| 1. |
- Espath, Luis, et al.
(författare)
-
Statistical learning for fluid flows : Sparse Fourier divergence-free approximations
- 2021
-
Ingår i: Physics of fluids. - : AIP Publishing. - 1070-6631 .- 1089-7666. ; 33:9
-
Tidskriftsartikel (refereegranskat)abstract
- We reconstruct the velocity field of incompressible flows given a finite set of measurements. For the spatial approximation, we introduce the Sparse Fourier divergence-free approximation based on a discrete L & nbsp;projection. Within this physics-informed type of statistical learning framework, we adaptively build a sparse set of Fourier basis functions with corresponding coefficients by solving a sequence of minimization problems where the set of basis functions is augmented greedily at each optimization problem. We regularize our minimization problems with the seminorm of the fractional Sobolev space in a Tikhonov fashion. In the Fourier setting, the incompressibility (divergence-free) constraint becomes a finite set of linear algebraic equations. We couple our spatial approximation with the truncated singular-value decomposition of the flow measurements for temporal compression. Our computational framework thus combines supervised and unsupervised learning techniques. We assess the capabilities of our method in various numerical examples arising in fluid mechanics.
|
|
| 2. |
- Hult, Henrik, et al.
(författare)
-
Algorithmic trading with Markov chains
-
Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- An order book consists of a list of all buy and sell offers, represented by price and quantity, available to a market agent. The order book changes rapidly, within fractions of a second, due to new orders being entered into the book. The volume at a certain price level may increase due to limitorders, i.e. orders to buy or sell placed at the end of the queue, or decrease because of market orders or cancellations. In this paper a high-dimensional Markov chain is used to represent the state and evolution of the entire order book. The design and evaluation of optimal algorithmic strategies for buying and selling is studied within the theory of Markov decision processes. General conditions are provided that guarantee the existence of optimal strategies. Moreover, a value-iteration algorithm is presented that enables finding optimal strategies numerically. As an illustration a simple version of the Markov chain model is calibrated to high-frequency observations of the order book in a foreign exchange market. In this model, using an optimally designed strategy for buying one unit provides a significant improvement, in terms of the expected buy price, over a naive buy-one-unit strategy.
|
|
| 3. |
- Kammonen, Aku, 1984-, et al.
(författare)
-
Adaptive random fourier features with metropolis sampling
- 2019
-
Ingår i: Foundations of Data Science. - : American Institute of Mathematical Sciences. - 2639-8001. ; 0:0, s. 0-0
-
Tidskriftsartikel (refereegranskat)abstract
- The supervised learning problem todetermine a neural network approximation $\mathbb{R}^d\ni x\mapsto\sum_{k=1}^K\hat\beta_k e^{{\mathrm{i}}\omega_k\cdot x}$with one hidden layer is studied asa random Fourier features algorithm. The Fourier features, i.e., the frequencies $\omega_k\in\mathbb{R}^d$,are sampled using an adaptive Metropolis sampler.The Metropolis test accepts proposal frequencies $\omega_k'$, having corresponding amplitudes $\hat\beta_k'$, with the probability$\min\big\{1, (|\hat\beta_k'|/|\hat\beta_k|)^\gamma\big\}$,for a certain positive parameter $\gamma$, determined by minimizing the approximation error for given computational work.This adaptive, non-parametric stochastic method leads asymptotically, as $K\to\infty$, to equidistributed amplitudes $|\hat\beta_k|$, analogous to deterministic adaptive algorithms for differential equations. The equidistributed amplitudes are shown to asymptotically correspond to the optimal density for independent samples in random Fourier features methods.Numerical evidence is provided in order to demonstrate the approximation properties and efficiency of the proposed algorithm. The algorithm is testedboth on synthetic data and a real-world high-dimensional benchmark.
|
|
| 4. |
|
|
| 5. |
- Kammonen, Aku, 1984-, et al.
