process-based modelling approaches presented in the literature are scarce. In this paper, we study a unique set of Swedish spatio-temporal ambulance call data, which consist of the spatial (GPS) locations of the calls (within the four northernmost regions of Sweden) and the associated days of occurrence of the calls (January 1, 2014-December 31, 2018). Motivated by the nature of the data, we here employ log-Gaussian Cox processes (LGCPs) for the spatiotemporal modelling and forecasting of the calls. To this end, we propose a K-means clustering based bandwidth selection method for the kernel estimation of the spatial component of the separable spatio-temporal intensity function. The temporal component of the intensity function is modelled by means of Poisson regression, using different calendar covariates, and the spatiotemporal random field component of the random intensity of the LGCP is fitted using the Metropolis-adjusted Langevin algorithm. Spatial hot-spots have been found in the south-eastern part of the study region, where most people in the region live and our fitted model/forecasts manage to capture this behaviour quite well. Also, there is a significant association between the expected number of calls and the day-of-the-week, and the season-ofthe-year. A non-parametric second-order analysis indicates that LGCPs seem to be reasonable models for the data. Finally, we find that the fitted forecasts generate simulated future spatial event patterns which quite well resemble the actual future data. (C) 2020 The Author(s). Published by Elsevier B.V.
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| 6. |
- Fedosova, K., et al.
(författare)
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High energy asymptotics for the perturbed anharmonic oscillator
- 2021
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Ingår i: Complex Variables and Elliptic Equations. - : Informa UK Limited. - 1747-6933 .- 1747-6941.
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Tidskriftsartikel (refereegranskat)abstract
- In this article, we study the spectral properties of the perturbation of the generalized anharmonic oscillator. We consider a piecewise Holder continuous perturbation and investigate how the Holder constant can affect the eigenvalues. More precisely, we derive several first terms in the asymptotic expansion for the eigenvalues.
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| 7. |
- Gulliksson, Mårten, 1963-, et al.
(författare)
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An Iterative Approach to Ill-Conditioned Optimal Portfolio Selection
- 2020
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Ingår i: Computational Economics. - : Springer. - 0927-7099 .- 1572-9974. ; 56, s. 773-794
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Tidskriftsartikel (refereegranskat)abstract
- Covariance matrix of the asset returns plays an important role in the portfolioselection. A number of papers is focused on the case when the covariance matrixis positive definite. In this paper, we consider portfolio selection with a singu-lar covariance matrix. We describe an iterative method based on a second orderdamped dynamical systems that solves the linear rank-deficient problem approxi-mately. Since the solution is not unique, we suggest one numerical solution that canbe chosen from the iterates that balances the size of portfolio and the risk. The nu-merical study confirms that the method has good convergence properties and givesa solution as good as or better than the constrained least norm Moore-Penrose solu-tion. Finally, we complement our result with an empirical study where we analyzea portfolio with actual returns listed in S&P 500 index.
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| 8. |
- Laksman, Efraim, 1983, et al.
(författare)
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The stochastic opportunistic replacement problem, part III: improved bounding procedures
- 2020
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Ingår i: Annals of Operations Research. - : Springer Science and Business Media LLC. - 1572-9338 .- 0254-5330. ; 292:2, s. 711-733
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Tidskriftsartikel (refereegranskat)abstract
- We consider the problem to find a schedule for component replacement in a multi-component system, whose components possess stochastic lives and economic dependencies, such that the expected costs for maintenance during a pre-defined time period are minimized. The problem was considered in Patriksson et al. (Ann Oper Res 224:51–75, 2015), in which a two-stage approximation of the problem was optimized through decomposition (denoted the optimization policy). The current paper improves the effectiveness of the decomposition approach by establishing a tighter bound on the value of the recourse function (i.e., the second stage in the approximation). A general lower bound on the expected maintenance cost is also established. Numerical experiments with 100 simulation scenarios for each of four test instances show that the tighter bound yields a decomposition generating fewer optimality cuts. They also illustrate the quality of the lower bound. Contrary to results presented earlier, an age-based policy performs on par with the optimization policy, although most simple policies perform worse than the optimization policy.
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| 9. |
- Arkeryd, Leif, 1940, et al.