(författare)
-
SMALLER GENERALIZATION ERROR DERIVED FOR DEEP COMPARED TO SHALLOW RESIDUAL NEURAL NETWORKS
-
Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Estimates of the generalization error are proved for a residual neural network with $L$ random Fourier features layers $\bar z_{\ell+1}=\bar z_\ell + \mathrm{Re}\sum_{k=1}^K\bar b_{\ell k}e^{\mathrm{i}\omega_{\ell k}\bar z_\ell}+\mathrm{Re}\sum_{k=1}^K\bar c_{\ell k}e^{\mathrm{i}\omega'_{\ell k}\cdot x}$. An optimal distribution for the frequencies $(\omega_{\ell k},\omega'_{\ell k})$ of the random Fourier features $e^{\mathrm{i}\omega_{\ell k}\bar z_\ell}$ and $e^{\mathrm{i}\omega'_{\ell k}\cdot x}$ is derived. This derivation is based on the corresponding generalization error for the approximation of the function values $f(x)$. The generalization error turns out to be smaller than the estimate ${\|\hat f\|^2_{L^1(\mathbb{R}^d)}}/{(LK)}$ of the generalization error for random Fourier features with one hidden layer and the same total number of nodes $LK$, in the case the $L^\infty$-norm of $f$ is much less than the $L^1$-norm of its Fourier transform $\hat f$. This understanding of an optimal distribution for random features is used to construct a new training method for a deep residual network that shows promising results.
|
|
| 6. |
- Kiessling, Jonas, et al.
(författare)
-
A Computable Definition of the Spectral Bias
- 2022
-
Ingår i: Proceedings of the 36th AAAI Conference on Artificial Intelligence, AAAI 2022. - : Association for the Advancement of Artificial Intelligence. ; 36:7, s. 7168-7175
-
Konferensbidrag (refereegranskat)abstract
- Neural networks have a bias towards low frequency functions. This spectral bias has been the subject of several previous studies, both empirical and theoretical. Here we present a computable definition of the spectral bias based on a decomposition of the reconstruction error into a low and a high frequency component. The distinction between low and high frequencies is made in a way that allows for easy interpretation of the spectral bias. Furthermore, we present two methods for estimating the spectral bias. Method 1 relies on the use of the discrete Fourier transform to explicitly estimate the Fourier spectrum of the prediction residual, and Method 2 uses convolution to extract the low frequency components, where the convolution integral is estimated by Monte Carlo methods. The spectral bias depends on the distribution of the data, which is approximated with kernel density estimation when unknown. We devise a set of numerical experiments that confirm that low frequencies are learned first, a behavior quantified by our definition.
|
|
| 7. |
- Kiessling, Jonas, 1980-
(författare)
-
Approximation and Calibration of Stochastic Processes in Finance
- 2010
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis is a study of approximation and calibration of stochastic processes with applications in finance. It consists of an introduction and four research papers. The introduction is as an overview of the role of mathematics incertain areas of finance. It contains a brief introduction to the mathematicaltheory of option pricing, as well as a description of a mathematicalmodel of a financial exchange. The introduction also includessummaries of the four research papers. In Paper I, Markov decision theory is applied to design algorithmic trading strategies in an order driven market. A high dimensional Markov chain is used to model the state and evolution of the limit order book. Trading strategies are formulated as optimal decision problems. Conditions that guarantee existence of optimal strategies are provided, as well as a value-iterative algorithm that enables numerical construction of optimal strategies. The results are illustrated with numerical experiments on high frequency data from a foreign exchange market. Paper II focuses on asset pricing with Lévy processes. The expected value E[g(XT )] is estimated using a Monte Carlo method, when Xt is a d-dimensional Lévy process having infinite jump activity and a smooth density. Approximating jumps smaller then a parameter ε > 0 by diffusion results in a weak approximation, Xt, of Xt. The main result of the paper is an estimate of the resulting model error E[g(XT )] − E[g(T )], with a computable leading order term. Option prices in exponential Lévy models solve certain partia lintegro-differential equations (PIDEs). A finite difference scheme suitable for solving such PIDEs is studied in Paper III. The main resultsare estimates of the time and space discretization errors, with leading order terms in computable form. If the underlying Lévy process has infinite jump activity, the jumps smaller than some ε > 0 are replacedby diffusion. The size of this diffusion approximation is estimated, as well as its effect on the space and time discretization errors. Combined, the results of the paper are enough to determine how to jointly choose the grid size and the parameter ε. In Paper IV it is demonstrated how optimal control can be used to calibrate a jump-diffusion process to quoted option prices. The calibration problem is formulated as an optimal control problem with the model parameter as a control variable. The corresponding regularized Hamiltonian system is solved with a symplectic Euler method.