(författare)
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Stationary solutions to the two-dimensional Broadwell model
- 2020
-
Ingår i: Documenta Mathematica. - 1431-0643 .- 1431-0635. ; 25, s. 2023-2048
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Tidskriftsartikel (refereegranskat)abstract
- Existence of renormalized solutions to the two-dimensional stationary Broadwell model in a square with given indata in L-1 is proven. Averaging techniques from the continuous velocity case being unavailable when the velocities are discrete, the approach is based on direct L-1-compactness arguments using the Kolmogorov-Riesz theorem.
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| 10. |
- Westerberg, Marcus, 1990-
(författare)
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Prostate cancer incidence, treatment and mortality : Empirical longitudinal register-based studies and methods for handling missing data
- 2022
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The diagnostic activity for prostate cancer has increased substantially in Sweden, primarily due to increased use of prostate-specific antigen (PSA) testing in asymptomatic men, and this has led to a large increase in diagnoses. There have also been changes in the diagnostic workup, guidelines, treatment strategies, and more effective treatments have been introduced in different phases of the disease. This thesis aims to increase the understanding of consequences of changes in diagnostic activity and treatment, with a focus on empirical studies, methodological development, and handling of missing data.In paper I, the survival of men with metastatic prostate cancer was investigated across calendar time periods by use of Kaplan-Meier analyses and Cox regression. The median survival from diagnosis increased with six months comparing men diagnosed 1998-2001 with men diagnosed 2010-2015, while median PSA decreased.In paper II, a discrete time multivariate longitudinal model was combined with a proxy for the unobserved level of diagnostic activity to produce prognoses of incidence and mortality. Simulations indicated that a higher diagnostic activity was associated with fewer men diagnosed with metastatic disease and fewer prostate cancer deaths.In paper III, we looked for clinical variables predictive of the survival of men with castration-resistant prostate cancer (CRPC). A new data base was created including longitudinal data on prescriptions of hormonal treatment, PSA, and cause of death. We found that PSA doubling time and PSA at time of CRCP were highly predictive and could be used for treatment decision.In paper IV, we estimated annual incidence of metastatic prostate cancer using different methods for handling missing data in metastatic status (M stage). Missing data in M stage was high and varied over calendar time and risk groups, yet each method indicated a downward trend in incidence. Although men with unknown metastatic status cannot be assumed to have nonmetastatic disease in general, this may be reasonable among those with tumour characteristics that indicate a low risk of metastases.In paper V, the estimation of multivariate longitudinal models was considered in a context where some events are observed on a coarser level (e.g. grouped) at some time points, causing gaps in the data. The likelihood function, score and observed information were derived under an independent coarsening mechanism. A simulation study was conducted comparing properties of several estimators including direct maximum likelihood and Monte Carlo Expectation Maximisation.
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| 11. |
- Sabahno, Hamed, et al.
(författare)
-
New machine-learning control charts for simultaneous monitoring of multivariate normal process parameters with detection and identification
- 2023
-
Ingår i: Mathematics. - : MDPI. - 2227-7390. ; 11:16
-
Tidskriftsartikel (refereegranskat)abstract
- Simultaneous monitoring of the process parameters in a multivariate normal process has caught researchers’ attention during the last two decades. However, only statistical control charts have been developed so far for this purpose. On the other hand, machine-learning (ML) techniques have rarely been developed to be used in control charts. In this paper, three ML control charts are proposed using the concepts of artificial neural networks, support vector machines, and random forests techniques. These ML techniques are trained to obtain linear outputs, and then based on the concepts of memory-less control charts, the process is classified into in-control or out-of-control states. Two different input scenarios and two different training methods are used for the proposed ML structures. In addition, two different process control scenarios are utilized. In one, the goal is only the detection of the out-of-control situation. In the other one, the identification of the responsible variable (s)/process parameter (s) for the out-of-control signal is also an aim (detection–identification). After developing the ML control charts for each scenario, we compare them to one another, as well as to the most recently developed statistical control charts. The results show significantly better performance of the proposed ML control charts against the traditional memory-less statistical control charts in most compared cases. Finally, an illustrative example is presented to show how the proposed scheme can be implemented in a healthcare process.
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| 12. |
- Bolin, David, 1983, et al.