|
|
| 8. |
- Kiessling, Jonas, et al.
(författare)
-
Artificial Intelligence Outperforms Kaplan-Meier Analyses Estimating Survival after Elective Treatment of Abdominal Aortic Aneurysms
- 2023
-
Ingår i: European Journal of Vascular and Endovascular Surgery. - : Elsevier BV. - 1078-5884 .- 1532-2165. ; 65:4, s. 600-607
-
Tidskriftsartikel (refereegranskat)abstract
- Objective: Long term differences in survival after elective repair of abdominal aortic aneurysms (AAAs) between open surgical repair (OSR) and endovascular aneurysm repair (EVAR) are unclear, and hitherto artificial intelligence has not been used for this purpose. The aim was to compare the precision of survival estimates between the Kaplan -Meier (KM) method and the artificial intelligence derived method Neural Multi-Task Logistic Regression (N-MTLR), and to compare survival estimates as a function of patient age and time since surgery between OSR and EVAR using N-MTLR. Methods: All AAAs between 2003 and 2018 in Denmark were identified in the Danish vascular registry. Survival was estimated using the KM and N-MTLR methods, and prediction performance was estimated with the Brier score. Results: 7 912 patients were included in the study, n = 6 569 (83%) men, median age 72 years (range 35 -92), with a median follow-up time of 45.7 months (range 0 -120). The two treatment cohorts, OSR n = 5 495 (69%), and EVAR n = 2 417 (31%), differed significantly in patient characteristics. The Brier score for KM increased from 0.044 to 0.244, and for N-MTLR from 0.044 to 0.206, from 90 days to 10 years. The N-MTLR method was more accurate than KM from 90 days to 10 years after surgery, p < .025. N-MTLR demonstrated significant increased probability for survival for OSR in patients aged 58 -76 years at five years, and 65 -73 at 10 years after surgery, and the opposite was found for the benefit of EVAR in patients aged 72 -85 years at one year, 85 -90 years at five years, and for 85 -90 year olds at 10 years after surgery. Conclusion: N-MTLR outperforms KM for the entire post-operative follow-up time. This N-MTLR model has the potential to render more precise patient specific survival estimates and establish survival differences between subgroups of patients that KM is unable to detect, demonstrated here for different age groups.
|
|
| 9. |
- Kiessling, Jonas
(författare)
-
Calibration of a Jump-Diffusion Process Using Optimal Control
- 2012
-
Ingår i: Numerical Analysis Of Multiscale Computations. - Berlin, Heidelberg : Springer Berlin/Heidelberg. - 9783642219429 ; , s. 259-277
-
Konferensbidrag (refereegranskat)abstract
- A method for calibrating a jump-diffusion model to observed option prices is presented. The calibration problem is formulated as an optimal control problem, with the model parameters as the control variable. It is well known that such problems are ill-posed and need to be regularized. A Hamiltonian system, with non-differentiable Hamiltonian, is obtained from the characteristics of the corresponding Hamilton-Jacobi-Bellman equation. An explicit regularization of the Hamiltonian is suggested, and the regularized Hamiltonian system is solved with a symplectic Euler method. The paper is concluded with some numerical experiments on real and artificial data.
|
|
| 10. |
|
|