(författare)
-
The Rational SPDE Approach for Gaussian Random Fields With General Smoothness
- 2020
-
Ingår i: Journal of Computational and Graphical Statistics. - : Informa UK Limited. - 1061-8600 .- 1537-2715. ; 29:2, s. 274-85
-
Tidskriftsartikel (refereegranskat)abstract
- A popular approach for modeling and inference in spatial statistics is to represent Gaussian random fields as solutions to stochastic partial differential equations (SPDEs) of the form , where is Gaussian white noise, L is a second-order differential operator, and is a parameter that determines the smoothness of u. However, this approach has been limited to the case , which excludes several important models and makes it necessary to keep beta fixed during inference. We propose a new method, the rational SPDE approach, which in spatial dimension is applicable for any , and thus remedies the mentioned limitation. The presented scheme combines a finite element discretization with a rational approximation of the function to approximate u. For the resulting approximation, an explicit rate of convergence to u in mean-square sense is derived. Furthermore, we show that our method has the same computational benefits as in the restricted case . Several numerical experiments and a statistical application are used to illustrate the accuracy of the method, and to show that it facilitates likelihood-based inference for all model parameters including beta. for this article are available online.
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| 13. |
- Kuronen, M., et al.
(författare)
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Hierarchical log Gaussian Cox process for regeneration in uneven-aged forests
- 2022
-
Ingår i: Environmental and Ecological Statistics. - : Springer Science and Business Media LLC. - 1352-8505 .- 1573-3009. ; 29:1, s. 185-205
-
Tidskriftsartikel (refereegranskat)abstract
- We propose a hierarchical log Gaussian Cox process (LGCP) for point patterns, where a set of points x affects another set of points y but not vice versa. We use the model to investigate the effect of large trees on the locations of seedlings. In the model, every point in x has a parametric influence kernel or signal, which together form an influence field. Conditionally on the parameters, the influence field acts as a spatial covariate in the intensity of the model, and the intensity itself is a non-linear function of the parameters. Points outside the observationwindowmay affect the influence field inside the window. We propose an edge correction to account for this missing data. The parameters of the model are estimated in a Bayesian framework using Markov chain Monte Carlo where a Laplace approximation is used for the Gaussian field of the LGCP model. The proposed model is used to analyze the effect of large trees on the success of regeneration in uneven-aged forest stands in Finland.
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| 14. |
- Fridolfsson, Jonatan, 1992, et al.
(författare)
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Physical activity spectrum discriminant analysis—A method to compare detailed patterns between groups
- 2021
-
Ingår i: Scandinavian Journal of Medicine and Science in Sports. - : Wiley. - 0905-7188 .- 1600-0838. ; 31:12, s. 2333-2342
-
Tidskriftsartikel (refereegranskat)abstract
- Investigating physical activity (PA) patterns as a detailed intensity spectrum instead of crude intensity categories have improved the ability to analyze the relationship between measured PA and health variables. The aim of this methodological study was to introduce and investigate the utility of using detailed PA intensity spectrum compared to crude PA intensity categories for comparison of PA between groups and between repeated measures. The study sample consisted of two groups of children, where one group was scheduled for extended physical education (PE) by daily classes while the other group followed usual PE schedule. Accelerometer data was processed into traditional crude PA intensity categories and into detailed PA intensity spectrum. Multivariate partial least squares regression for discriminant analysis (PLS-DA) was applied for PA intensity spectrum group comparison and compared to traditional univariate statistical analysis. Repeated measures were investigated using independent PLS-DA as well as multilevel PLS-DA for paired analysis. While traditional analysis of crude PA intensity categories was unable to find any group differences, multivariate analysis of the PA intensity spectrum identified statistically significant differences. By the extension of multilevel PLS-DA for paired comparison, a clear difference in the PA intensity spectrum was demonstrated between repeated measures. In conclusion, analysis of detailed PA intensity spectrum demonstrates utility for comparing detailed PA data between groups and between repeated measures in interventional and observational research.
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| 15. |
- Gerken, Jan, 1991, et al.
(författare)
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Geometric deep learning and equivariant neural networks
- 2023
-
Ingår i: Artificial Intelligence Review. - : Springer Nature. - 1573-7462 .- 0269-2821. ; 56:12, s. 14605-14662
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Tidskriftsartikel (refereegranskat)abstract
- We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds M using principal bundles with structure group K and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces M= G/ K , which are instead equivariant with respect to the global symmetry G on M . Group equivariant layers can be interpreted as intertwiners between induced representations of G, and we show their relation to gauge equivariant convolutional layers. We analyze several applications of this formalism, including semantic segmentation and object detection networks. We also discuss the case of spherical networks in great detail, corresponding to the case M= S2= SO (3) / SO (2) . Here we emphasize the use of Fourier analysis involving Wigner matrices, spherical harmonics and Clebsch–Gordan coefficients for G= SO (3) , illustrating the power of representation theory for deep learning.
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| 16. |
- Jagers, Peter, 1941
(författare)
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Branching Processes: A Personal Historical Perspective.
- 2020
-
Ingår i: Statistical Modeling for Biological Systems: In Memory of Andrei Yakovlev. - Cham : Springer International Publishing.
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- The chapter, based on an Oberwolfach talk, gives a - personally biased - sketch of the development of branching processes, from the mid 19th Century to recently, emphasizing relations to bioscience and demography, and to society and culture in general.
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| 17. |
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| 18. |
- Bayisa, Fekadu, et al.
(författare)
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Regularised Semi-parametric Composite Likelihood Intensity Modelling of a Swedish Spatial Ambulance Call Point Pattern
- 2023
-
Ingår i: Journal of Agricultural Biological and Environmental Statistics. - : Springer. - 1085-7117 .- 1537-2693. ; 28, s. 664-83
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Tidskriftsartikel (refereegranskat)abstract
- Motivated by the development of optimal dispatching strategies for prehospital resources, we model the spatial distribution of ambulance call events in the Swedish municipality Skelleftea during 2014-2018 in order to identify important spatial covariates and discern hotspot regions. Our large-scale multivariate data point pattern of call events consists of spatial locations and marks containing the associated priority levels and sex labels. The covariates used are related to road network coverage, population density, and socio-economic status. For each marginal point pattern, we model the associated intensity function by means of a log-linear function of the covariates and their interaction terms, in combination with lasso-like elastic-net regularized composite/Poisson process likelihood estimation. This enables variable selection and collinearity adjustment as well as reduction of variance inflation from overfitting and bias from underfitting. To incorporate mobility adjustment, reflecting people's movement patterns, we also include a nonparametric (kernel) intensity estimate as an additional covariate. The kernel intensity estimation performed here exploits a new heuristic bandwidth selection algorithm. We discover that hotspot regions occur along dense parts of the road network. A mean absolute error evaluation of the fitted model indicates that it is suitable for designing prehospital resource dispatching strategies. Supplementary materials accompanying this paper appear online.
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| 19. |
- Belomestny, Denis, et al.
(författare)
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Weak solutions to gamma-driven stochastic differential equations
- 2023
-
Ingår i: Indagationes Mathematicae. - 0019-3577. ; 34:4, s. 820-829
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Tidskriftsartikel (refereegranskat)abstract
- We study a stochastic differential equation driven by a gamma process, for which we give results on the existence of weak solutions under conditions on the volatility function. To that end we provide results on the density process between the laws of solutions with different volatility functions.
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| 20. |
- Brehier, C. E., et al.
(författare)
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SPLITTING SCHEMES FOR FITZHUGH-NAGUMO STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
- 2023
-
Ingår i: Discrete and Continuous Dynamical Systems-Series B. - 1531-3492. ; 29:1, s. 214-44
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Tidskriftsartikel (refereegranskat)abstract
- . We design and study splitting integrators for the temporal discretization of the stochastic FitzHugh-Nagumo system. This system is a model for signal propagation in nerve cells where the voltage variable is the solution of a one-dimensional parabolic PDE with a cubic nonlinearity driven by additive space-time white noise. We first show that the numerical solutions have finite moments. We then prove that the splitting schemes have, at least, the strong rate of convergence 1/4. Finally, numerical experiments illustrating the performance of the splitting schemes are provided.
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| 21. |
- Brubaker, B., et al.
(författare)
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Colored five-vertex models and Demazure atoms
- 2021
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Ingår i: Journal of Combinatorial Theory Series A. - : Elsevier BV. - 0097-3165 .- 1096-0899. ; 178
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Tidskriftsartikel (refereegranskat)abstract
- Type A Demazure atoms are pieces of Schur functions, or sets of tableaux whose weights sum to such functions. Inspired by colored vertex models of Borodin and Wheeler, we will construct solvable lattice models whose partition functions are Demazure atoms; the proof of this makes use of a Yang-Baxter equation for a colored five-vertex model. As a byproduct, we will construct Demazure atoms on Kashiwara's B-infinity crystal and give new algorithms for computing Lascoux-Schutzenberger keys. (C) 2020 Elsevier Inc. All rights reserved.
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| 22. |
- Bågmark, Kasper, 1995, et al.
(författare)
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An energy-based deep splitting method for the nonlinear filtering problem
- 2023
-
Ingår i: Partial Differential Equations and Applications. - : Springer Science and Business Media LLC. - 2662-2963 .- 2662-2971. ; 4:2
-
Tidskriftsartikel (refereegranskat)abstract
- The purpose of this paper is to explore the use of deep learning for the solution of the nonlinear filtering problem. This is achieved by solving the Zakai equation by a deep splitting method, previously developed for approximate solution of (stochastic) partial differential equations. This is combined with an energy-based model for the approximation of functions by a deep neural network. This results in a computationally fast filter that takes observations as input and that does not require re-training when new observations are received. The method is tested on four examples, two linear in one and twenty dimensions and two nonlinear in one dimension. The method shows promising performance when benchmarked against the Kalman filter and the bootstrap particle filter.
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| 23. |
- Cohen, David, et al.
(författare)
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Drift-preserving numerical integrators for stochastic Poisson systems
- 2021
-
Ingår i: International Journal of Computer Mathematics. - : Informa UK Limited. - 0020-7160 .- 1029-0265. ; 99:1
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Tidskriftsartikel (refereegranskat)abstract
- We perform a numerical analysis of a class of randomly perturbed Hamiltonian systems and Poisson systems. For the considered additive noise perturbation of such systems, we show the long-time behaviour of the energy and quadratic Casimirs for the exact solution. We then propose and analyse a drift-preserving splitting scheme for such problems with the following properties: exact drift preservation of energy and quadratic Casimirs, mean-square order of convergence 1, weak order of convergence 2. These properties are illustrated with numerical experiments.
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| 24. |
- Eisenmann, Monika, et al.
(författare)
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Error estimates of the backward Euler-Maruyama method for multi-valued stochastic differential equations
- 2022
-
Ingår i: Bit Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 62:3, s. 803-48
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we derive error estimates of the backward Euler-Maruyama method applied to multi-valued stochastic differential equations. An important example of such an equation is a stochastic gradient flow whose associated potential is not continuously differentiable but assumed to be convex. We show that the backward Euler-Maruyama method is well-defined and convergent of order at least 1/4 with respect to the root-mean-square norm. Our error analysis relies on techniques for deterministic problems developed in Nochetto et al. (Commun Pure Appl Math 53(5):525-589, 2000). We verify that our setting applies to an overdamped Langevin equation with a discontinuous gradient and to a spatially semi-discrete approximation of the stochastic p-Laplace equation.
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| 25. |
- Kovacs, Mihaly, 1977, et al.
(författare)
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Hilbert-Schmidt regularity of symmetric integral operators on bounded domains with applications to SPDE approximations
- 2023
-
Ingår i: Stochastic Analysis and Applications. - : Informa UK Limited. - 0736-2994 .- 1532-9356. ; 41:3, s. 564-90
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Tidskriftsartikel (refereegranskat)abstract
- Regularity estimates for an integral operator with a symmetric continuous kernel on a convex bounded domain are derived. The covariance of a mean-square continuous random field on the domain is an example of such an operator. The estimates are of the form of Hilbert-Schmidt norms of the integral operator and its square root, composed with fractional powers of an elliptic operator equipped with homogeneous boundary conditions of either Dirichlet or Neumann type. These types of estimates, which couple the regularity of the driving noise with the properties of the differential operator, have important implications for stochastic partial differential equations on bounded domains as well as their numerical approximations. The main tools used to derive the estimates are properties of reproducing kernel Hilbert spaces of functions on bounded domains along with Hilbert-Schmidt embeddings of Sobolev spaces. Both non-homogeneous and homogeneous kernels are considered. In the latter case, results in a general Schatten class norm are also provided. Important examples of homogeneous kernels covered by the results of the paper include the class of Matern kernels.
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